What are valid arguments?

The definition of valid argument is the following:

An argument with a form that guarantees us that it’s impossible for the premises to be true and the conclusion to be false at the same time.

An example of a valid argument is the following:

1. All dogs are mammals.
2. If all dogs are mammals, then all dogs are animals.
3. Therefore, all dogs are animals.

This argument is valid because it has the following valid argument form:

1. A.
2. If A, then B.
3. Therefore, B.

A and B can stand for any two statements, and they can both be replaced by any two statements. Any argument with this form is logically valid, such as the following:

1. All dogs are reptiles.
2. If all dogs are reptiles, then all dogs are animals.
3. Therefore, all dogs are animals.

In that case A and B stand for the following two statements:

A: All dogs are reptiles.

B: All dogs are animals.

Notice that “valid argument” does not mean “good argument.” It is merely one important requirement of a good deductive argument. The problem with the above argument is that the first premise is clearly false—it is false that all dogs are reptiles. We know that all dogs are mammals.

Why deductive arguments need to be valid

One lesson that we learn from formal logic is the importance for deductive arguments to have a valid argument form. The premises of a valid argument are a reason to believe the conclusion because if we assume the premises of a valid argument to be true, then we will also have to assume the conclusion to be true. If a deductive argument is invalid, then the premises don’t give us a reason to believe the conclusion to be true because the conclusion could be false, even if the premises are true.

Other than having a valid argument form, good deductive arguments must also have sufficiently justified premises. We need the premises to be justified (perhaps by everyone already agreeing with them in a debate). We do not necessarily need to know that the premises of our arguments are true because we often have to base our opinions on limited information. What we should believe and what is actually true are not always identical. We should believe our best scientific theories to be true, even though it is possible that they will be found to be false at some point in the future.

Most people seem to know that the premises of their deductive arguments need to be justified, but many people don’t know that their deductive arguments need a valid argument form. If we should believe that a valid deductive argument has true premises, then we should also believe the conclusion is true (because a valid argument with true premises also has a true conclusion). In that case the argument should be persuasive.

One problem is that many people who don’t properly understand the implications of valid argument forms won’t fully understand that an argument can have true premises and a false conclusion. They might think that all arguments with true premises are good arguments, or they might simply not know why some arguments with true premises fail to be good arguments.

Consider the following invalid argument:

1. All pythons are reptiles.
2. If all pythons are snakes, then all pythons are reptiles.
3. Therefore, all snakes are reptiles.

Does this look like a good argument to you? Many people are likely to think so, but it’s not. The premises and conclusion are all true, but that’s not good enough. The problem is that the argument has an invalid argument form. Even if we assume the premises are true, we could still assume the conclusion to be false. Someone who doesn’t think all snakes are reptiles will not be persuaded by the argument.

The argument form of the above invalid argument is the following:

1. A.
2. If B, then A.
3. Therefore, B.

Another argument with this argument form, true premises, and a false conclusion is the following:

1. All pythons are reptiles.
2. If all pythons are lizards, then all pythons are reptiles.
3. Therefore, all pythons are lizards.

It is true that “all pythons are reptiles” and it is true that “if pythons are lizards, then all pythons are reptiles,” but it is false that “all pythons are lizards.” Actually, all pythons are snakes, not lizards.

To repeat—an argument with true premises can still have a false conclusion. Invalid deductive arguments are not good arguments, even though they can be persuasive. We need to make sure our deductive arguments have a valid argument form or they won’t be good arguments.

Examples of valid argument forms

Knowing how to prove an argument is logically valid can take a logic class, but we don’t need to take an entire logic class to know that certain argument forms are logically valid. Consider the following five valid argument forms:

Modus ponens

1. If A, then B.
2. A.
3. Therefore, B.

Modus tollens

1. If A, then B.
2. Not-B.
3. Therefore, not-A.

Disjunctive syllogism

1. Either A or B.
2. Not-A.
3. Therefore, not-B.

Hypothetical syllogism

1. If A, then B.
2. If B, then C.
3. Therefore, if A, then C.

Constructive dilemma

1. Either A or B.
2. If A, then C.
3. If B, then D.
4. Therefore, either C or D.

How to improve our arguments

In order to improve our arguments, we can memorize the above five valid argument forms and make sure our arguments have one of them. In order to learn to improve our arguments, we will want to get a lot of practice doing this. We can start small by considering beliefs we think are obviously true and how we know they are true.

Consider that we know that rocks exist. That will be our conclusion. How do we know that? Because we can see them and touch them. We can now formulate the argument as the following:

1. We can see and touch rocks.
2. Therefore, rocks exist.

This argument is invalid, and it clearly does not use any of the valid argument forms. The question we can now to ask is, “What does seeing and touching rocks have to do with knowing rocks exist?” Modus ponens seems like it would be appropriate here. The missing premise can be “if we see and touch rocks, then rocks exist.” The argument is now the following:

1. We see and touch rocks.
2. If we see and touch rocks, then rocks exist.
3. Therefore, rocks exist.

Complex arguments

Keep in mind that some arguments seem much more complex than those above, but those arguments can actually be taken to be more than one argument. In general, complex arguments are actually one main argument with premises that are also argued for.

For example, consider the following argument:

1. We see and touch rocks.
2. Our experience of sight and touch is a reliable way to know about what exists.
3. If we see and touch rocks, then rocks exist.
4. Therefore, rocks exist.

It might seem like the second premise is extraneous. The argument uses modus ponens as long as we eliminate the second premise. However, the second premise is informative and should not be removed from the argument. The reason for that is that the second premise is actually a justification for the third premise. We could take the justification to be part of a second argument, and reformulate it as the following two arguments:

1. We see and touch rocks.
2. If we see and touch rocks, then rocks exist.
3. Therefore, rocks exist.

1. Our experience of sight and touch is a reliable way to know about what exists.
2. Therefore, if we see and touch rocks, then rocks exist.

Notice that the second argument (for the second premise) is not a valid argument. Once again, we can use modus ponens and reformulate it. It is then missing the premise “if our experience of sight and touch is a reliable way to know about what exists; then if we see and touch rocks, then rocks exist.” The second argument will now be the following:

1. Our experience of sight and touch is a reliable way to know about what exists.
2. If our experience of sight and touch is a reliable way to know about what exists; then if we see and touch rocks, then rocks exist.
3. Therefore, if we see and touch rocks, then rocks exist.

Conclusion

Good deductive arguments must be logically valid. We are unlikely to properly understand valid argument form on our own without any educational resources, and it can take a lot of work to fully understand it. Even so, we can learn quite a bit about valid argument form quickly. In particular, we can learn about five different valid argument forms and try to make sure our arguments have one of them. Sometimes an argument will have premises that we should also argue for, but any deductive arguments for those premises should also have a valid argument form.

The first argument

Many traditional theists believe the following:

1. God is willing and able to eliminate evil (because God is omnipotent, omniscient, and all good).
2. If God is willing and able to eliminate evil, then evil does not exist (because God would have immediately eliminated it). (“Evil” here merely refers to the fact that the world could be a better place.)
3. Evil exists. (Insofar as the world could be a better place.)

The problem is that these beliefs are inconsistent. The first two beliefs imply that no evil exists.

I think the problem of evil can be used as a good argument against the relevant view of theism and can be formulated in the following way:

1. The three beliefs listed above are inconsistent.
2. If multiple beliefs are inconsistent, then we know one of them is false.
3. Therefore, we know one of the three beliefs listed above are false.

The problem of evil is not an argument against the existence of any type of God. Perhaps God is not all good, or God is not all powerful, etc. Those are potential solutions to the problem of evil.

A theist’s response

Some theists are arguing that the problem of evil can’t be used against any type of theism because it would require the atheist to believe in objective morality, but objective morality requires God.

But what’s objective morality have to do with anything? Notice that the argument against the relevant view of theism did not mention objective morality. The argument is merely that people with a certain view of theism have inconsistent beliefs. There is no way to attack the argument by referring to objective morality.

Also notice that the argument is not against the existence of every type of God. There are theists who could very well give the argument in order to refute the relevant type of theism.

Even so, notice that the relevant traditional theistic beliefs refer to the existence of evil and God’s absolute goodness. How could anyone who rejects moral objectivity know anything about evil or absolute goodness?

My answer: Even a person who rejects objective morality could critique the theist’s beliefs by showing how they are inconsistent. A theist believes they know something about absolute goodness and evil. What they believe about those things seems inconsistent. For example, the theist will claim that it’s evil for a parent or guardian to leave their child in a closet all week to starve to death. It’s also evil for no one to help the child who is easily able to do so. Why doesn’t God ever help children like this? Why would it be morally wrong for us to not help, but morally permissible for God? I would think the same actions would be right and wrong for God. After all, what makes the action morally wrong seems to be that it causes needless suffering and death.

It is true that the atheist must be familiar with theistic beliefs concerning objective morality, but it is the theist’s own beliefs that can be found to be inconsistent.

The second argument

Some skeptics could have a different argument against traditional theism using the problem of evil. Consider the following:

1. Certain traditional theists believe that “God is willing and able to eliminate evil,” “if God I willing and able to eliminated evil, then evil doesn’t exist,” and that “God couldn’t make the world a better place because it’s the best of all possible worlds.”
2. However, these theists are wrong that “God couldn’t make the world a better place.”
3. Therefore, the relevant traditional theists have a false belief.

I think this argument is plausible, and it does require the assumption of objective morality. Again, an atheist can argue that objective morality seems to be perfectly compatible with atheism.

A theist’s objection

Does the above argument have anything to do with objective morality? Yes. A theist could object to the second argument in the following way:

1. The second premise (that states “these theists are wrong that God couldn’t make the world a better place because this isn’t the best of all possible worlds”) requires the assumption that objective morality exists. (Otherwise the world can’t really be a better place.)
2. Objective morality requires God.
3. Therefore, the second argument is inconsistent.

Reply 1 – It should be noted that I reject the belief that “Objective morality requires God.” I know of no good argument in support of that premise. It should be pointed out that many atheists do believe in objective morality, and they believe that objective morality is compatible with atheism. I take “objective morality” to refer to “moral realism.” There are clearly many moral realist atheists, and the most reputable meta-ethical theories developed by philosophers have nothing to do with God. Such philosophers are not necessarily atheists. The point is that whether God exists or not, we think we know something about morality.

For more information about why I don’t think objective morality requires God, go here. If a theist is sure that God is necessary for objective morality by using premises I will likely agree with, then I would like to hear their argument.

Reply 2 – This argument is logically invalid. The second argument did not state that no type of God exists. The relevant traditional theistic viewpoint (that God is omnipotent, omniscient, and all good) is being shown to be inconsistent with what we know about there being evil in the world, but that doesn’t imply that no type of God exists.

Moreover, even if it was inconsistent for an atheist to believe in objective morality, the same exact argument could be given by a theist. Not all theists believe God is all good or all powerful. Perhaps God exists, but doesn’t have thoughts or have much of anything in common with human beings. It is simply irrelevant who makes the argument. The argument will stand or fall on its own.

Let’s assume that atheism is incompatible with objective morality, and that an atheist gives the second argument against a traditional type of theism. Perhaps that atheist does know that objective morality exists, even though the atheist doesn’t know God exists. In that case the argument could still be based on the atheist’s knowledge of morality. The argument itself contains no inconsistent premises. Of course, the atheist could believe something false—perhaps that God doesn’t exist and that God isn’t required for objective morality. That is a separate issue.

Finally, the theist could argue that morality can only exist if their specific type of God exists (an omnipotent, omniscient, and all good God). However, I know of no good arguments that prove such a thing. Anyone that thinks they know of a good argument using premises I will actually agree with should let me know about it.

Conclusion

Anyone’s atheism should be irrelevant to the debate over the problem of evil as it has been described here. One type of argument against a traditional type of theism doesn’t require any assumptions about objective morality at all, but another does. The traditional theist who argues that the skeptic must assume objective morality exists will not be able to soundly refute either of those arguments using that as a premise. It is totally irrelevant to the first argument against a traditional type of theism, and it is somewhat relevant to the other. However, both of those theistic objections are unsound.

There are atheists who believe in objective morality and will argue against a traditional type of theism by arguing that evil exists. However, those arguments do not require that we reject theism entirely. Even if God was required for morality, the relevant atheists need not claim that the problem of evil proves no Gods exist at all. They merely need to argue that the problem of evil could prove certain types of traditional theism to require a false belief.

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• My Defense of the Argument From Evil, an Argument Against God
• Victor Reppert on the Argument from Evil as a Reductio
• William Lane Craig’s Moral Argument for God
• Copan’s Critique of Atheistic Objective Morality

What are cognitive biases?

Critical thinking and logic are concerned with proper reasoning—the way we should reason about things. But human beings are not made to think properly all the time. What we call “cognitive bias” refers to consistent ways people reason poorly. Cognitive bias concerns how people actually reason as opposed to how they should reason.

“Cognitive bias” does not refer to prejudice, racism, a liberal preference in the media, or anything like that. Instead, it refers to how we consistently think in certain ways that often causes us to reason poorly. The human mind was not made to reason perfectly. To reason well can require a great deal of effort and people aren’t usually going to spend hours doing formal logic to figure out if their beliefs are well-reasoned or not. How we actually reason is often intuitive, immediate, and effortless. We use shortcuts and jump to conclusions based on limited information without thinking much at all. This works well for the most part. Spending too much time and energy reasoning about whether we should take a shower would obviously not be a good way to live our lives.

We now know of over a hundred cognitive biases (such as the confirmation bias, outcome bias, and halo effect), and these biases are likely to cause us to use fallacious reasoning—or to find the fallacious arguments given by others to be persuasive.

It shouldn’t be a surprise that we tend to think our own reasoning is perfectly good, even when it isn’t. The most important lesson we can learn from cognitive bias research is that they are part of being human and we can’t stop being biased. Instead, we need to share our thoughts and reasoning with others because we are much better at spotting the poor reasoning of others than our own. Others are likely to help spot the mistakes we make in our reasoning process, especially if they have a different view of the world.

Examples of biases and fallacies

Confirmation bias

Consider the following argument:

1. All dogs are mammals.
2. If all dogs are animals, then all dogs are mammals.
3. Therefore, all dogs are animals.

People are more likely to think this is a good argument because the conclusion is obviously true. However, it’s actually an invalid argument. The confirmation bias causes us to be likely to think any argument in favor of a belief we agree with is well-reasoned, even when it’s actually fallacious.

The above argument has the following argument form:

1. A.
2. If B, then A.
3. Therefore, B.

Another argument with this argument form is the following:

1. All dogs are animals.
2. If all dogs are reptiles, then all dogs are animals.
3. Therefore, all dogs are reptiles.

Both premises are true, but the conclusion is false. Any argument with this form can have true premises and a false conclusion. That means that the premises fail to give us a good reason to believe the conclusion. However, good arguments do give us a good reason to believe the conclusion.

The confirmation bias causes us to take any confirmation for our beliefs too seriously and it causes us to take any counter-evidence against our beliefs much less seriously than we should. We are also likely to fail to realize that some people disagree with what we believe and fail to learn about any counter-evidence against our beliefs.

For example, liberals are more likely to think that any type of environmental regulation is a good idea and fail to realize when they aren’t; and conservatives are more likely to think any type of environmental regulation isn’t a good idea and fail to realize when they are. People are more likely to read articles that support their beliefs, and they are more likely to dismiss or marginalize any evidence they read against their beliefs. Liberals would be more likely to read about the importance of an environmental regulation rather than the problems with it, but they are also likely to think any problems with an environmental regulation aren’t really a “big deal.”

The confirmation bias is likely to cause us to find any fallacious argument for a conclusion we agree with to be a good argument when given by others, but it can also cause us to sincerely engage in fallacious reasoning of our own. In particular, it often causes us to engage in “one-sided reasoning.” We are more likely to reason that any evidence for a belief justifies the belief, even when there’s also evidence against the belief. We should consider “both sides” of a debate when we want to know what we should believe.

Consider the following one-sided argument:

1. Some people say that robbing a bank to take their money would be wrong because it could cause people to needlessly die.
2. However, the money we get from robbing a bank could be used to help pay the medical bills for poor people.
3. Therefore, we should rob a bank.

The problem is that robbing a bank is clearly wrong because it could cause people to needlessly die, even though we could use the money to help some people. We can’t just dismiss the reasons against robbing a bank and think any reason is good enough.

Outcome bias

The outcome bias causes us to think that decisions are morally right or wrong based on the consequences rather than the reasoning process. Consider a person who drives drunk. If no one gets hurt, we are more likely to forgive her and judge her less harshly than if she ends up causing a car accident that kills someone. Her decision to drive drunk was poorly-reasoned either way, and getting someone killed did not actually make the decision more unethical.

It is important to consider what consequences our actions are likely to have, but the actual consequences do not determine how ethical our decisions are. A person who decides to drive a car safely and ends up killing a child who jumps in front of the car does nothing wrong whatsoever, even though it ends up killing someone. And a person who drives drunk is doing something wrong, even if no one gets hurt.

One obvious case of the outcome bias is how we judge revolutionaries. When revolutionaries fail, we are much more likely to see them as terrorists or murderers; but if the revolutionaries succeed, then we are more likely to see them as heroes and to call them “freedom fighters.” The founding fathers of the USA are considered to be freedom fighters, but any revolutionaries against the USA at any later point in time are considered to be terrorists.

The outcome bias is related to the one-sidedness fallacy because we are likely to take the outcome of a decision too seriously and to take the actual reasoning process less seriously than we should. If the decision was well-made, but caused something bad to happen, then we will likely think the decision was unethical, even though it was well-reasoned. If a decision was poorly-reasoned, but caused something good to happen, then we are more likely to judge the decision less harshly.

Consider the following one-sided argument:

1. Martha decided to go for a walk and got mugged.
2. The area is known to be safe and it is the first time anyone got mugged in the area for over a year, but Martha is a small woman and knew that such a thing could happen.
3. Therefore, Martha should have known better and she shouldn’t have went for a walk.

The problem is that Martha would not be said to have done anything wrong if she went for a pleasant walk without being mugged. Perhaps it’s one of her best opportunities to get some exercise. The area is known to be safe and it is part of human life to take small chances. Driving a car safely can get someone killed, but it’s not wrong to drive a car. Going for a walk in a safe area is similarly a reasonable decision to make, even though there is a slight possibility of getting mugged.

Halo effect

The halo effect refers to the fact that we are likely to see a person as nearly all-good or all-bad based on our limited knowledge of the person. The halo effect is highly related to the power of first impressions, but the bias is not limited to first impressions. We change our mind about a person over time and we are still likely to see that person as mostly all-good or all-bad.

If someone is attractive, then we are more likely to have a good first-impression of that person and to see that person as mostly all-good. We are then more likely to think that person is more qualified for any job, has superior abilities, and will be more likely to give to charity.

We are more likely to trust and take seriously anyone we see as mostly all-good than one we see as mostly all-bad. If a person is well-groomed, articulate, well-dressed, and attractive; then we are more likely to think that person’s arguments are well-reasoned, even if they aren’t. We are also more likely to think someone we dislike gives poorly-reasoned arguments, even if they aren’t.

The halo-effect is also likely to cause us to reason fallaciously using “hasty generalizations” and “ad-hominems.”

Hasty generalizations – We are often likely to make judgments based on insufficient data. Knowing that a person is attractive does not tell us how qualified she is, what abilities she has, or how ethical she is. We should try to make those judgments based on much better information than that. However, that is exactly how we tend to actually think about attractive people.

An example of a hasty generalization is the following:

1. The President has stated that he is against torture.
2. If the President has stated that he is against torture, then he is a good person and is qualified for the job.
3. Therefore, the President is a qualified for the job.

The problem is that the limited information presented here about the President is not sufficient to know how qualified she is for the job.

A person’s limited information about a politician is likely to greatly influence her entire view of the person. If the limited information sounds good, then the politician is more likely going to be viewed as all-good. Of course, liberals are also more likely to view Democratic politicians as mostly all-good, and conservatives are more likely to view Republican politicians as all-bad.

Ad hominems – We are often likely to see a person as mostly all-bad based on limited information and jump to conclusions based on that view. If we see a person as mostly all-bad, then we are also likely to view their argument as poorly-reasoned based on very little information. To fallaciously base our judgments on a negative view of a person is called the “ad hominem” fallacy.

An example of an ad hominem is the following:

1. John wants us to legalize marijuana because it is much more harmless than cigarettes or alcohol.
2. However, John smoked marijuana when in college.
3. If John makes poor judgments like that, then we can’t take his argument seriously.
4. Therefore, we should reject John’s argument.

We are given a reason to think we should legalize marijuana. The argument is not going to be less reasonable based on any potentially negative characteristic of John. We should not reject the argument just because of that potentially negative characteristic. It’s irrelevant.

Conclusion

Cognitive biases often cause us to reason fallaciously and to find fallacious arguments to be persuasive, but cognitive bias is based on effortless automatic thought processes that are often beneficial to us. It’s not always entirely clear when our automatic process will lead us into fallacious reasoning, and we ultimately can’t stop it from happening. However, we can look out for our biases and try to make our reasoning process public, so that other people have a chance to correct any mistakes we make in our thinking.

You can follow Ethical Realism on Facebook or Twitter.

Related

• Amazon.com: Thinking, Fast and Slow by Daniel Kahneman
• Charlie Rose: Daniel Kahneman on Thinking, Fast and Slow (youtube)
• @Google Presents: Daniel Kahneman (youtube)
• List of Cognitive Biases
• Nonrational Forms of Persuasion & Manipulation
• Cognitive Bias: What they are, why they’re important (youtube)

I will introduce the plausible view that we can know about scientific issues, and the common view that many people have that we can’t know about philosophical issues. I will explain why we can’t reject philosophy without being inconsistent, and I will give examples of various plausible philosophical beliefs and reasonable philosophical arguments.

Introduction

Can we know about scientific facts?

Most of us believe that scientists are experts. They know more about biology, chemistry, physics, and psychology than the rest of us because they conduct experiments and know about a lot of the relevant data. There are scientific arguments that are given for various hypotheses using the data because we can find out that certain hypotheses are supported by the data better than others. But are philosophers also experts? Many people think philosophers aren’t, and very few people seem to take the arguments given by philosophers very seriously. Many people think philosophy is nonsense. That we shouldn’t trust what philosophers have to say.

At one point natural science was called “natural philosophy” and it was considered to be part of the philosophical domain. Now people think science is clearly better than philosophy, which is now restricted to certain major domains of inquiry, such as logic (proper argumentation), epistemology (the nature of knowledge), ethics (the nature of morality), and metaphysics (the nature of reality).

Consider the following beliefs:

1. Scientists are experts. (They know more about their domain of expertise than the rest of us.)
2. Everyone’s opinion about science is not equal. (Some scientific beliefs are supported by the evidence better than others.)
3. We can know something about scientific issues. (We can know some hypotheses are better than others.)

None of these beliefs are scientific beliefs. Science can’t tell us what science “should be like” or the difference between science and pseudoscience. It can’t tell us the difference between a justified or an unjustified belief. That’s what “philosophy of science” is about—the nature of science. All of these beliefs are highly related to the nature of knowledge itself. “What is knowledge? Can we know anything? If so, what?” If scientists know more about science than we can, then we are assuming that we can know what knowledge is, that we can know something about science, and that some people know more about scientific facts than the rest of us.

Can we know about philosophical facts?

Although most people think it’s obvious that scientists can know more about science than the rest of us, many do not think it’s obvious about philosophy. In fact, many people reject philosophy. People endorse “scientism” when they think science is the only way to know anything—and many scientistically-minded people think philosophy fails to be science. Therefore, there is no “philosophical” way to know anything. There can be no philosophical experts. We can’t really know anything about the philosophical domains. Everyone’s opinion is equal concerning philosophical issues.

The point is that many people don’t think anyone can know more about the nature of reasoning, knowledge, morality, or reality than anyone else. Philosophers might be experts concerning “philosophical trivia,” but no philosophical belief is actually better than another.

People who reject philosophy believe that there are people out there who call themselves “philosophers.” They say that they are “doing philosophy.” They can learn about the history of philosophical arguments and ideas. However, they think there is no real philosophical progress. For example, we can’t know anything about the nature of argumentation. We can’t even know that certain beliefs about logic are false. We can’t know that “it is not the case that all men are mammals” is logically equivalent to saying “some men are not mammals.”

Consider the following beliefs:

1. Philosophers aren’t experts. (Philosophers don’t know more than the rest of us concerning philosophical domains.)
2. Everyone’s opinion is equal concerning philosophical issues. (No philosophical belief is more justified than another.)
3. We can’t know anything about philosophical issues. (We can’t know if one philosophical statement is better than another.)

None of these beliefs are “scientific” and they aren’t self-evident. They are beliefs that fall within the philosophical domain. Again, the main philosophical domain at interest here concerns the nature of knowledge. The first belief listed is not plausible if it’s possible for some people to know more about philosophical issues than other people. The second and third beliefs about philosophy are self-defeating because they are such strong statements against philosophy, but they are philosophical beliefs.

If we can know philosophers aren’t experts, then how can we know that “we can know philosophers aren’t experts”? Perhaps everyone is an equally good philosopher, but how could that happen?

If everyone’s opinion is equal concerning philosophical issues, then the belief “everyone’s opinion is equal concerning philosophical issues” is no more justified than the belief that “some opinions concerning philosophical domains are better than others.” If one person knows something is true and someone else only thinks she knows, then their opinions are not equal. Whether or not everyone’s opinion is equal is a philosophical issue because the nature of knowledge is a philosophical issue.

If we can’t know anything about philosophical issues, then we can’t know that “we can’t know anything about philosophical issues.” What exactly the limits of knowledge are is a philosophical issue because the nature of knowledge is a philosophical issue.

The real issue here is whether or not we can know anything about the reality of philosophical issues. Many people don’t think we can. Could we prove that we can’t know anything about philosophical issues? That itself would seem impossible because it’s a philosophical issue. If we know nothing about philosophical issues, then we actually do know something about philosophical issues.

More examples of philosophical beliefs and arguments

One reason to think we can know something about philosophical issues is that there are good examples of things we think we know about them already. People who think they know something about a philosophical issue yet deny that anyone knows anything about philosophical issues are being inconsistent.

One reason to think that we can learn more about philosophical issues (and to think there can be philosophy experts) is that there can be reasonable philosophical arguments. One way to give philosophical arguments is by showing how we think we know something already, and what we think we already know implies that something else must also be true.

Perhaps one reason that so many people distrust philosophy is because they think philosophers are doing something esoteric and far-removed from everyday life (unlike scientists). However, I believe that everyone actually has philosophical beliefs and engages in philosophical reasoning. Their beliefs are often justified, and their reasoning is often reasonable. What philosophers do is not so different, but they keep it up, and philosophers are aware of several arguments for and against various issues.

I will give examples of various things we think we know concerning issues from four major philosophical domains (logic, epistemology, ethics, and metaphysics).

Logic

We think we know that a belief can’t be factually true and false at the same time. If I say, “It’s raining” and you are in another place and say, “It’s not raining,” we are not actually contradicting one another because we are merely saying “it’s raining where I am.” However, to say that “there is life on another planet in the universe” and that “there isn’t life on another planet in the universe” would be to give contradictory statements. When two factual statements contradict one another, at least one of the statements must be false.

I think this belief is plausible. If I am right that this belief is plausible, then we can know something about a philosophical issue (that at least one philosophical belief is plausible). However, anyone who thinks we can’t know anything about philosophical issues can’t consistently believe it to be plausible.

A simple argument using this belief as a premise is the following:

1. A belief can’t be factually true and false at the same time.
2. Some people believe that it’s factually true that there is life on another planet in the universe.
3. Other people believe that it’s factually true that there is no life on another planet in the universe.
4. If “there is life on another planet in the universe” is true and “there is no life on another planet in the universe” is also true, then a belief would be factually true and false at the same time.
5. Therefore, “there is life on another planet in the universe” is false and/or it’s not the case that “there is no life on another planet in the universe” is false.

I think this argument is reasonable. However, anyone who rejects philosophy must not find this argument to be reasonable. The only way for this argument to be reasonable is if we can know certain things about philosophy (such as the premise concerning logic mentioned earlier).

However, it’s a bit worse than that for those who reject philosophy entirely. Scientists who reject logic won’t be able to support their hypotheses with good arguments. Scientists are committed to giving logical arguments (good arguments using criteria given by logicians), even though logic itself is a philosophical domain. We don’t observe that two factual beliefs can’t be true at the same time. It is a plausible belief about logic anyway.

Anyone who rejects philosophy will have to also reject logic. They will have to reject the idea that we can know anything about proper ways to reason about things, or that a good argument must meet certain criteria.

Epistemology

We think that we can know something about the future by knowing about the past. For example, you can know that rocks that are dropped two seconds from now (on our planet) will fall to the ground based on the fact that all similar objects that were dropped in the past also fell to the ground.

I find this belief to be plausible. If I am right that it’s plausible, then we can know something about philosophy (because we can know that a philosophical belief is plausible). However, anyone who rejects philosophy will not be able to consistently believe it to be plausible .

An argument using this belief as a premise is the following:

1. All cats observed by scientists throughout history were mammals.
2. We can know something about the future by knowing about the past.
3. If we can know something about the future by knowing about the past and all cats observed by scientists throughout history were mammals, then all cats are probably mammals.
4. Therefore, all cats are probably mammals.

This argument could very well be one given by a scientist, but notice that one of the premises is a philosophical one. Sometimes scientists rely on philosophical premises. There is no absolute boundary between science and philosophy.

It is inconsistent for scientists to assume any philosophical belief is justified while simultaneously claiming that we can’t know anything about philosophy. The fact that we can know something about the future by knowing about the past has never been proven by a scientist. Assume for a moment that a scientist did prove it. How could she prove it? Perhaps by seeing if future data resembled past data in the past. But how does that prove it? Does the fact that future data resembles past data prove that we can know something about the future from the past? If so, we can know something about the future by knowing about the past, but that’s circular reasoning.

She might as well argue the following:

1. We can know something about the future by knowing about the past.
2. Therefore, we can know something about the future by knowing about the past.

This is not a good way to argue. It’s no better than just repeating an assertion. People can repeat any assertion, but that doesn’t mean we should agree that it’s true.

Ethics

We think we know that we shouldn’t cause people intense pain unless we have an overriding reason to do so. For example, it’s wrong to kick a two year old child really hard while having a pleasant conversation with the child because there is no overriding reason to do so.

I find this belief to be plausible. If I am right that it’s plausible, then we can know something about a philosophical issue. However, anyone who doesn’t think we can know anything about philosophical issues will not be able to consistently believe it to be plausible.

An argument using this belief as a premise is the following:

1. We shouldn’t cause people intense pain unless we have an overriding reason to do so.
2. Torturing people who are caught smoking marijuana would cause them intense pain.
3. We don’t have an overriding reason to torture people who are caught smoking marijuana.
4. Therefore, we shouldn’t torture people who are caught smoking marijuana.

I find this argument to be reasonable. If I am right, then we can know something about a philosophical issue (because we can know that a philosophical argument is reasonable). However, anyone who rejects philosophy will not be able to consistently believe it to be reasonable.

Metaphysics

We think we know that other people have mental activity. I know that I see things, hear things, feel things, and have thoughts because I experience all of that for myself. However, I also know other people also see things, hear things, feel things, and have thoughts.

I find this belief to be plausible. If I am right, then we know something about a philosophical issue (because we would know a certain philosophical belief to be plausible). However, anyone who doesn’t think we can know anything about philosophical issues could not consistently believe it to be plausible.

An argument using this premise is the following:

1. Other people have mental activity.
2. If other people have mental activity, then other people who have functioning eyes can see me when I am standing in front of them.
3. Therefore, other people who have functioning eyes can see me when I am standing in front of them.

I believe this argument to be reasonable. If I am right, then we know something about a philosophical issue (because we would know a certain philosophical argument to be reasonable). However, anyone who doesn’t think we can know anything about philosophical issues could not consistently believe it to be reasonable.

What is philosophical knowledge like?

Scientific facts are considered to be “proven” at some point. We hypothesized that germs existed and we eventually found them under a microscope. Perhaps philosophical facts can’t be proven like that. Philosophical and scientific progress can be somewhat different, but related.

In both cases we think we know certain facts. For example, we think we know that rocks exist. Scientists and philosophers should generally use those plausible factual beliefs as “evidence” rather than reject them as “unproven prejudice.” We have to start from somewhere.

In both cases we can accept a hypothesis until we have an overriding reason to reject it. The hypothesis must be consistent with the data and it must not be significantly worse than alternatives.

In both cases we should reject a hypothesis when we believe it to be “falsified.” At this point the hypothesis seems to contradict the facts we think we know about. We have to decide if the hypothesis is less plausible than the factual beliefs it contradicts. For example, we don’t necessarily reject the laws of physics when they don’t predict the motion of the stars perfectly. Instead, we might think the factual knowledge we have about outer space is incomplete. We now believe there is “dark matter” involved. In that case the “facts” we thought we knew about outer space (such as the position of stars) was not more plausible than the laws of physics we think we know about.

In other words the progress found in both philosophy and science is often of elimination. We think we know that certain beliefs are better supported than others, and sometimes a belief is “rejected” because it’s not consistent with our understanding of things. The main difference between the progress found in science and philosophy is that there might be a point when science proves something to be true once and for all, which might never happen in philosophy. (This point is up for debate.)

Conclusion

Many people think we can’t know anything about philosophical issues, but such a belief is inconsistent because what we can know itself is a philosophical issue. Moreover, people who claim that we can’t know about philosophical issues are also likely to think they do know about certain philosophical domains—they are likely to think they know certain things about logic, epistemology, ethics, and metaphysics. That is also inconsistent of them.

When we find out our beliefs are inconsistent concerning factual matters, we know one of our beliefs is false. Otherwise we’ve given up on logic entirely and no argument will matter any longer. It would be impossible to have a good reason to believe anything.

When we find out our beliefs are inconsistent, we should try find out which ones are false. In this case I think the belief that “we can’t know anything about philosophical issues” is quite implausible and should be rejected. Some of our philosophical beliefs are much more plausible and should be taken seriously.

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Abstract

The focus of this book is propositional logic. I discuss the meaning of “logic,” the importance of logic, logical connectives, truth tables, natural deduction, and rules of inference.

Why is logic education important? The main question here is what the real point of logic education is. The real point of logic is not to teach people how to be logic professors, or to increase test scores, or to impress potential employers. Philosophers and mathematicians were very interested in understanding logic long before it was taught in universities precisely because of how important it is. Why is logic so important? The answer is that logic helps us better understand good arguments—it helps us differentiate between good and bad reasons to believe something. We should want to have well-justified beliefs. We want to know what we should believe. Understanding good argumentation helps us understand when we should believe something, and understanding logic helps us understand good argumentation.

How exactly does logic help us understand good argumentation? There are many necessary characteristics that good arguments must have, and logic tells us what some of those characteristics are. Logic also helps us better understand concepts that are relevant to good argumentation.

What is a good argument?

Good arguments are good reasons to believe something is likely true. If we know of a good argument to believe something, then we should believe it. For example:

1. All dogs are mammals.
2. If all dogs are mammals, then all dogs are living organisms.
3. Therefore, all dogs are living organisms.

People should agree that “all dogs are living organisms.” We know the premises are true (that “all dogs are mammals” and “if all dogs are mammals, then all dogs are living organisms”). We know that if the premises are true, then the conclusion has to be true as well. The premises can’t be true and the conclusion false at the same time because the argument is logically valid.

What characteristics do good arguments have?

Ultimately good arguments must have sufficiently justified premises, and the premises should be appropriately relevant to the conclusion. Even so, there is much to be said about this criteria. Many of the necessary characteristics of good arguments are covered by the various issues discussed in logic classes—logical form, logical validity, the distinction between inductive and deductive reasoning, argument interpretation, and informal fallacies. Examples about what various general logical issues can teach us about good argumentation includes the following:

1. Logical form – Understanding logical form is of paramount importance to understanding good deductive argumentation, but ordinary language makes it very difficult to discuss logical form. It is much easier to understand logical form and how it relates to good argumentation after learning about logical form in a logic class. For example, “If Socrates is a man, then he is mortal” has the logical form “If A, then B.”
2. Logical validity – Logically valid deductive arguments have premises that guarantee the truth of the conclusion (assuming they are true). An invalid deductive argument gives us no reason to think the conclusion is true. In that case the premises can be true and the conclusion can be false at the same time.
3. The distinction between inductive and deductive reasoning – All good deductive arguments are logically valid, but good inductive arguments aren’t. Inductive arguments are not meant to be valid because the premises are only supposed to make the conclusion probable. A good inductive argument is unlikely to have true premises and a false conclusion, but it can happen. For example, the evidence scientists use to support scientific theories is inductive and it is possible that the theories are actually false (imperfectly accurate). The predictions made by scientists could always turn out to be false, but they are likely to be true when they are well-justified. Even dropped objects could fail to fall in the future. Even so, we should agree that dropped objects will fall in the future anyway.
4. Argument interpretation – It is important to fully understand people’s arguments and to know how to clarify their arguments. It’s important that we know what exactly the premises and conclusions are. Sometimes understanding an argument also requires us to identify unstated assumptions and some creativity could be required. It is impossible to properly debate with someone who doesn’t understand your arguments. An argument can’t be properly refuted unless it is understood well.
5. Informal fallacies – Informal fallacies are errors in reasoning other than having an invalid argument form. Interpreting arguments uncharitably is one common example called the “straw man fallacy.”

There are innumerable specific examples about how each general issue can apply to good argumentation. One example is that people often refute the conclusion of an argument by arguing against a premise. However, it is possible for a different argument to be given for any given conclusion. Refuting a premise of an argument does not simultaneously refute the conclusion. Consider the following argument:

1. If the President of the United States is a lizard, then the President is a mammal.
2. The President is a lizard.
3. Therefore, the President is a mammal.

In this case both premises are false, but the conclusion is true. Stating that the premises are false gives us no reason to think the conclusion is false. Such an argument could look like the following:

1. “The President is a lizard” is false.
2. Therefore, we should reject that “the President is a mammal.”

This argument is clearly invalid. The premise is true, but the conclusion is false. This example shows how understanding validity can help us understand why certain arguments fail to be good arguments. Actual people do argue this way now and then, so learning about it in a logic class could actually help people come to a realization that they might not think about otherwise.

Many examples were given concerning how logic classes can help us better understand good argumentation. However, there are potentially other characteristics of good arguments that are not discussed in logic classes. The philosophical domain called “epistemology” concerns the nature of knowledge, rationality, and justified belief. There are certain details about what counts as sufficient justification and rational thought that logic does not cover, but is covered by epistemology instead. Epistemology is the domain concerning how open-minded we should be to avoid being close-minded, how skeptical we should be to avoid being gullible, and how much evidence a belief requires in order to be sufficiently justified. Also, logicians don’t tell us if any beliefs are self-evident, if intuition is ever a good reason to believe something, or when we can rationally assume a premise to be true without argument. Those are issues concerning epistemology.

Why is good argumentation important?

Logic alone can’t tell us why good argumentation is important. It’s a philosophical question. Answers include the following:

1. We want to know what’s true. Good arguments can tell us what is likely true.
2. We often can’t believe what we should believe unless we understand good augmentation. Understanding good argumentation helps us know what we should believe, and it helps us prove to other people what they should believe. Moreover, we should believe certain things because we know about good arguments. Believing what is true at random is not appropriate. We should believe what is likely true based on a good reasoning process.
3. Our beliefs can motivate us to behave in certain ways, and false beliefs are more likely to motivate us to behave in inappropriate ways. For example, we found out that lead is poisonous and we try to make sure children’s toys no longer contain lead for that reason. People felt free to put lead in children’s toys until their beliefs were corrected.
4. Many people want to manipulate us to believe certain things, and understanding good argumentation can help us spot the faulty arguments that are used to manipulate us. For example, charlatans want to sell us products that don’t actually work. Many medical products are sold that don’t actually do what they are said to do. People waste their money on those products when they get duped.

Update (3/8/13): I clarified how epistemology relates to good argumentation and rationality.

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Many people say that those who claim that bigfoot, ghosts, and gods exist are making “extraordinary claims” and we should reject the existence of these things because we don’t have enough evidence for them.

What’s “extraordinary” mean? It refers to claims that conflict with what we think we know about the world. Many claims are extraordinary because they are extreme (likely false) or potentially impossible.

Extreme

Some claims are extraordinary because they are so extreme. Extreme claims are those that should find to be unlikely given our understanding of the world. Such claims can be about rare characteristics or situations that are unlikely to actually describe anything.

An example of a rare characteristic that is unlikely given our understanding of the world is that someone is nine foot tall. This claim is extreme because the tallest person we know about was 8 foot 11 inches and died in 1940. Saying that someone will be born this year in New York City and will grow to be 8 foot 5 inches tall is similarly extreme because people don’t grow to be that tall very often.

An example of a rare situation that is unlikely given our understanding of the world is that there’s a celestial teapot going around the Sun in our solar system. This claim is extreme because all the teapots we know about are on the planet Earth, they exist here because they are made by humans, humans don’t go to outer space very often, and we don’t know how any teapots would end up going around the Sun.

Potentially impossible

What’s potentially impossible is what might not be able to exist given our understanding of the world. What is impossible is what can’t be true. There are different types of impossibility, such as the following:

1. Physical impossibility – What can’t be true because it would require the laws of nature to be violated. For example, jumping to the Moon is physically impossible. Some people say that “miracles” are physically impossible, but still occur due to divine or supernatural intervention.
2. Metaphysical impossibility – What can’t be true in any reality. What never happens in any possible world. Not even the supernatural could violate what’s metaphysically impossible. For example, finding a world where water isn’t H₂O is plausibly metaphysically impossible.
3. Logical impossibility – Logical contradictions. What can’t be true because of logical constraints. For example, it’s logically impossible for Socrates to be both mortal and immortal.

The sense of “impossible” that deals with extraordinary claims is “metaphysical impossibility.” To claim that something is true that we know to be metaphysically impossible is absurd, but to claim that something is true that we suspect could be metaphysically impossible (because of our understanding of the world) is extraordinary. Anything we know to be physically impossible is potentially metaphysically impossible—It might be metaphysically impossible for the laws of nature to be violated. Perhaps there are no possible worlds where the laws of nature are violated.

Miracles are extraordinary precisely because miracles are physically impossible by definition. However, miracles might still be metaphysically possible—we don’t know for certain that there’s nothing supernatural or divine. The claim that miracles exist is extraordinary because the assumption that everything happens because of laws of nature has helped make natural science the best source of knowledge about the world we have. For similar reasons claims that ghosts exist is extraordinary (because we don’t know that it’s metaphysically possible to have a mind without a body). Also, the claim that gods exist is extraordinary because it might be metaphysically impossible for them to exist.

Why would it be impossible for gods to exist? It might depend on the definition of gods. Pantheists claim that gods are identical to the universe. We know that the universe exists, and it could seem trivial that a god exists if we define it as the universe. That is not the type of god we find extraordinary in the sense relevant to this discussion. However, the following types of gods are extraordinary:

1. A god that can have a mind without a body. All minds seem to require bodies. To have a mind without a body would violate our scientific understanding of the world, and it could be metaphysically impossible.
2. A god that is all-knowing (omniscient) or all-powerful (omnipotent). Science has not confirmed that anything has any of these characteristics, and the scientific evidence we have indicates that knowledge and power is always limited from physical constraints. To be omniscient or omnipotent is as extreme as anything can be, and they might be metaphysically impossible.
3. A supernatural god that can violate the laws of nature is extraordinary in the relevant sense because we don’t know for certain that it’s metaphysically possible for anything to do such a thing. If miracles happen, then they are very rare (and extreme). So far there have been no scientifically confirmed miracles, and making predictions based on the assumption that everything will happen in accordance with the laws of nature has been very successful. Natural science is the most successful study of the world and it always works under the assumption that the laws of nature will not be violated. Natural science is successful precisely because it is so good at making predictions.

What about ordinary claims?

The reason that “extraordinary claims” are singled out is because some claims are so ordinary that we might suspect they are true just because of our understanding of the world. Let’s say we know a neighbor lives in a house. To say that the person who lives in the house is less than ten feet tall is an ordinary claim, and we should believe it. It is probably true because we know of no one who has ever lived taller than nine feet. It is so common for people to be less than ten feet that it is unlikely for such a prediction to ever fail. Such a prediction could very well have a perfect track record for the entirety of human history (and the future of human kind).

Other ordinary claims have a decent probability of being true. For example, to predict that a neighbor is less than seven feet tall is still a decent prediction, even though some people are taller than that.

Ordinary claims are known to be metaphysically possible. We know it is metaphysically possible for a person to be under ten feet tall.

Of course, there are gray areas that are neither ordinary nor extraordinary. Scientists hypothesize about laws of nature that might not exist. Such laws of nature might actually encourage scientists to make predictions that would violate the actual laws of nature (and be metaphysically impossible as a result). However, scientists should never hypothesize that a law of nature exists that we know contradicts the actual laws of nature. A hypothesis that repeatedly fails to make certain predictions will be rejected precisely because we will find out that it violates the actual laws of nature that actually exist.

There a burden of proof against extraordinary claims

People know something about the world and there are many scientific observations that are relevant to the claims we make. If someone makes an extraordinary claim (a claim that contradicts what we think we know about the world), then we should reject the claim until we are given a good reason to change our mind. The burden of proof is the requirement that those who make extraordinary claims have in order to rationally change our minds.

The existence of ghosts and tiny faeries are extraordinary. The scientific view of the world states that minds can’t exist without bodies, and that bigger brains are required to have the intelligence of a human being. Ghosts have minds without bodies; and faeries are very small, but have the intelligence of a human being. These entities literally violate what we think we know about the world. We should be hesitant to believe in ghosts and faeries. We should require at least some evidence that these entities exist before deciding they exist.

We can certainly imagine having a good reason to change our mind and think ghosts or faeries exist. Perhaps we can literally find a tiny faerie that talks and interacts with us over a period of years. In that case the faerie’s existence could be the best explanation for our experiences (as opposed to a hallucination or dream).

Once we find out that ghosts or faeries exist, that will require us to revise our understanding of the world. We might find out that minds don’t require bodies, or that a tiny brain can be capable of having a human level of intelligence. Once we find out that ghosts or faeries exist, such a claim will no longer be extraordinary for those of us who find out that they exist. If anyone finds out that ghosts or faeries exist, then such people will not require a burden of proof that they exist because that debate will be over. Of course, anyone who believes in ghosts of faeries should stay open to the possibility that they are wrong, and the debate can be resumed at some point.

Do extraordinary claims require extraordinary evidence?

Extraordinary claims don’t require some strange type of evidence that is so different from an ordinary type of evidence. Extraordinary claims can be proven to be true in much the same way as any other type of claim. In fact, extraordinary claims are not totally different from other types of claims. They are far fetched, but all claims are part of a single continuum that ranges from “far fetched” to “obviously true.”

Obviously true claims require no evidence, such as the claim that the person who lives in a house is under ten feet tall. The reason is simply because what we know about the world already proved it to be true. The evidence has already been attained, and no further evidence is required.

Extraordinary claims require extraordinary evidence in the sense that they require more evidence than usual before they are properly proven to be true. Claims that violate what we think we know about the world require more evidence because certain other claims we think were proven to be true are incompatible with them. For example, we think we know that minds require bodies, and the existence of ghosts violates that. We already have evidence that minds require bodies. In fact, we think we know certain types of bodies can have minds (such as that of a living mammal) and others can’t have mind (such as that of a plant). So, we think we know that there are only certain types of bodies that can have minds.

Before we decide that ghosts exist, we should require better evidence that ghosts exist than the evidence we have that minds require bodies. We should not reject one well-justified belief (such as the belief that minds require bodies) just because we want to accept an extraordinary claim that contradicts the belief (such as the belief that ghosts exist). However, it is theoretically possible to have better evidence that ghosts exist than that minds require bodies. If we find a way to interact and talk with ghosts, perhaps we will eventually know that ghosts exist.

We do have some evidence that ghosts exist. People say they have seen ghosts, talked to them, or been touched by them. People have had observations, and they speculated that ghosts are an explanation of those observations. The problem is that there can be alternative explanations of our ghost experiences. One common alternative explanation is “misidentification.” We can simply think a ghost caused an observation when something else actually caused it. Hallucinations and dreams are also possible alternative explanations when more ordinary explanations are ruled out.

Should we literally assume that something extraordinary doesn’t exist until it is proven to exist? Yes, we should assume a claim to be false when it contradicts what we think we know to be true. We should assume that a person didn’t jump to the Moon, and we should assume that the snake oil doesn’t actually cure all ills precisely because it would defy reasonable expectation.

Related:

• An Argument Against God, a Teapot, and Garvey’s Objection (Part 1)
• What is the Burden of Proof?
• Do Default Positions Exist?

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This is part 6 of a series. Links to the other parts of the series are above.

The straightforward way to construct proofs using natural deduction is called the “direct method.” Every line of that type of proof is validly deduced from the premises and rules of inference. Every line of such a proof could be considered to be true as long as we consider the premises to be true. However, there are two other strategies: The conditional proof and the indirect proof. Both of these types of proofs introduce an additional premise that is assumed to be true “for the sake of argument.”

Conditional proof

If a proof contains a conditional statement as a premise, conclusion, or as a proven statement based on the premises, then we can add an additional premise afterward—the first part of the conditional. For example, “A” is the first part of the conditional “A → B.” Let’s assume “A” is added as an “assumption for conditional proof.” In that case the letters “ACP” are put on the right-hand side of the line it’s added on. This line can be added to the proof at any time, but it is often added right after the other premises are stated. The assumed premise is then used to derive a conditional statement.

Then once a conditional is derived using the assumed premise, we have a conditional proof and the final line of the proof has “CP” on the right-hand side. All lines using the assumption are also cited. However, the conditional proof is not necessarily the entirety of the proof—more might have to be proven to deduce the conclusion of the original argument that we want to prove to be valid.

The lines of conditional proofs that are used to derive the conditional conclusion can’t be used by any other part of the proof. These lines require a tentative assumption and are not validly deduced by the premises of the original argument. The assumption itself is not validly deduced from the original argument, and the rest of the proof requires that assumption, other than the conclusion of the conditional proof.

Example

Consider the following argument:

1. If all humans are mortal, then no humans are gods and no humans are angels.

2. If no humans are gods, then no human is omnipotent and no human is omniscient.

3. Therefore, either all humans are mortal or it’s not the case that no human is omnipotent.

This argument has the following form in propositional logic:

1. A → (B ∧ C)

2. B → (D ∧ E)

3. ∴¬A ∨ D

A: All humans are mortal.

B: No humans are gods.

C: No humans are angels.

D: No human is omnipotent.

E: No human is omniscient.

The following is the proof that this argument is valid (using conditional proofs):

Lines 3, 4, and 5 contain a single conditional proof. The assumption for conditional proof (A) is allowed because line 1 contains a conditional, and the first part of the conditional is “A.”

Lines 7, 8, and 9 contain another conditional proof. The assumption for conditional proof (B) is allowed because line 2 contains a conditional, and the first part of the conditional is “B.”

Because lines 3-5 and 7-9 contain conditional proofs, those lines are not actually validly deduced from the original premises. If they were, line 5 would have already proven “B,” so the second conditional proof would have been unnecessary. These lines are merely tentatively assumed to be validly deduced for the sake of argument.

The conclusions of each conditional proof are proven, and they are used with a hypothetical syllogism to prove “A → D.” We know “A → B; B → D; ∴ A → D” is a valid argument because it has the same form as the hypothetical syllogism (p → q; q → r; ∴p → r”).

Finally, keep in mind that the conclusion of our argument can also justify the use of an assumption for a conditional proof as long as it’s a conditional statement. For example, if an argument concludes “A → (B ∧ C),” then we can have “A” as an assumption for a conditional proof.

Indirect proof

Indirect proofs are also known as a “reductio ad absurdum” (i.e. “reduction to the absurd”). Indirect proofs can be used to prove any argument is valid. Indirect proofs have three additional steps:

1. There’s an additional premise—a statement that is tentatively assumed to be true. This is the “assumption for indirect proof” and “AIP” is written on the right-hand side of the line of the assumption. This statement is one we will actually hope to prove to be false.

2. The assumption is used to derive a contradiction (p ∧ ¬p). The contradiction must appear on a line, and it is often explicitly derived on a single line using the rule of conjunction.

3. Once the contradiction is derived, the negation of the assumption is proven. “IP” is written on the right-hand side of that line along with the numbers of all lines that use the assumed premise.

Once again, the lines of an indirect proof require tentative assumptions, and they are not proven to be validly deduced from the original premises. They can’t be used by other parts of the proof for that reason. Only the final line of an indirect proof is actually proven (and it is validly deduced from the original premises).

Note that indirect proofs often assume the negation of the conclusion. Once it is proven that assuming the negation of the conclusion leads to a contradiction, the conclusion is actually proven to be true. For example, we can assume that the conclusion of modus ponens is false. In that case we assume “p → q,” “p” and “¬q” to be true. But “p → q” and “¬q” proves “¬p” to be true via modus tollens. We now know that the argument is valid because assuming the conclusion is false leads to a contradiction (“p ∧ ¬p”).

Example

Consider the following argument:

1. Either killing people is sometimes wrong or always wrong.

2. If the killing people is always wrong, then killing people when necessary for self-defense is wrong, and killing people when necessary for self-defense is not wrong.

3. If killing people is sometimes wrong, then not all homicide is murder.

4. Therefore, sometimes homicide is not murder.

The logical form of this argument is the following:

1. P ∨ Q

2. P → (R ∧ ¬R)

3. Q → S

4. ∴S

P: Killing people is sometimes wrong.

Q: Killing people is always wrong.

R: Killing people when necessary for self-defense is wrong.

S: Sometimes homicide is not murder.

A proof that this argument is valid using an indirect proof is the following:

The indirect proof occurs on lines 5, 6, and 7. It was necessary to show that “Q” is true in order to use line 3 (Q → S) with modus ponens to reach the conclusion (S).

Note that the lines of the indirect proof are not actually taken to be validly deduced in the long run. They are only tentatively assumed to be validly deduced. The assumption is actually the negation of what is proven.

Update: I said something about how lines of a proof are “proven true,” but they are only proven true if the premises are true. This point was clarified above. Other minor corrections were made.

This is part 5 in a series. There are links to the other parts of the series above.

Natural deduction is used to give proofs of validity by showing all the steps in reasoning required. In this case natural deduction uses rules of inference to allow us to reach conclusions from statements of propositional logic.

Rules of inference

Rules of inference include rules of implication (valid argument forms) and rules of replacement (statements with equivalent logical form).

The rules are the following:

Rules of implication

Each of these rules states a valid argument form. For example, modus ponens states that the following argument form is valid:

1. p → q

2. p

3. ∴q

“∴” merely states that the final statement is a conclusion. Whenever we know “p → q” and “p” are true, we can conclude “q.” Each of these lower-case letters can stand for any statement, no matter how complex. Each letter must stand for the same statement when concerning the same argument. For example, “p” can stand for “all humans are mammals” and “q” can stand for “all humans are living organisms” for an entire argument. In that case we can develop the following valid argument in the English language using modus ponens:

1. If all humans are mammals, then all humans are living organisms.
2. All humans are mammals.
3. Therefore, all humans are living organisms.

Rules of replacement

Each of the rules of replacement state that certain statements are logically equivalent. You can validly conclude either part of a rule of replacement from the other part—the first part of the “::” symbol can be concluded from the second part, and vise versa.

For example, double negation states that “p” and “¬¬p” are logically equivalent. If you know “p,” then you can conclude “¬¬p” and vice versa. “p” can stand for “all humans are mammals.” In that case we can conclude that “it’s not the case that ‘all humans are mammals’ is false.”

Some rules of replacement actually have more than one type of equivalence. For example, DeMorgan’s rule has two different ways it can be used.

How to construct a proof

Proving validity using natural deduction requires the following steps:

1. Find the logical form of an argument.

2. Write the logical form of the premises.

3. Write the conclusion on the same line as the last premise after the “/” symbol.

4. Use those premises and the rules of inference to reach the logical form of the argument’s conclusion.

I will present two examples of proofs.

Example 1

Consider the following argument:

All humans are lizards. If all humans are lizards, then all humans are reptiles. If all humans are reptiles, then they are cold-blooded. Therefore, all humans are cold-blooded.

The logical form of this argument is the following:

1. P
2. P → Q
3. Q → R
4. ∴R

Each letter is capitalized and represents a specific statement in English:

P: All humans are lizards.

Q: All humans are reptiles.

R: All humans are cold-blooded.

The proof that the argument is valid looks like the following:

The premises are written on lines 1, 2, and 3. The conclusion is written after the final premise on line 3.

Line 4 concludes “Q” from lines 1 and 2 using modus ponens. “1, 2, MP” is written on the right-hand side to make that clear. “P; P → Q; ∴Q” is valid because it uses modus ponens. (It has the same form as “p; p → q; ∴q.”)

Line 5 concludes “R” from lines 3 and 4 using modus ponens. “3, 4, MP” is written on the right-hand side to make that clear. “Q → R; Q. ∴R” is valid because it uses modus ponens. (It has the same form as “p; p → q; ∴q.”)

Line 5 has the conclusion of the original argument, so we have proven that the argument is valid. We can derive the conclusion from the premises and rules of inference.

Example 2

Consider the following argument:

All humans are mammals, and if all dogs are warm-blooded and have thoughts, then it is not the case that all dogs are reptiles or insects. If all humans are cold-blooded, then it is not the case that all humans are mammals. If it is not the case that all humans are cold-blooded, then no humans are lizards. Therefore, no humans are lizards.

The logical form of this argument is the following:

1. A ∧ [(B ∧ C) → ¬(D ∨ E)]
2. F → ¬A
3. ¬F → G
4. ∴G

the letters stand for the following:

A: All humans are mammals.

B: All dogs are warm-blooded.

C: All dogs have thoughts.

D: All dogs are reptiles.

E: All dogs are insects.

F: All humans are cold-blooded.

G: No humans are lizards.

The proof that this argument is valid is the following:

Lines 1, 2, and 3 contain the premises. The conclusion is also written on line 3.

Line 4 concludes “A” by simplification using line 1. “A ∧ [(B ∧ C) → ¬(D ∨ E)]; ∴A” is a valid argument because of simplification. “A ∧ [(B ∧ C) → ¬(D ∨ E)]; ∴A” has the same form as “p ∧ q; ∴p.”

Line 5 concludes “¬¬A” by using line 4 and double negation. “A; ∴¬¬A” is a valid argument because they are equivalent. “A :: ¬¬A” has the same form as “p :: ¬¬p.”

Line 6 concludes “¬F” by using lines 2 and 5, and modus tollens. “F → ¬A; ¬¬A. ∴¬F” is a valid argument because of modus tollens. “F → ¬A; ¬¬A; ∴¬F” has the same form as “p → q; ¬q.; ∴¬p.”

Line 7 concludes “G” by using lines 3 and 6, and modus ponens. “¬F → G; ¬F; ∴G” is a valid argument because of modus ponens. “¬F → G; ¬F; ∴G” has the same form as “p → q; p; ∴q.”

“G” is the conclusion of the argument, so the argument is valid. We were able to deduce the conclusion from the premises and rules of inference.

Arguments are reasons given to believe something. For example—“All men are mortal. Socrates is a man. Therefore, Socrates is mortal.” In this case the first two statements of the argument are a reason to believe that Socrates is mortal (the conclusion). If we know that “all men are mortal” and that “Socrates is a man,” then we can also know that “Socrates is mortal.”

Is it rude to present the above argument about why we should believe that Socrates is mortal? Doesn’t seem like it. Is it part of a shouting match or hostile disagreement? That seems unlikely. In this case the argument could be considered to be “rational persuasion.” Being capable of giving good arguments and seeing flaws in poor arguments is central to thinking rationally and being capable of critical thinking.

Even so, arguments can be disrespectful and manipulative. Imagine that someone argues, “Republicans just want to keep assault weapons legal because they want to use them to murder people.” That is insulting to Republicans and it fails to account for the best arguments given by Republicans to keep assault weapons legal. We could imagine this manipulative argument to be used during a shouting match or hostile disagreement. Manipulative arguments are used during so-called political debates and they are used as propaganda for a television commercials. Manipulative arguments should not be considered to be rational persuasion.

Arguments should generally be intended to be rational persuasion. There’s nothing disrespectful about trying to persuade people to believe something based on good reasoning. In fact, science, logic, and philosophy would all be impossible without rational persuasion. The only reason that science can tell us anything about the world is because scientists can tell us how the evidence available is a good reason to believe something is true about the world (or refutes certain beliefs about the world).

It is true that any argument—including good arguments—can make people feel bad. A lot of people feel like they need to win every debate and that being proven wrong makes them look bad. They hate being proven wrong. They could get angry at anyone who disagrees with them and anyone who gives rational arguments with the “wrong conclusions.” Good arguments can motivate a great deal of anger and hostility, and they can be part of a “shouting match” or “hostile disagreement” when someone doesn’t take kindly to them. Insults are often thrown around by those who don’t like others who try to prove them wrong.

However, we should be ashamed of ourselves to reacting badly to good arguments (or even an attempt to give good arguments). It is perfectly respectful to expect people to want to hear good arguments—to want to know what we should believe and why we should believe it. Human beings are capable of rationality and generally do care about what we should believe. To assume that a person doesn’t want to know what we should is disrespectful. Such a person would be seen as “irrational.”

Moreover, psychologists have proven that people are very biased. Even scientists. Perhaps the worst bias we all suffer from is the “confirmation bias”—we too quickly accept evidence that supports our beliefs and we don’t take counter-evidence as seriously as we should. One reason that science is so successful is because of peer review—scientists who give poor arguments and have biased experiments will likely be refuted by others at some point. Science is self-correcting in this way. But it’s not just scientists that can benefit from peer review. We are all likely to justify our beliefs inadequately and reason poorly now and then. But if we tell other people about our reasoning process by giving arguments for our beliefs, then they are likely to be able to find flaws in our reasoning. We are much better at finding flaws in the reasoning of others than in the reasoning of ourselves.

In conclusion, arguments can be disrespectful and they can result in a hostile disagreement, but we should try to give rationally persuasive arguments anyway. It is perfectly respectful to assume that people in general want to know what we should believe, and giving arguments to others is a good way for them to help find flaws in our reasoning.

Related

• Corruption of Philosophical Language
• The Difference Between Sophistry & Philosophy
• The Marginalization of Philosophy
• Why A Teen Who Talks Back May Have A Bright Future

You can follow Ethical Realism on Facebook or Twitter.

 What is the default position? What I mean by a default position in this context is that there’s a better reason to think something is true than the alternative when considered in isolation of our knowledge of the world. To think that atheism is the default position would then mean a person who has no knowledge of the world would have more reason to think gods don’t exist than to think at least one god exists. We would then say that theists have the burden of proof in a debate to give people at least some reason to think that at least one god really does exist—otherwise everyone will continue to have more reason to think gods don’t exist than to think they do.

What I call “atheism” in this context is literally the view that gods don’t exist. Many atheists say they are in some sense undecided and have no opinion about whether gods really exist or not. That is not what I am talking about.

For any given belief, it is often reasonable to think that the default position is simply being undecided—to be skeptical that particular belief is true. We would then say that both sides of the debate have the burden of proof. Both sides should give arguments.

In everyday life the default position is not necessarily relevant to us because we already know a lot about the world and we might already be familiar with various arguments. A theist could very well have a good reason to believe in God and an atheist could very well have a good reason to believe no gods exist. In that case the default position will do little to change their mind. We would have to debunk the arguments they have to give them a good reason to change their mind.

A reason to think theism is the default position

The reason to think something is the default position is based on mathematical or logical principle of probability rather than because of what we know about the world. The assumption is that we can at least imagine that we know nothing about the world while still giving arguments involving the probability of something being true. Why think theism is the default position? Because there’s an unlimited number of possible gods and one could argue that to claim that none of them exist is a stronger position than saying that at least one of them exists.

Consider some gods that might exist: Yahweh, Zeus, Thor, Quetzalcoatl, and the Force.

Each god has at least somewhat different characteristics. Some of them are much different. Yahweh is an all-powerful person, gods like Zeus are much more powerful than humans are, and the Force is not a person at all with no thoughts or intentions.

We will represent each god with a letter (A) for Yahweh, (B) for Zeus, (C) for Thor, (D) Quetzalcoatl, and (E) the Force. “Not-A” will translate to mean “Yahweh does not exist” and so on.

We could then say that theism only requires that at least one of these gods exist:

(1) A or B or C or D or E…

Atheism requires that none of them exist:

(2) Not-A, not-B, not-C, not-D, and not-E…

The odds of (1) seems higher than (2), so theism appears to be the default position.

There’s a rule of probability that states that “A or B” is more likely true than “A.” For example, the odds of an alien species from another planet being a silicon-based life form or a carbon-based life form is higher than the odds that the alien will be a silicon-based life form.

Also, assuming that we know nothing about the odds of “A or B” being true, it seems plausible to assume that “A or B” is more likely true than “not-A.” For example, it is plausible that the odds that an alien species from another planet being a silicon-based life form or a carbon-based life form is higher than the odds that the alien isn’t a silicon-based life form.

Also consider an undiscovered species of mammal and an undiscovered species of fish. It seems plausible to say that it’s more probable that there is either an undiscovered species of mammal or an undiscovered species of fish than the odds that there is no undiscovered species of mammal.

An objection

I believe that this simplistic formulation of the alternatives misrepresents the actual alternatives and the odds of them being true. The problem is that saying a god exists is actually several assertions. It’s not clear what exactly counts as a “god” because they are all so different, but a plausible list of godlike characteristics are the following:

1. It has thoughts.

2. It is much smarter than human beings.

3. It is much more powerful than human beings.

4. It is immortal or eternal.

5. It is a person.

6. It is always or often without a physical body.

7. It exists everywhere.

8. It exists outside of space and time.

9. It can interact with the world.

10. It is often or always invisible.

I don’t think all people will agree that god(s) have all ten of these characteristics, but there is a “family resemblance.” To count as a god, perhaps at least five of these characteristics have to be involved. Otherwise we might just be dealing with a powerful alien, a ghost, or an energy-based life form.

Many seem to think that Yahweh has all ten characteristics. He has thoughts, is much smarter than human beings, is much more powerful than human beings, is eternal, is a person, is often without a physical body, exists everywhere, exists outside space and time, can interact with the world, and is often invisible.

Zeus has five of the characteristics. He has thoughts, he is more powerful than human beings, he is immortal, he is a person, and he can interact with the world.

The Force has five of the characteristics as well. The Force is immortal, is always without a physical body, exists everywhere, can interact with the world, and it’s always invisible.

Rather than claiming that theists are saying “A or B or C or D or E” it would be more accurate to say that they are saying something more like the following:

(A and B and C and D and F) or (A and B and C and F and G) or (A and B and F and G and H) or (A and F and G and H and I) or (A and F and G and H and J)

Because the theistic position was misrepresented, it is no longer clear that it is more probable than atheism prior to empirical investigation. The odds of each god existing is much less probable once we take them to require multiple assertions to be true. The odds of “A” being true is higher than the odds of “A and B and C and D and E” being true.

The fact that we can formulate what initially looks like a single assertion as multiple assertions isn’t enough to say that it should be done. However, I believe it should be done in this case because we need to know that everything we consider to be a god really is a god. Only by realizing that the concept of gods is ambiguous because it’s based on a family resemblance can we determine what atheists need to reject. Some things that have been said to be gods, such as the Sun, Earth or Universe, should be excluded from the list of gods that atheists must reject (unless such things are also said to have certain godlike characteristics from this list).

Moreover, it’s not entirely clear how we should define gods for the purpose of the debate over their existence. I believe it can be fair for an atheist to assume that all gods must have at least certain godlike characteristics to be considered relevant to the debate. Various non-god supernatural beings, such as ghosts, energy-based life forms, angels, and demons should be excluded. In that case being an atheist could be compatible with belief in the Sun, the Earth, the Universe, ghosts, energy-based life forms, angels, and demons.

A reason to think atheism the default position

Let’s say that we define “god” in a non-arbitrary way. For example, the following five characteristics are plausibly characteristics that every god has in order to count as a god:

1. It is immortal or eternal.

2. It is always or often without a physical body.

3. It can interact with the world.

4. It is often or always invisible.

5. It is not a ghost.

In that case we can characterize theism as stating that something has all five of these characteristics:

(A and B and C and D and E)

The atheist will then deny that something has all four of the characteristics:

Not-(A and B and C and D and E)

In this case atheism seems more probable than theism when taken apart from our knowledge of the world. In isolation, the odds of five assertions being true is lower than that at least one of those assertions is false. In fact, in that context the odds of two assertions being true is lower than at least one of them being false. At least prior to empirical investigation, (A and B) is less likely true than not-(A and B). For example, the odds of a woman being both a lawyer and a feminist is less likely true than a woman not being both a lawyer and a feminist.

Atheists often say that they don’t believe in gods just like theists don’t believe in faeries. It seems plausible to at least say that the default position is that faeries don’t exist because it too requires multiple assertions (i.e. there is a person that is (1) very small and (2) can fly.) It seems rational for theists and atheists to assume faeries don’t exist (until good enough arguments are presented). It also seems unreasonable to demand that a person who rejects the existence of faeries prove that faeries don’t exist. There are hundreds of fictional types of creatures and it would be silly to think everyone has a duty to disprove that they all exist.

Some potential objections

Not everyone will be convinced by my argument, and I will discuss six objections that can be raised.

1. How do we know the correct way to formulate these beliefs? I have suggested that the correct way to formulate the assertion that a god exists is as actually having multiple assertions. The mere fact that we can give a god a single name and say simply “Thor exists” does not mean that it really should only count as a single assertion to determine the odds of it being true. As was stated earlier, it’s important that we know the characteristics that we consider gods to have and how we can differentiate gods from other things. The characteristics that the argument states gods have is meant to help us understand why we should think of a being as a god and to differentiate it from non-god things.

2. Aren’t there many types of gods? Doesn’t the atheist have to deny that any of them exist? The argument that atheism is the default position only argues against one conception of god, but that conception is compatible with several different kinds of gods. Since the conception of god used by the argument is compatible with so many types of gods, it literally does tell us why we should reject several different kinds of gods. The gods it does not tell us to reject are assumed not to be gods at all. Perhaps the argument did not quite define gods properly by giving the correct list of characteristics, but the argument can easily be changed by defining gods by using a set of different characteristics instead.

3. Doesn’t the fact that the atheist must reject multiple types of gods increase the odds that theism is true? The atheist rejects everything that has every necessary godlike characteristic, but there’s perhaps infinite different gods must be rejected. Doesn’t that mean that the existence of at least one of the gods is more probable? I think not. The odds of a women existing is not more likely just because a women can exist and have long hair, and a woman can exist and have short hair, etc. Adding more characteristics to a list does not increase the probability of the entire list of characteristics actually describing something, even though there are many different ways we can add more characteristics to a list.

4. Wouldn’t the default position be that human beings and rocks don’t exist? At this point it is not clear that faeries or gods need to be singled out. I think we should agree that humans, dogs, rocks, and chairs are all unlikely to exist when taken in isolation of our knowledge of the world. The default is then that none of these things exist. Even so, we certainly can give good arguments that these things exist. The belief that humans, dogs, rocks, and chairs exist is highly justified when we consider our knowledge of the world. Additionally, no one will debate over whether humans, dogs, rocks, and chairs exist insofar as everyone already agrees that they do. When it’s time to actually debate shared assumptions are not required to be proven, at least for practical reasons.

5. What if the default position is that nothing exists? Everything that can exist has more than one characteristic, so the probability I discuss would indicate that the default position is that they don’t exist. This could be taken to be a reason to think that we shouldn’t believe anything about the world unless it’s proven, but in that case perhaps we can’t believe anything counts as evidence for a belief about the world at all. We couldn’t accept that anyone has experiences or observations that could be used as a reason to believe anything about the world. My response would be that what is discussed as a default position here is incomplete and certain assumptions will be required for any debate to exist. In fact, during actual debates anything both sides agree with will not need to be proven. Perhaps there is something “nonrational” about having assumptions, but it would be impractical to demand that no one ever has assumptions. It seems important that we can hypothesize about the world and test our hypotheses while tentatively assuming the hypothesis is true.

Two objections to default positions

Perhaps the strongest objection to the arguments for default positions is against default positions as I have defined them in general. Both of the arguments for default positions that I presented require us to accept that we can assess the odds of certain beliefs being true prior to any knowledge of the world, but perhaps arguments given in that situation about such probabilities are not meaningful after all. Consider the following two objections to default positions:

1. What good does a default position do us when we never argue in a bubble where we know nothing about the world? Perhaps the default position does indicate that those who argue against the default position has more hurdles to face in order to give the skeptic good reasons to think they don’t exist. Characteristics that are non-controversial (such as having a mind) should be accepted by atheists, but certain characteristics are controversial (such as having a mind without a body). It’s those characteristics that will actually be relevant to the debate over theism.

2. How can we discuss probabilities in isolation of our knowledge of the world? The probabilities I discuss are not set in stone. Sometimes “A and B” has the same probability as “B.” For example, when A has a 0% chance of being true. We need to know how we can actually compare the probabilities of various things in order to take them seriously. This is probably the strongest objection to the arguments, but I think probabilities can count for something if we can find a way to agree about what is likely true about probabilities. Perhaps the atheist should give the theist the benefit of the doubt and say that each controversial characteristic of a god can be initially assumed to be 99% probable until the relevant data is discussed. Giving the theist the benefit of the doubt would be a good reason to reject some of the probabilities that I gave, so the probabilities and default position would have to be re-evaluated.

Conclusion

I have presented an argument that theism is the default position, and I explained why I don’t think it is a good argument. I also presented an argument that atheism is the default position and considered five objections to it, but it is unclear that any of these objections are serious. However, there are also two objections I discussed to any probabilistic argument involving the default position, and I believe that the second of those objections is serious. I can’t think of a good way to respond to that objection, but perhaps arguments involving the default position can be better developed in the future.

If a good argument can be presented that atheism is the default position, then atheism could be said to be more probable than theism when considered in isolation from our knowledge of the world. This could be taken to be an a priori argument for atheism—an argument that does not require us to know anything about the world. However, this argument would merely be a reason to prefer atheism based on probability when taken in isolation of what we know about the world, so it would not be the final word. It would not absolutely prove that gods don’t exist. The implication would be that atheism can be rational until theism is proven to be more likely true.

The default position has little to no bearing once arguments are given. If there is a good enough argument to believe in gods, then atheists should change their mind. Until then we should expect atheists and theists to continue to debate. There can be arguments for and against the existence of gods, and those not persuaded by those arguments will provide objections to the arguments given by the opposing side.

Update (1/30/2013): I added three new objections to the argument that atheism is the default position. Other parts of the essay were changed to reflect my increasing doubt based on those objections.

Update (1/31/2013): I added more information about the first objection I raised against the argument that theism is the default position, I improved the discussion of the first two objections to the argument that atheism is the default position, I added the third discussed objection against the argument that atheism is the default position, and I made it clear that there are two objections to arguments about default positions in general.

Related

1. Do Default Positions Exist?
2. What is the Burden of Proof?
3. An Argument Against God, a Teapot, and Garvey’s Objection (Part 1)

You can follow Ethical Realism on Facebook or Twitter.

Based on my experiences, I also believe that people tend to be much worse at rational debate in general than they realize. They not only give very bad arguments on occasion without realizing it, but it is often unclear how they think their arguments have any bearing on a debate whatsoever. This is likely because they don’t quite understand how arguments and debates function. I will explain the very minimal requirements of rational debate here.

Rational debate has little to do with debate teams or political debates as seen on TV. Debate is a discussion about what we should believe between two or more people who disagree on some subject. One side in a rational debate gives arguments for some belief and defends that belief from objections. The other side of the rational debate gives arguments for why we should reject the belief and defends their position from objections. Rational debate requires that we do not try to trick or manipulate our opponents. Instead, we must be sincere about what we should believe and why.

A simple diagram of debate looks like the following:

We can consider the initial argument, objection, and defense to all be the necessary steps of both sides of a debate. Each side should have its own initial argument and their arguments should be defended from objections. However, we should keep in mind that there can actually be several initial arguments for a position, several objections to each argument, and several ways to defend an argument from objections. Debates can become very lengthy and complicated for that reason. Some philosophical debates have lasted for thousands of years between several different authors.

I will discuss each major step of debate—the initial arguments, objections, and defenses. Examples will be given of each.

Initial arguments

People are motivated to participate in a rational debate when they both disagree about what we should believe. For that reason both sides of the debate present an argument for what they think we should believe.

Let’s consider a potential topic for rational debate. Wendy might think we should use the death penalty and Casper might think we shouldn’t. In that case Wendy would argue that the death penalty should be used and defend her view from objections; and Casper would argue that the death penalty shouldn’t be used and defend his view from objections.

The conclusion of Wendy’s argument is “we should use the death penalty,” and Casper’s conclusion is a rejection of that belief—that we shouldn’t use it.

The initial arguments are the reasons given for us to believe the conclusions. For example, each side could argue the following:

Wendy’s initial argument

1. Evil people deserve to die.
2. If evil people deserve to die, then we should use the death penalty.
3. Therefore, the death penalty should be used.

Casper’s initial argument

1. We shouldn’t kill people unless it’s necessary to save lives.
2. Therefore, the death penalty shouldn’t be used.

Objections

After the initial arguments are presented, we have an argument to think something is true and an argument to think it’s false. At this point in the debate we can’t be sure that either of these arguments is successful. If both arguments are perfectly good, then we will have a good reason to think the death penalty should be used, but we have another good reason to think it shouldn’t be used. We will then have a good reason to think that two contradictory statements are both true, but we know that two contradictory statements can’t be true.

That’s where objections come in. These are arguments given against the arguments given by the opposing side. In this context objections are not against the conclusions of the opposing side—they are only against the arguments. In particular, an objection tells us that an argument given by the opposing side is not a good argument. There are two main reasons to think that an argument is not a good argument. One, a premise could be unjustified. Two, the premises might not properly support the conclusion.

An example of each type of objection is the following:

Casper’s objection against Wendy’s initial argument

1. All evil people can be reformed.
2. If all evil people can be reformed, then they don’t deserve to die.
3. Therefore, evil people don’t deserve to die.

This objection gives us a reason to think Wendy’s first premise should be rejected. That premise is needed for her argument to give us a reason to agree with her conclusion. If we reject the premise, then Wendy’s argument will not give us a good reason to think that we should use the death penalty.

Wendy’s objection to Casper’s argument

1. Perhaps the death penalty is necessary to save lives.
2. So, even if “we shouldn’t kill people unless it’s necessary to save lives” is true, perhaps we should use the death penalty anyway.
3. Therefore, Casper’s argument fails to give us a good reason to believe that the death penalty shouldn’t be used.

This time no premise in particular is objected to, but it is explained that Casper’s argument isn’t a good reason to think the death penalty shouldn’t be used—even if his premise is true.

Defenses

A defense is a reason to think an objection isn’t a good argument. A defense could be said to be an objection to an objection. During the debate Wendy will be required to tell us if she thinks her argument can be rationally defended from Casper’s objection or if she retracts her initial argument, and Casper will be required to do the same.

An example of defenses include the following:

Wendy’s defense of her initial argument

1. Casper doesn’t give us a good reason to believe that evil people can be reformed.
2. If Casper doesn’t give us a good reason to believe that evil people can be reformed, then Casper doesn’t give us a good reason to believe that evil people don’t deserve to die.
3. Therefore, Casper doesn’t give us a good reason to believe that evil people don’t deserve to die.

This time Wendy’s argument is a bit weaker because it’s merely stated that one of Casper’s premises is not justified. This seems like an appropriate response to an argument in a debate that is particularly controversial. It could also be mentioned that the premise is controversial and our best science does not support it. We can prove pretty much anything in a debate if we are allowed to require everyone to agree with our controversial premises, so it is necessary to try to find premises that we believe our opponents will agree with.

However, we should also note that it is important for both sides of a debate to try to find premises that everyone can agree with. If no premise is ever accepted during a debate, then the debate will never end. A person can be asked to justify premises with additional arguments over and over again forever. It would be unfair to ask one side to argue endlessly in this way.

Casper’s defense of his initial argument

1. Incarceration is just as effective as the death penalty at saving lives.
2. If incarceration is just as effective as the death penalty at saving lives, then the death penalty is not necessary to save lives.
3. If the death penalty is not necessary to save lives, then we should reject that “perhaps the death penalty is necessary to save lives.”
4. Therefore, we should reject that “perhaps the death penalty is necessary to save lives.”

It seems fair to assume that Casper’s initial argument contained an unstated premise—that the death penalty isn’t necessary to save lives. People don’t always state all of the premises of their arguments, and that premise can be justified in order to defend his argument from Wendy’s objection. In that case one premise of Wendy’s objection in particular can be refuted.

Conclusion

Although I have discussed the basic requirements of rational debate, there is a lot more that could be said. Even so, this is a good starting place. If we want to know what we should believe, we should know arguments both for and against the belief when possible. We also need objections and defensive arguments. We need to know if we have a good reason to reject an argument, and we need to know why exactly it’s a good reason to reject the argument.

Related:

• Five Tips for Better Debates

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Truth tables are an important tool for evaluating statements and arguments. We can create our own truth tables using following steps:

1. Translate statements of ordinary language.

2. Break all complex statements into smaller parts.

3. Determine how many columns are required.

4. Determine how many rows are required.

5. Determine the truth values of statement letters.

6. Determine the truth values of complex statements.

I will illustrate how to follow these steps by using an example. In particular, I will show how we can make a truth table of an argument to find out if the argument is logically valid.

Step 1: Translate statements of ordinary language.

We can make a truth table of one or more statements. Arguments can be presented on a truth table by presenting all the statements from the argument. We will make a truth table of an argument, but that requires that we first translate an argument of ordinary language into propositional logic.

Consider the following argument written in English:

If Mary will either go to the store or stay at home, then Mark will go to the store and buy milk. Mary will not go to the store or stay at home. Therefore, it’s not the case that Mark will go to the store and buy milk.

We can identify the premises and conclusion:

1. Premise 1: If Mary will either go to the store or stay at home, then Mark will go to the store.

2. Premise 2: Mary will not go to the store or stay at home.

3. Conclusion: It’s not the case that Mark will go to the store.

We can then replace each statement with a propositional variable and translate these statements into propositional logic:

(A ∨ B) → C

¬(A ∨ B)

¬C

A: Mary will go to the store.

B: Mary will stay at home.

C: Mark will go to the store.

Step 2: Break all complex statements into smaller parts.

We can break statements into smaller parts by removing logical connectives. Each logical connective must be removed one at a time until only statement letters are left. Whenever a logical connective is removed, it should be the one with the least amount of parentheses around it.

First statement

Look at the first propositional statement of the argument: (A ∨ B) → C

First remove the conditional (→) to create the following simpler statement and statement letter: A ∨ B, C

We still need to break “A ∨ B” into smaller parts. Next remove the disjunction (∨) to get the following two statement letters: A, B

Second statement

Look at the second propositional statement of the argument: ¬(A ∨ B)

First we need to remove the negation (¬) to get the following simpler statement: A ∨ B

We still need to break “A ∨ B” into smaller parts. We need to remove the disjunction to get the following statement letters: A, B

Third statement

Look at the final statement of the argument: ¬C

We need to remove the negation to get the following statement letter: C

Overview

A list of all statements including the smaller parts include the following:

1. (A ∨ B) → C

2. C

3. A ∨ B

4. A

5. B

6. ¬(A ∨ B)

7. ¬C

Step 3: Determine how many columns are required.

Columns are the vertical areas of the truth table. We need one column for each statement of the argument and the smaller parts of those statements. If you look at the list of all the statements and smaller parts, you will see that there are seven statements that the table will need.

The top row of the truth table will contain these statements and look like the following:

Step 4: Determine how many rows are required.

Rows are the horizontal areas of the truth table. A number of rows is required in order to determine every possible truth value, which depends on the number of statement letters used:

One statement letter: 2

Two statement letters: 4

Three statement letters: 8

Four statement letters: 16

Five statement letters: 32

The argument uses three statement letters (A, B, C), so we need 8 rows of truth values.

The truth table will now look like the following:

Step 5: Determine the truth values of statement letters.

The last statement letter alternates from being true and false every other row starting with true (T, F, T, F…). The second to last statement letter alternates from being true and false every two rows (T, T, F, F, etc.) The third from last statement letter alternates from being true and false every four rows (T, T, T, T, F, F, F, F, etc.) The first statement of each table should be true in the first half of the rows and false on the bottom half.

We can now determine when A, B, and C are true or false:

Step 6: Determine the truth values of complex statements.

There are truth tables that determine when a statement with a logical connective is true or false. We can summarize these tables as saying the following:

• Conjunction (∧) – “p ∧ q” is only true when p is true and q is true.
• Negation (¬) – “¬p” has the opposite truth value of “p.”
• Disjunction (∨) – “p ∨ q” is only false when both p and q are false.
• Conditional (→) – “p → q” is only false when p is true and q is false.
• Equivalence (↔) – “p ↔ q” is only true when p and q have the same truth value.

The first statement with a connective that we need to determine the truth values for is “A ∨ B.” It will only be false when both A and B are false because it’s a disjunction:

The second statement we need the truth values for is “(A ∨ B) → C.” It will only be false when “ A ∨ B” is true and “C” is false because it’s a conditional:

The third statement we need to find truth values for is “¬(A ∨ B).” It will have the opposite truth values as “A ∨ B” because it’s a negation:

The fourth and final statement we need to find truth values for is “¬C.” It will have the opposite truth values as “C” because it’s a negation:

The reason we usually want to make a truth table of an argument is to find out if it’s logically valid. It is logically valid if it’s impossible for it to have true premises and a false conclusion at the same time. It is logically invalid if it is possible for it to have true premises and a false conclusion at the same time.

We can look at the truth table to find out if it’s valid. Look at all the rows where both premises are true and see if the conclusion is ever false on those rows. If so, it’s invalid. Otherwise it’s valid.

The premises are the following:

(A ∨ B) → C

¬(A ∨ B)

The conclusion is the following:

¬C

The premises are only true on the same row on the bottom two rows, and the conclusion is false on one of those rows. Therefore, the argument is logically invalid:

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Truth tables are visual aids to help us determine all the truth value possibilities of various statements. Learning about truth tables can help us better understand logic. Truth tables are used to define logical connectives, and to help us identify various distinctions (such as tautologies, self-contradictions, consistent statements, equivalent statements, and valid arguments).

Logical connectives

The five connectives used in propositional logic are the following: “and” (∧), “not” (¬), “or” (∨), “implies” (→), and “if and only if” (↔). Each of the logical connectives has a precise definition, which is provided by a truth table:

Conjunction

Lower case letters represent “predicate constants.” These lower case letters stand for any possible statement, such as “rocks exist” or “if rocks exist, then bananas are pink.”

The first row contains various statements (“p,” “q,” and “p ∧ q”). “p ∧ q” roughly translates to mean “both p and q.” For example, “p” can mean “rocks exist” and “q” can mean “bananas exist.” In that case “p ∧ q” means “rocks and bananas exist.”

The following image tells us how to read the truth table:

There is a column (vertical area) under each statement, which contains every possible truth value. The column under “p” has “T, T, F, F” (true, true, false, false). The column under “q” is “T, F, T, F” (true, false, true, false). The column under “p ∧ q” contains “T, F, F, F” (true, false, false, false).

Every row (horizontal area) beneath the statements contains every combination of truth values. The first row of truth values states that “p,” “q,” and “p ∧ q” are all true. The second row states that “p” is true, “q” is false, and “p ∧ q” is false. The third states that “p” is false, “q” is true, and “p ∧ q” is false. The fourth states that “p,” “q” and “p ∧ q” are all false.

We can replace “p” and “q” with statements of the English language to clarify how the truth table works. “p” could stand for “life used to exist on Mars” and “q” could stand for “life will exist on Mars in the future.” We don’t currently know if either of those statements are true, but we can talk about all the possibilities.

Each row states the following possibilities:

• Row 1: It’s true that “life used to exist on Mars.” It’s true that “life will exist on Mars in the future.” In that case it’s also true that “Life used to exist on Mars, and that life will exist on Mars in the future.”

• Row 2: It’s true that “life used to exist on Mars.” It’s false that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”
• Row 3: It’s false that “life used to exist on Mars.” It’s true that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”
• Row 4: It’s false that “life used to exist on Mars.” It’s false that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”

The truth table makes it clear that “p ∧ q” is only true when both “p” is true and “q” is true. For example, “humans are mammals and they are animals” is true because “humans are mammals” is true and “humans are animals” is true.

Consider what happens if one of these statements is false. “p” can mean “humans are reptiles” and “q” can mean “humans are animals.” In that case we will have the statement “humans are reptiles and they’re animals.” That statement is false because one part of the statement is false.

Negation

Roughly speaking, “p” is any possible statement and “¬p” means “it’s not the case that p.”

Each box on the top row contains a logical statement. (In this case “p” or “¬p.”) Each box below a statement tells us the possible truth values of that statement. “p” can be true or false, and “¬p” can be false or true.

Each row of boxes below the logical statements contains the possible combinations of truth values of the statements above. The first row down says “p” is true and “¬p” is false. Whenever “p” is true, “¬p” will be false. For example, “p” can stand for “rocks exist.” In that case the statement is true, and “¬p” is false because it stands for “it’s not the case that rocks exist.”

The final row says “p” is false and “¬p” is true. Whenever “p” is false, “¬p” will be true. For example, “p” could stand for “1+1=3,” which is false. In that case “¬p” is true because it means “it’s not the case that 1+1=3.”

Truth tables provide every possible combination of truth values that logical statements can have. The only two truth values needed here are true and false, so there are only two rows beneath the logical statements.

Disjunction

“p ∨ q” roughly translates as “either p or q.” For example, “p” can be “dogs are mammals” and “q” can be “dogs are reptiles.” In that case “p or q” will be “dogs are mammals or dogs are reptiles.

The truth table indicates that every “p ∨ q” statement is true unless both “p” and “q” are false, which is shown on the final row down. For example “p” can be “dogs are reptiles” and “q” can be “dogs are lizards.” In that case “p ∨ q” stands for “either dogs are reptiles or they’re lizards.” That statement is false.

Conditional

“p → q” roughly translates as “if p, then q.” For example, “if humans are mammals, then humans are animals.”

The truth table indicates that “p → q” is true unless “p” is true and “q” is false. “p” can be “the President of the USA is a human” and “p” can be “the President of the USA is a reptile.” In that case “p → q” will mean “if the President of the USA is a human, then the President of the USA is a reptile.” That statement is false.

We can also consider a true statement where “p” is false and “q” is false. For example, “p” can be “the President of the USA is a lizard” and “q” can be “the President of the USA is a reptile.” In that case “p → q” will be “if the President of the USA is a lizard, then the President of the USA is a reptile.” That statement is true.

Finally, let’s consider a conditional statement where “p” is false and “q” is true. “p” can stand for “the President of the USA is a lizard” and “q” can stand for “the President of the USA is an animal.” In that case the statement is “if the President of the USA is a lizard, then the President of the USA is an animal.” That statement is true.

Equivalence

“p ↔ q” roughly translates as “p if and only if q.” For example, “p” can stand for “1+2=3” and “q” can stand for “2+1=3.” In that case “p ↔ q” stands for “1+2=3 if and only if 2+1=3.”

The table above makes it clear that “p ↔ q” is only true when “p” and “q” have the same truth values. They must both be true or false. If not, the statement is false.

Consider when “p” stands for “dogs are animals” and “q” stands for “dogs are reptiles.” In that case “p ↔ q” stands for “dogs are animals if and only if dogs are reptiles.” That statement is false.

A complex truth table

We can use the above truth tables to create a more complex truth table. For example, “Socrates is a human and he’s not a vampire.” We can translate this into logical form as “A ∧ ¬B.” (We use capital letters because they stand for something specific.) In this case “A” stands for “Socrates is a human” and “B” stands for “Socrates is not a vampire.”

The truth table for this is the following:

We have “¬B” on the truth table because simpler statements must be resolved before we can find the truth values of more complicated statements. “¬B” is contained in “A ∧ ¬B.”

The truth value for “¬B” is the opposite of the truth vale for “B,” so we just write in the opposite values there.

The truth value for “A ∧ ¬B” will be true whenever both “A” and “¬B” are true. There’s only one place on the truth table where “A ∧ ¬B” is true.

Tautologies

A tautology is a statement that’s always true because of it’s logical form. For example, “there are life forms on other planets or there are no life forms on other planets.” That statement has the form “P ∨ ¬P.”

We can identify a tautology by looking at the truth table because all the possible truth values of a tautology are true. When looking at all the above truth tables, you will notice that none of the statements are tautologies because there’s always one possibility of each of the above statements to be false.

A truth table of a tautology:

“P ∨ ¬P” is true whenever at least on of those statements is true. That’s why it’s always true—Whenever P is false, ¬P is true and vise versa.

Self-contradictions

A self-contradiction is a statement that’s always false. For example, “there are life forms on other planets and there are no life forms on other planets.” That statement has the form “A ∧ ¬A.”

We can identify a self-contradiction on a truth table by seeing when a statement is always false. None of the above truth tables contain self-contradictions because none of those statements are always false.

A truth table of a self-contradiction:

“P ∧ ¬P” is false whenever either “P” or “¬P” is false. One of those simple statements is always false.

Consistent statements

Statements are logically consistent as long as they can all be true at the same time, and contradictory (or inconsistent) whenever they can’t be. “P” and “Q” are consistent because it’s possible they are both true, but “P” and “¬P” are inconsistent because it’s not possible that they’re both true.

Consider the following two statements:

¬P → Q

¬(P ∨ Q)

We can make a truth table for them:

“¬P” must be given truth values before we can find the truth values of “¬P → Q” because it is part of that more complex statement. We need to find the truth values for “P ∨ Q” before “¬(P ∨ Q)” because it’s also part of that more complex statement.

“¬P → Q” will only be false when “¬P” is true and “Q” is false. That only happens on the bottom row.

“¬(P ∨ Q)” will only be true when “P ∨ Q” is false. That only happens on the bottom row as well.

The truth table above shows that “¬P → Q” and “¬(P ∨ Q)” are contradictory statements because they’re never both true at the same time.

Equivalent statements

Equivalent statements always have the same truth values. For example, “all humans are mammals and all humans are animals” is logically equivalent to “all humans are animals and all humans are mammals.” “P ∧ Q” is logically equivalent to “Q ∧ P.”

Consider the following two statements:

¬(P ∧ Q)

¬P ∨ ¬Q

We can make a truth table for them:

This truth table shows why they are equivalent—they always have the same truth values. Whenever “¬(P ∧ Q)” is false “¬P ∨ ¬Q” is also false, and they are both always true at the same time.

Valid arguments

An argument is logically valid whenever it’s impossible for the premises to be true and the conclusion false at the same time. They are logically invalid whenever it is possible for the premises to be true and the conclusion to be false at the same time.

Argument 1

Consider the following argument:

1. Socrates is a human and Socrates isn’t a vampire.

2. If Socrates is a human and Socrates isn’t a vampire, then Socrates is mortal.

3. Therefore, Socrates is mortal.

We can translate this argument into the following logical statements:

A ∧ ¬B

(A ∧ ¬B) → C

C

Each letter stands for a specific statement:

A: Socrates is a human.

B: Socrates is a vampire.

C: Socrates is mortal.

The truth table for this argument is the following:

There’s only one spot on the table where both premises are true, and the conclusion is also true. Therefore, this argument is logically valid.

Argument 2

Consider the following argument:

1. Socrates is an animal.
2. If Socrates is a mammal, then Socrates is an animal.
3. Therefore, Socrates is a mammal.

We can translate this argument into logical form:

A

B → A

B

The truth table for this argument is the following:

“B → A” is only false when “B” is true and “A” is false. That’s only on the second row from the bottom.

This truth table proves that the argument is invalid because there’s a row of true premises and a false conclusion.

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1. What is Logic?

Find out a little about what you’d learn in a logic class. People use the word “logic” a lot, but they are rarely familiar with the concept of “logic” used by logicians. The term as logicians and philosophers understand it is quite important and has a lot to do with thinking in better ways. This seems to be my most popular post of all time, which surprised me quite a bit. I didn’t expect so many people to want to learn about logic. It gives me hope for the human race. (This was mentioned in Stone Links in the New York Times.)

2. Not All Good Arguments Are Logically Sound

I found out that telling people that not all good arguments are logically sound makes them angry. It doesn’t mean that logic isn’t important. You need to understand logic to even know why I’d say such a thing. I suspect this is my second most popular post of all time. (This was also mentioned in Stone Links.) I also wrote a sequel—Not All Good Arguments Are Logically Sound Part 2.

3. What Are Facts? Do Facts Exist?

I was surprised when someone told me that facts didn’t exist. Based on the concept of “facts” often used by philosophers, I thought it was obvious that facts do exist. This explains the philosophical concept and why I think they exist. This is also one of my most popular posts (perhaps in part because it was mentioned by The Browser).

4. What is Argument Mapping?

After finding out about evidence indicating that argument mapping classes are the most effective at teaching critical thinking, I decided to learn more about it. I then decided I wanted to help other people find out about what argument mapping has to offer. In particular, it can be very helpful at helping us make certain distinctions. You might also be interested in knowing the differences between Argument Maps vs Other Argument Diagrams.

5. 10 Myths About Logic

I discussed common false beliefs people have about logic. The fact that few people learn much about logic really does have a strong impact. If they took logic classes, these are some of the myths I hope would be debunked.

6. Nonrational Forms of Persuasion & Manipulation

The effects of poor reasoning, cognitive biases, manipulation, and propaganda are widespread. One of the reasons we need philosophy and logic is to help fight against these potentially harmful forces.

7. Critical Thinking Concepts Everyone Should Know About

A major project I gave myself this year was to define several philosophical concepts in a way I thought just about everyone could understand, and to organize the definitions into a dictionary. This post was written as I was making progress with that project, which is now available as the Comprehensible Philosophy Dictionary. The dictionary will be updated again soon, and more updates will probably be made in the future. I found out that there seems to be an endless number or terms used by philosophers.

8. The Difference Between Sophistry & Philosophy

Although philosophy has made many wonderful contributions to the world (such as the invention of natural science and computers), it is quite possible that sophistry has an even greater influence on civilization. And I don’t think that’s a good thing.

9. Do Default Positions Exist?

I’ve heard atheists talk about “default positions” quite a bit, and I became curious about exactly what they thought a “default position” was. They often say that atheism is the “default position.” That led me to study the work of one of the first atheists to use the term.

10. What is the Burden of Proof?

I realized that default positions are intimately tied to what philosophers call the “burden of proof.” Atheists often say that theists have the “burden of proof.” The assumption is that it is rational to be an atheist until the existence of a god is proven.

Honorable Mention: Top 10 Posts on Ethical Realism from 2011

• What is Argument Mapping?

Introduction

Argument maps are visual representations of arguments to help people better understand them. A meta-analysis of various studies found that classes with lots of argument map practice are the most effective type of critical thinking class to help improve critical thinking skills.

We want to know how to create and identify good and bad arguments insofar as we want to know what we should believe. Good arguments give us a good reason to believe something. Some arguments fail to give us a good reason to believe something, and we should make sure not to form beliefs on the basis of bad arguments. Argument maps can help us identify good and bad arguments, but they are insufficient to do so. There are many issues that should be studied in order to better understand the difference between good and bad arguments, such as logical fallacies and argument interpretation. Argument mapping might help us better understand informal fallacies and argument interpretation, but the focus here is merely the distinctions argument maps help us understand.

Argument maps are mainly helpful because they help us better understand the distinctions between the following:

1. Premises and conclusions
2. Arguments with one and multiple premises
3. Multiple arguments and multiple premises
4. Simple and extended arguments
5. Supporting arguments and objections
6. Objections to conclusions, premises, and forms of reasoning

I will present examples of argument maps that illustrate each of these distinctions.

Chapter 1: The distinction between premises and conclusions

A rational argument is supposed to give us a reason to believe something is true. Arguments require both premises and conclusions. Premises are statements (true or false sentences) that are used as a reason (or part of a reason) to believe something. Conclusions are statements that are supported by premises.

Consider the following argument:

We should believe that rocks exist because we can see them.

In this case the word ‘because’ makes it clear that “we can see them” is the reason to believe the conclusion, so it’s a premise. The phrase “we should believe that” helps indicate that the conclusion is “rocks exist.” We can then rephrase the argument with a statement representing the premise and the conclusion as the following:

1. We can see rocks.
2. Therefore, rocks exist.

An argument map can represent the rephrased version of the argument as the following:

Consider the following:

1. Each statement of the argument is put into a separate box and that the word ‘therefore’ was removed from the conclusion.
2. The argument map is a visual representation of the argument, and there are various features to help us understand how the argument works.
3. The word ‘conclusion’ is written above the box containing the conclusion, and the word ‘premise’ is written above the box containing the premise.
4. The box containing the premise is blue.
5. An arrow points from the premise to the conclusion.
6. The word ‘supports’ is written alongside the arrow.

Chapter 2: The distinction between arguments with one and multiple premises

Although some arguments only have one premise, almost all good arguments actually require more than one premise. The argument above only uses one premise, but it could be considered to be rationally persuasive for those of us who make a certain assumption—that if we can see rocks, then rocks exist. The assumption should be made explicit to make it clear why seeing rocks is sufficient to conclude that rocks exist. Only then will we have a truly good argument.

Now that we have identified an assumption, we can add it to the argument as an additional premise in order to form the following argument:

1. We can see rocks.
2. If we can see rocks, then rocks exist.
3. Therefore, rocks exist.

The argument map for this argument is the following:

This time there are two premises and each are in a separate blue box. There is a single line connecting both premises that lead to the arrow to make it clear that both premises are part of a single reason to believe the conclusion is true. In this case the assumption that both premises are true is sufficient to assume the conclusion is true. If either premise is false, then the other premise will not be sufficient for us to know that the conclusion is true.

Chapter 3: The distinction between multiple arguments and multiple premises

The above argument map featured a single argument that had multiple premises, but sometimes there are also multiple arguments for the same conclusion. Consider the following passage:

We should believe that rocks exist because we can see them, and also because we can touch them.

In this case the passage presents us with two separate arguments:

1. We can see rocks.
2. Therefore, rocks exist.

1. We can touch rocks.
2. Therefore, rocks exist.

A single argument map can represent both of these arguments at the same time:

This time there is no line connecting both premises to a single arrow. Instead, each premise has its own arrow pointing separately to the conclusion. This makes it clear that each premise is a separate argument for the conclusion. They are considered to be different reasons to believe the conclusion to be true.

Chapter 4: The distinction between simple and extended arguments

Simple arguments in this context are those with a single conclusion. Extended arguments have more than one conclusion—at least one premise is also a conclusion that is supported by another argument.

Consider the following passage:

We should believe that rocks exist because we can see rocks; and if we can see rocks, then they exist. Moreover, sight is a reliable way to know what exists, so we should believe that if we see rocks, then they exist.

In this case we could rephrase this passage as the following two arguments:

1. We can see rocks.
2. If we can see rocks, then they exist.
3. Therefore, rocks exist.

1. Sight is a reliable way to know what exists.
2. Therefore, if we see rocks, then they exist.

These two arguments are represented by a single argument map:

In this case the main conclusion (that is not a premise) is in a white box with ‘conclusion’ written over the box, but the premise that is supported by an argument is in a blue box with ‘premise’ written over it.

Chapter 5: The distinction between supporting arguments and objections

All the arguments above are supporting arguments. They are meant to tell us why we should believe something. Objections are attempts to tell us why we should reject something. An objection is meant to tell us when a statement is false or unjustified. Unjustified statements are those that we currently have no good reason to believe to be true, and such statements could be either true or false. However, all statements we prove to be false are also proven to be unjustified.

Consider the following passage:

Karla believes that rocks exist because she sees them, but we should believe that rocks don’t exist because that’s what my parents believe.

In this case the passage contains the following two arguments:

1. We see rocks.
2. Therefore, rocks exist.

1. My parents believe that rocks don’t exist.
2. Therefore, rocks don’t exist.

These two arguments can be represented with a single argument map:

The argument map above makes it clear that there is an argument for and against a conclusion. This time the premise of the supporting argument is in a blue box as usual, but the premise of the opposing argument is in a pink box. Additionally, the box containing the objection has an arrow pointing to the conclusion and the word ‘opposes’ is written alongside the arrow to make it clear that the premise is an argument against the conclusion.

When having a debate we are interested in arguments both for and against a certain statement, and argument maps are capable of representing multiple such arguments at once.

Finally, the above argument map makes it clear that there is no winning side of the debate at this point in time. There’s an argument for and against a conclusion, but it is not yet clear which side has the better argument. Of course, we know that the opposing argument should be rejecting because what my parents believe is not relevant to the debate.

Chapter 6: The distinction between objections to conclusions, premises, and forms of reasoning

I discussed an objection to a conclusion above, but objections can also be given against premises and forms of reasoning. I will now present an example of an objection to a premise and an objection against a form of reasoning.

An objection to a premise

Consider the following passage:

Jenny believes that rocks exist because we see them; and if we see them, then they exist. However, we should reject that if we see rocks, then they exist because we can see things that don’t exist when we are dreaming.

This passage contains the following two arguments:

1. We see rocks.
2. If we see rocks, then they exist.
3. Therefore, rocks exist.

1. We see things that don’t exist when we are dreaming.
2. Therefore, we should reject the belief that if we see rocks, then they exist.

These arguments can both be represented in a single argument map:

This argument map makes it clear what particular premise is being opposed. Keep in mind that this objection is not against the conclusion. If this objection succeeds, then we will wonder if there’s a better reason to believe that rocks exist than the one presented here.

If anyone argues for a conclusion that we believe to be false, then we might want to provide an objection to that argument in addition to an argument against that conclusion. If there is only an argument for and against the conclusion, then we will still need to know if either argument is a good argument. If there is only an objection against the supporting argument, then we still need to know why we should reject the conclusion.

For example, imagine that someone argues that all dogs are reptiles because dogs have cold blood and scales. In this case we can object to the conclusion by saying that dogs aren’t reptiles because they are warm-blooded, have mammary glands, and give birth to live young. Additionally, we can object to the supporting argument by stating that biologists have unanimously confirmed that dogs are not cold-blooded and don’t have scales.

An objection to a form of reasoning

An argument has a logically valid form of reasoning when assuming the premises are true would guarantee that the conclusion is also true. It’s impossible for logically valid arguments to have true premises and a false conclusion. An example of a logically valid argument is, “If dogs are reptiles, then dogs are animals. Dogs are reptiles. Therefore, dogs are animals.” In this case a premise is false, but the form of reasoning is valid. We object to the form of reasoning of an argument when it’s not valid or has contradictory premises.

Consider the following passage:

Mark argues that we should believe that rocks exist because life used to be on Mars and life never used to be on Mars. However, Mark’s argument requires us to accept two contradictory premises and two contradictory statements can’t be true at the same time, so at least one of Mark’s premises has to be false.

This passage contains the following two arguments:

1. Life used to be on Mars.
2. Life never used to be on Mars.
3. Therefore, rocks exist.

1. Both those premises form a contradiction.
2. It’s impossible for two contradictory premises to be true at the same time.
3. Therefore, at least one of the above premises are false.

An argument map can represent both of these arguments:

This time the objection is against both premises at the same time, so there are two arrows pointing at each of them. We don’t currently know which premise is false, but we know one of them is because they form a contradiction. One premise literally says the other premise is false.

The reason that this objection is against the form of reasoning is because the form of reasoning contains a contradiction. A contradictory form of reasoning occurs when two statements state “A and not-A.” In other words, they state that something is both true and false at the same time. “A” could be any statement. For example, it could be that “there’s intelligent life on another planet.” In that case the contradictory statements would be “there’s intelligent life on another planet, and it’s not the case that there’s intelligent life on another planet.”

Related:

• A Study Finds That Formal Logic Can Help High School Students
• Argument Mapping Classes Are The Most Effective At Improving Critical Thinking

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Four different kinds of argument diagrams.

The four types of argument diagrams are Hurley diagrams, Toulmin diagrams, ordinary argument diagrams, and argument maps. In order to understand how they differ, let’s consider a supporting argument and objection they can all represent.

Supporting argument

First, let’s consider the supporting argument:

1. Socrates is a man.
2. All men are mortal.
3. Therefore, Socrates is mortal.

Hurley diagrams

Hurley diagrams are briefly taught in the most popular logic textbook: A Concise Introduction to Logic by Patrick Hurley.

Each circle represents a different statement of the argument. The statements of the argument were already numbered above. 1 and 2 are the first two premises, and 3 is the conclusion. The bracket makes it clear that both premises are a single argument for the conclusion.

Note that Hurley diagrams are not capable of representing a supporting argument and objection at the same time.

Toulmin diagrams

Toulmin diagrams require that we don’t simply give deductive arguments, and that some sort of justification is discussed other than a mere deduction. For that reason we write down the “grounds” that support the deductive premises. The deductive premises are mentioned in the “warrant” area. The conclusion is written as the “claim.”

Notice that Toulmin diagrams don’t make it clear when we have an extended argument. Sometimes a specific premise is justified by an argument, but this diagram does not make it clear when a specific premises is supported by an additional argument. In the specific case of this argument the grounds is technically a justification to the premise that states “all men are mortal.”

Ordinary argument diagrams

Ordinary argument diagrams are almost identical to Hurley diagrams. Just like the Hurley diagrams, each circle represents a different statement, the first two circles represent the premises, and the final one represents the conclusion. The main difference is that a line is often used to make it clear that the premises support the conclusion instead of a bracket. Additionally, ordinary argument diagrams are capable of modeling both arguments and objections within a single diagram, which Hurley diagrams can’t do.

Argument maps

Argument maps are much like ordinary argument diagrams, except the statements are written in boxes, the boxes are labeled, the arrows are labeled, and premises are given a color. It is made clear that both premises are a single argument for the conclusion because they are connected by a curved line that leads to a single arrow. We know that the premises are given as a reason to believe the conclusion because the term “supports” is written alongside the arrow.

Objections

Now let’s consider how the four different diagrams can represent an objection to a supporting argument.

The supporting argument is the following:

1. If Socrates is a dog, then Socrates has a tail.
2. Socrates is a dog.
3. Therefore, Socrates has a tail.

The objection to the supporting argument is the following:

4. Socrates is a human.
5. No humans are dogs.
6. Therefore, Socrates is not a dog.

Hurley diagrams

Hurley diagrams are incapable of representing objections as distinct from supporting arguments. It’s not possible for them to represent a supporting argument and objection within a single diagram.

Toulmin diagrams

The Toulmin model requires that we give additional support to the supporting argument because it requires more than a mere deduction from premises, it asks us to give a qualifier (that not all dogs actually have tails because it could be removed), and it is capable of giving the objection in the “rebuttal” section.

However, note that the Toulmin model does not make the following clear:

1. The “grounds” is an additional argument to the premise that states that “if Socrates is a dog, then he has a tail.” This would be useful information, but it’s not provided.
2. The objection is technically against the premise that states “Socrates is a dog.” This would be useful information, but it’s not provided either.

Ordinary diagrams

The ordinary argument diagram is also capable of representing the supporting argument and objection simultaneously, and it is capable of making it clear that the objection is against a specific premise (premise 2: Socrates is a dog). Premise 4 and 5 are the premises of the objection. In this case the objection has an arrow with a strike through it, which is pointing towards the premise they are against. It is clear that there is a single objection consisting of two premises because of the line that connects both of the premises.

Argument maps

The argument map is capable of representing the positive argument and objection within a single diagram. The boxes containing the objection are pink to make it clear that they’re part of an objection, and the word “opposes” is written alongside the arrow to make it clear that they are against a specific premise that the arrow points to. It is clear that the objection consists of a single argument with two premises because there’s a line connecting them both to a single arrow.

Why argument maps seem to be the best.

Just based on the argument diagrams above, argument maps are capable of doing everything the other diagrams do, but they are much easier to read because of the colors, the fact that the premises are written in boxes, and because the boxes and arrows are labeled. In particular, argument maps can help make the following clear:

1. when multiple premises form a single argument.
2. when a statement is a premise or a conclusion.
3. what statement is supported by premises.
4. when premises are used as an objection.
5. what specific statement an objection is against.

Of the four types of diagrams, Hurley diagrams and Toulmin diagrams are the most inferior:

Hurley diagrams make it clear when there are multiple premises for a single argument, when a statement is a premise or conclusion, and what statement is supported by premises. However, it doesn’t tell us what premises are used as an objection or the specific statement the objection is against.

Toulmin diagrams can tell us when multiple premises form a single argument, when a statement is a premise or conclusion, and when premises are used to form an objection. However, Toulmin diagrams don’t always tell us when a specific statement is supported by an extended argument, or what specific statement an objection is against.

Here is a table that summarizes many of the results:

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Related:

• Introduction to Argument Mapping
• Argument Mapping Classes Are The Most Effective At Improving Critical Thinking

‘Translation’ refers to the act of converting statements of natural language to statements of a symbolic logical system. In this case I will discuss how to convert statements of English into statements of propositional logic. Translation requires us to know logical connectives used in propositional logic, and ways we use logical connectives in English.

Not all propositions use logical connectives. In order to translate such a proposition, replace it with a capital letter. For example “1+1=2” is a proposition that can be translated as “A.”

What are logical connectives?

Logical connectives are ways of saying when propositions are true. For example “or” is a way of saying at least one of two propositions are true. “The President of the USA is a man or a dog” means that either the proposition “the President of the USA is a man” is true or the proposition “the President of the USA is a dog” is true. We can then replace the propositions with capital letters to get “A or B.” However, the word “or” is also replaced in symbolic logic with the symbol “∨,” so we would write it out as “A ∨ B.”

There are five connectives used in propositional logic: “not” (¬), “and” (∧), “or” (∨), “implies” (→), and “if and only if” (↔):

Negation

The “negation” connective is “¬,” which means “is false.” We also state negation by saying “is not true” or sometimes simply as “not.” For example, “the President of the USA is not a dog” should be taken to mean “it is false that the President of the USA is a dog.” It can be translated into propositional logic as “¬A.” In this case “A” stands for “ the President of the USA is a dog.”

More examples of translating English statements that use negation:

1. It’s false that only two people exist. [“A” stands for “only two people exist.” Translation: “¬A.”]
2. Not all dogs are fish. [This means “it's false that all dogs are fish.” “B” stands for “all dogs are fish.” Translation: “¬B.”]
3. It is not the case that Samantha went to the store. [This means “it's false that Samantha went to the store. “C” stands for “Samantha went to the store.” Translation: “¬C.”]
4. Evolution is false. [This will be taken to mean “the central claims of the theory of evolution are not consistent with our observations.” “D” stands for “the central claims of the theory of evolution are consistent with our observations.” Translation: “¬D.”]
5. People are wrong who think that nothing exits. [This means “it's false that nothing exists.” “E” stands for “nothing exists.” Translation “¬E.”]
6. It’s absurd to think that “2+2=5.” [This implies that “2+2=5” is false. “F” stands for “2+2=5.” Translation: “¬F.”]

Conjunction

The conjunction connective is “∧,” which means “and.” We also state conjunctions by saying “but.” For example, “the President of the USA has an PhD, but dislikes chocolate” should be taken to mean “the President of the USA has a PhD, and the President of the USA dislikes chocolate.” This can be translated as “A ∧ B.” Both “A” and “B” must represent statements that are true or false, so “B” must represent “the President of the USA dislikes chocolate” rather than “dislikes chocolate.”

More examples of translating English statements that use conjunctions are the following:

1. Jill is tall and Jeff is short. [“A” stands for “Jill is tall” and “B” stands for “Jeff is short.” Translation: “A ∧ B.”]
2. Jennifer is a human and a mammal. [“C” stands for “Jennifer is a human” and “D” stands for “Jennifer is a mammal.” Translation: “C ∧ D.”]
3. At least two people exist, but no unicorns exist. [“E” stands for “at least two people exist” and “F” stands for “no unicorns exist.” Translation: “E ∧ F.”]
4. It is true that “1+1=2” and it is true that “2+2=4.” [“G” stands for “1+1=2” and “H” stands for “2+2=4.” Translation: “G ∧ H.”]
5. Evolution is true and we should believe it. [“I” stands for “the central claims of evolution are consistent with our observations” and “J” stands for “we should believe the central claims of evolution.” Translation: “I ∧ J.”]

Disjunction

The disjunction connective is “∨” and it means “or.” We also state this by saying “and/or” or “unless.” It means that one or both of two propositions are true. For example, “Julia went to the store unless she went to school” can be translated as “A ∨ B.” In this case “A” stands for “Julia went to the store” and “B” stands for “Julia went to school.”

Sometimes disjunctions are meant to be exclusive—only one of the two propositions are true. For example, “the President is a man or a woman” could imply that the President is a man or a woman, but not both a man and a woman. In that case we can translate the disjunction as saying “either A or B, but not both-A-and-B.” This can be written in symbolic form as “(A ∨ B) ∧ ¬(A ∧ B). (“A” stands for “the President is a man” and “B” stands for “the President is a woman.”) We use parentheses to group propositions together that are intricately tied. The logical connective that is not in any parentheses is the “main connective.”

More examples of translating English statements that use disjunctions are the following:

1. Either the road leads back to town or the road leads to a farm. [“A” stands for “the road leads back to town” and “B” stands for “the road leads to a farm.” Translation: A ∨ B]
2. Either Jack or Jill is tall. [“C stands for “Jack is tall” and “D” stands for “Jill is tall.” Translation: “C ∨ D.”]
3. Humans are mammals unless they’re reptiles. [“E stands for “humans are mammals” and F” stands for “humans are reptiles.” Translation: “E ∨ F.”]
4. Either evolution is true or creationism is true. [“G” will stand for “the central claims of evolution are consistent with our observations” and “H” will stand for “the central claims of creationism are consistent with our observations.” Translation: “G ∨ H.”]

Material conditional

The material conditional is “→” and means “implies.” We also state material conditionals by saying “if… then” or “only if.” For example, “if all men are mortal, then Socrates is mortal.” We can translate this to be “A → B.” In that case “A” stands for “all men are mortal” and “B” stands for “Socrates is mortal.” The material conditional means that the first part is false or the second part is true.

People often talk about necessary and sufficient conditions, and they can be translated to be propositions using material conditionals. “A is necessary for B” means “B → A.” “A is sufficient for be” means “A → B.” For example, we could say that “the existence of the Sun is sufficient for the existence of light,” or that “the existence of atoms are necessary for the existence of humans.”

Also note that people sometimes say “A if B,” such as “light exists if the Sun exists.” In this case we have to switch the first and second parts of the material conditional to translate it into logical form. “A” can stand for “light exists” and “B” can stand for “the Sun exists.” The statement will then be translated as “B → A.”

More examples of translating English statements that use material conditionals are the following:

1. If Lisa kicked John, then John got hurt. [“A” stands for “Lisa kicked John.” “B” stands for “John got hurt.” Translation: “A → B.”]
2. George likes milk if he likes milkshakes. [“C” stands for “George likes milk.” “D” stands for “George likes milkshakes.” Translation: “D → C.”]
3. Birds are animals only if birds are living organisms. [“E” stands for “all birds are animals” and “F” stands for “birds are living organisms.” Translation: “E → F.”]
4. The fact that all humans are mammals implies that all humans are animals. [“G” stands for “all humans are mammals” and “H” stands for “all humans are animals.” Translation: “G → H.”]
5. Being human is sufficient to be a mammal. [“I” means “something is a human.” “J” means “something is a mammal.” Translation: “I → J.”]
6. Being a mammals is necessary for being a human. [“I” means “something is a human” and “J” means “something is a mammal." Translation: “J → I.”]

Material equivalence

The material equivalence is “↔” and it means “if and only if.” Sometimes we state it by saying “just in case.” It is used to state that two propositions have the same truth value—they’re both true, or they’re both false. For example, “Socrates is a person if and only if he’s a rational animal” could be translated as “A ↔ B.” In that case “A” stands for “Socrates is a person” and “B” stands for “Socrates is a rational animal.” Either Socrates is a person and a rational animal, or he is neither a person nor a rational animal.

More examples of translating English statements that use material equivalence are the following:

1. Jessica will win the game if and only if Bob loses the game. [“A” stands for “Jessica will win the game.” “B” stands for “Bob loses the game.” Translation: “A ↔ B.”]
2. Water is in the glass just in case H2O is in the glass. [“C” stands for “water is in the glass.” “D” stands for “ H2O is in the glass.” Translation: “C ↔ D.”]
3. Lizards are animals if and only if lizards are creatures. [“E” stands for “lizards are animals.” “F” stands for “lizards are creatures.” Translation: “E ↔ F.”]

Multiple connectives

Many statements use multiple connectives. Propositional logic requires that we keep all of the logical connectives possible. “Neither… nor” is a common example. “Humans are neither plants nor reptiles” means “it is not the case that humans are plants, and it is not the case that humans are reptiles.” This can be translated as “¬A ∧ ¬B.” In this case “A” stands for “all humans are plants” and “B” stands for “all humans are reptiles.”

Another example is “unless.” Although I translated “unless” to mean “or” above, there are other ways people can use the word and the context should be considered. It has been suggested that we should sometimes translate “A unless B” as “¬A ∨ ¬B” or “ B → ¬A.”

When multiple connectives are used, we often need to know which connective is the “main connective.” Identifying the main connective is needed when other connectives are grouped together inside parentheses. For example, “either evolution is consistent with our observations and a species can change over time, or creationism is consistent with our observations.” In this case a disjunction is the main connective and the conjunction is grouped together inside parentheses. “A” stands for “evolution is consistent with our observations, “B” stands for “a species can change over time,” and “C” stands for “creationism is consistent with our observations.” We can then translate the statement as “(A ∧ B) ∨ C.”

It is important to make it clear when two propositions contradict one another. We could need to translate a contradictory statement such as “no humans are reptiles, and some humans are reptiles.” We could translate “no humans are reptiles” as “A,” and “some humans are reptiles” as “B,” but that wouldn’t make it clear that both propositions form a contradiction. For that reason, we should make sure to make it clear that one is the negation of the other. We could simply translate this statement as saying “A ∧ ¬A.” In that case “some humans are reptiles” is taken to be “¬A” (that it’s false that no humans are reptiles).

More examples of translating English statements that use multiple connectives include the following:

1. If Sophia is not a dog, then Sophia is a human. [“A” stands for “Sophia is a dog.” “B” stands for “Sophia is a human.” Translation: “¬A → B.”]
2. Hansel will win the game if and only if Gretel doesn’t win. [“C” stands for “Hansel will win the game.” “D” stands for “Gretel will win the game.” Translation: “C ↔ ¬D”]
3. Neither Democrats nor Republicans will stop corporate welfare. [“E” stands for “Democrats will stop corporate welfare.” “F” stands for “Republicans will stop corporate welfare.” Translation: “¬E ∧ ¬F.”]
4. Either the President of the USA is a reptile, or he’s both a mammal and an animal. [“G” stands for “the President of the USA is a reptile.” “H” stands for “the President of the USA is a mammal.” “I” stands for “the President of the USA is an animal.” Translation: “G ∨ (H ∧ I).”]
5. Either all lizards are mammals or some lizards are not mammals. [“J” stands for “all lizards are mammals.” Translation: “J ∨ ¬J.”]

Scheme of abbreviation

When translating, it’s important to state what each letter stands for. The scheme of abbreviation is what tells us what each letter represents, and multiple statements can be translated using the same scheme of abbreviation. Translation requires that each letter represents the same thing within the scheme of abbreviation.

Consider the following three statements:

1. Jack is tall or Jill is short.
2. Jack is tall if and only if Jill is not short.
3. If Jill is short, then Jack is not tall.

We could use the following scheme of abbreviation for these three statements:

A: Jack is tall.

B: Jill is short.

We could then use this scheme of abbreviation to translate all three statements in the following way:

1. A ∨ B
2. A ↔ ¬B
3. B → ¬A

Arguments

Arguments are a series of two or more statements when any number of statements is meant to be a reason to believe another.  The statement that is being supported is the “conclusion” and the other statements are the “premises.” Arguments are often used for persuasion—people who agree with the premises are likely to agree with the conclusion of a logically valid argument.

Sometimes arguments are stated as a single statement. For example, “Socrates is a man; if Socrates is a man, then he’s mortal; therefore, Socrates is mortal.” When translating arguments, we want to separate each premise and conclusion. We also want to show the maximal number of premises. In this case we want to separate the premises and conclusion into three separate statements:

1. Socrates is a man.
2. If Socrates is a man, then Socrates is mortal.
3. Therefore, Socrates is mortal.

A translation and scheme of abbreviation for this argument is the following:

1. A
2. A → B
3. ∴ B

A: Socrates is a man.

B: Socrates is mortal.

Notice that “∴” is the symbol used to mean “therefore.” It indicates that the final statement is a conclusion. (Quite often, a line is used to separate premises from conclusions as well.)

One reason that translation is important is because we want to know when arguments stated in natural language are logically valid, and we can use propositional logic to know when an argument is logically valid.

The argument given above uses an argument form called “modus ponens.” Every argument with this form is logically valid. You can replace “A” and “B” with any two propositions, and the argument will be logically valid. For example, “A” could stand for “all dogs are mammals” and “B” could stand for “all mammals are animals.” The argument would then be “All dogs are mammals. If all dogs are mammals, then all dogs are animals. Therefore, all dogs are animals.”

It is important that deductive arguments are logically valid—if we know a logically valid deductive argument has true premises, then we should agree with the conclusion. However, not all logically valid arguments are good arguments. One of the most important reasons is that logical validity only concerns the logical form of the argument. A logically valid argument can have false premises.

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Related

• What is Logic?
• Logic Part 1: What is Propositional Logic?

1. The meaning of propositional logic.
2. Translation
3. Truth tables
4. Truth trees
5. Natural deduction

What is propositional logic?

Propositional logic is a symbolic language that lacks content, and uses letters to symbolize propositions. For example, consider the statement “the sky is red or blue.” We could replace “the sky is red” with “A” and “the sky is blue” with “B.” In that case we would end up with “A or B.”

Propositional logic is used to evaluate the validity of deductive arguments and the consistency of statements. Statements that are inconsistent are said to form “contradictions.” For example, the statement “the sky is blue” contradicts the statement “the sky is not blue.”

Propositional logic consists of logical connectives, rules of inference, and axioms. These will be discussed in more detail in later chapters. However, it is important to know what “statements” and “propositions” are, so they will be discussed in greater detail here.

What are statements?

Statements are sentences that are true or false. For example, “all mammals are animals” is a statement written in English. In this case the statement is true.

What are propositions?

Propositions consist in the meaning of statements. We could say that propositions are the “conceptual content” of statements. For example, “the sky is blue” is a statement written in English that refers to the same proposition as “le ciel est bleu,” which is a statement written in French.

Another example of two statements that refer to the same proposition is “all whales are mammals” and “if something is a whale, then it’s a mammal.”

Next: Logic Part 2: Translation

Related: What is Logic?

Formal logic

Formal logic is the traditional domain of logic in western philosophy. It is a domain that covers logical form, consistency, valid argumentation, and logical systems.

Logical form

Logical form allows us to symbolize statements by stripping statements of their content. For example, consider the statement “if it will rain today, then the roads will become slippery.” The logical form of this statement would be presented in propositional logic as “if A, then B.” In that case “A” stands for “it will rain today” and “B” stands for “the roads will be slippery.” Logical connectives are kept, such as “if,” “and,” “or,” and “not.”

Logicians don’t usually write statements as “if A, then B.” Instead, they usually use a symbol for logical connectives, such as “→.” We can state “if A, then B” as “A → B.”

Consistency

Two statements are consistent if it’s possible for them both to be true at the same time. For example, the statement “if it will rain today, then the roads will be slippery” is consistent with the statement “it will not rain today.” Logic provides us with a way to determine when statements are consistent, which is important to us because all true statements about the world are consistent. (Two true statements can never form a contradiction. For example, “Aliens live on another planet” and “aliens don’t live on another planet” form a contradiction, so one of the statements is false.)

We know that two statements are consistent as long as they can all be true at the same time, and contradictory when they can’t. Whenever two propositions contradict, one proposition can be symbolized as “A” and the other can be symbolized as “not-A.” For example, “it will rain today” contradicts “it will not rain today.”

Some statements are also self-contradictory, such as “one person exists and no people exist.” Many self-contradictions can be symbolized as “A and not-A.” These statements are always false.

Tautological statements are always true, such as “either the Moon revolves around the Earth or the Moon doesn’t revolve around the Earth.” Many tautologies can be symbolized as “A or not-A.”

Valid argumentation

A valid argument has an argument form that could never have true premises and a false conclusion at the same time. For example, “If it will rain today, then the roads will be slippery. It will rain today. Therefore, the roads will be slippery” is valid because it has the argument form “If A, then B. A. Therefore, B.” All arguments with this form are valid.

Logic gives us the tools to determine when an argument is logically valid. If a deductive argument is not logically valid, then it does not provide us with a good reason to agree with the conclusion. If the premises are true, then the conclusion could still be false.

An example of an invalid argument is “At least one person exists. If at least one person exists, then at least one mammal exists. Therefore, no mammals exist.” Although the premises are true, the conclusion is false. This argument does not do what arguments are supposed to do—provide us with a good reason to think the conclusion is true.

Logical systems

Logical systems have (1) a formal language that allows us to symbolize statements of natural language, (2) axioms, and (3) rules of inference.

1. A formal language is a way we can present the form of our statements involving logical connectives.
2. Axioms are rules, such as the rule that states that contradictions can’t exist.
3. Rules of inference are rules that state what premises can be used to validly infer various conclusions. For example, a rule known as “modus ponens” states that we can use “A” and “if A, then B” as premises to validly infer that “B.”

Logical systems are needed in order for us to best determine when statements are consistent or when arguments are valid.

Informal logic

Informal logic is domain that covers the application of rational argumentation within natural language—how people actually talk. What we call “critical thinking” is often said to involve informal logic, and critical thinking classes generally focus on informal logic. Informal logic mainly focuses on rational argumentation, the distinction between inductive and deductive reasoning, argument identification, premise and conclusion identification, hidden assumption identification, and error identification.

Rational argumentation

Arguments are a series of two or more statements including premises (supporting statements) and conclusions (statements that are supposed to be justified by the premises). For example, “All human beings that had lived in the distant past had died. Therefore, all human beings are probably mortal.”

The idea of rational argumentation is that it is supposed to give us a good reason to believe the conclusion is true. If an argument is good enough, then we should believe the conclusion is true. If an argument is rationally persuasive enough, then it would be irrational to think the conclusion is false. For example, consider the argument “All objects that were dropped near the surface of the Earth fell. Therefore, all objects that are dropped near the surface of the Earth will probably fall.” This argument gives us a good reason to believe the the conclusion to be true, and it would seem to be irrational to think it’s false.

The distinction between deductive and inductive reasoning

Deductive arguments are meant to be valid. If the premises are true, then the conclusion is supposed to be inevitable. Inductive arguments are not meant to be valid. If the premises of an inductive argument are true, then the conclusion is supposed to be likely true. If an inductive argument is strong and the premises are true, then it is unlikely for the conclusion to be false.

An example of a valid deductive argument was given above when valid arguments were discussed. Let’s assume that “if it will rain today, then the roads will be wet” and that “it will rain today.” In that case we have no choice but to agree that “the roads will be wet.”

An example of a strong inductive argument was given in the argument involving dropping objects. It is unlikely that dropped objects will not fall in the future assuming that they always fell in the past.

Argument identification

Knowing what arguments are and why people use them helps us know when people give arguments in everyday conversation. It can also be helpful to know the difference between arguments and other similar things. For example, arguments are not mere assertions. A person who gives a mere assertion is telling you what she believes to be true, but a person who gives an argument tells you why she believes we should agree that a conclusion is true.

Premise and conclusion identification

Knowing what premises and conclusions are helps us know how to know which are which in everyday conversation. For example, a person can say “the death penalty is wrong because it kills people.” In this case the premise is “the death penalty kills people” and the conclusion is “the death penalty is wrong.”

Hidden assumption identification

Knowing that an argument is meant to be rationally persuasive can help us realize when hidden assumptions are required by an argument. For example, the argument that “the death penalty is wrong because it kills people” requires the hidden assumption that “it’s always wrong to kill people.” Without that assumption the argument will not be rationally persuasive. If it’s not always wrong to kill people, then perhaps the death penalty is not wrong after all.

Error identification

Knowing about several errors of reasoning (i.e. fallacies) can help us know when people have errors of reasoning in arguments they present in everyday conversation. For example, the argument “my friend Joe never died, so no person will die in the future” contains an error. The problem with this argument is the unjustified assumption that we can know what will happen to everyone in the future based on what happened to a single person given a limited amount of time. This type of error is known as the “hasty generalization” fallacy.

What’s the difference between logic and epistemology?

Epistemology is the philosophical study of knowledge, justification, and rationality. It asks questions, such as the following:

1. What is knowledge?
2. Is knowledge possible?
3. What are the ways we can rationally justify our beliefs?
4. When it is irrational for a person to have a belief?
5. When should a person agree that a belief is true?

These issues are highly related to logic, and many philosophers have equated “logic” with “epistemology.” For example, the Stoic philosophers included epistemology in their domain of “logic.”

I believe that logic should now be considered to be part of the domain of epistemology. However, for educational purposes it is considered to be a separate subject and it’s not taught in epistemology classes.

Logic classes deal with argument form and certain rational criteria that applies to argumentation, but epistemology classes generally deal with somewhat abstract questions, as were listed above. Perhaps one of the most important issues that logic deals with much less than epistemology is justification—logic tends not to tell us when premises are justified and how well justified they are, but epistemology attempts to tell us when premises are justified, and when a premise is justified enough to rationally require us to believe it’s true.

Why do logic and epistemology classes teach different things? Perhaps because philosophers who have an interest in epistemology have historically not cared as much about logic and vice versa.

But why would philosophers who care about epistemology not care as much about logic? Perhaps because logic tends to be concerned with issues that can be answered with a much higher degree of certainty. We know what arguments are. We know that good arguments must apply certain rational criteria. We can determine when arguments are valid or invalid. We can determine that many arguments have hidden premises or various errors. However, we can’t determine the nature of knowledge, justification, and rationality with that degree of certainty. It is more controversial when a belief is justified and at what point a belief is justified enough to rationally require us to believe it’s true.

What is the essence of logic?

I don’t think that logic has an essence. It’s a domain concerned with certain rational criteria involved with argumentation, but not all criteria. Epistemology also covers related issues. What we consider to be logic or epistemology mainly has to do with a history of philosophers (and mathematicians) who label themselves as “logicians” or “epistemologists” and teach classes in the corresponding domains. These terms are used merely because they are convenient to us.

However, I think we can say that logic is a domain of epistemology that has a restricted focus, and that focus is mainly restricted to issues that we think we can answer with a much higher degree of certainty than usual. Logic and mathematics are now often taken to be part of the same domain, and both generally offer us with a degree of certainty higher than the natural sciences. Whenever scientific findings conflict logic, we are much more likely to think that our scientific findings are false than that our understanding of logic is false.

The same can not be said of epistemology once logic is removed from it. There are examples of epistemological issues that do seem to involve a great deal of certainty. I think we should be confident that we should believe that “1+1=2” and that it’s irrational to believe that “1+1=3.” Epistemology tells us what we should believe in that sense. However, there is also a great deal of uncertainty that is usually involved with epistemology. The big questions in epistemology are still very controversial.

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Related

• 10 Myths About Logic
• What You Need From Formal Logic
• Introduction to Critical Thinking & Argument Mapping
• Fallacy Files (external website)
• Introduction to Logic by Stefan Waner and Steven R. Costenoble (external website)

All secular arguments for creationism use the following reasoning:

1. Either creationism is true or evolution is true.
2. Evolution is false.
3. Therefore, creationism is true.

There are at least two major problems with this argument:

Premise 1 is unjustified.

Premise 1 (that either creationism is true or evolution is true) is unjustified because it requires us to assume that creationism and evolution are the only two ways we can explain where complex life forms came from (and our observations about how they seem to change). There could be other hypotheses as well. For example, complex life forms might have always existed exactly as they exist now (and perhaps there was no beginning of the universe). Premise 1 commits the false dilemma fallacy and is only persuasive to people who engage in overly simplistic “black or white” thinking.

Premise 2 is totally unjustified.

Premise 2 (that evolution is false) is totally unjustified. No argument has ever refuted evolution, and the evidence for evolution is much stronger than the evidence for creationism.

We can also summarize the reason that creationists think evolution is false as the following:

Creationist Argument Against Evolution

1. There are observations incompatible with evolution.
2. If there are observations incompatible with evolution, then evolution is false.
3. Therefore, evolution is false.

The main problem with this argument is that the first premise is false. There are no observations that are incompatible with evolution that we know about. However, we do have observations that seem to be incompatible with creationism. For example, we have a fossil record that seems to indicate that creatures that existed in the past changed over time until they became different creatures that exist now. We have a much better reason to reject creationism for being incompatible with our observations than evolution.

Let’s consider some of the so-called observations that creationists take to debunk evolution:

1. There is a complex organism (or body part of a complex organism), but we don’t know how it evolved. For example, the eye was once taken to be such an example. However, we now have a good idea about how the eye incrementally evolved from various beneficial adaptations. Additionally, the fact that we don’t know how something evolves in no way proves it did not evolve.
2. It is often claimed that something couldn’t possibly have evolved. For example, some people argued that the eye could not possibly evolve. However, we know of nothing that couldn’t have possibly evolved. Moreover, it is quite possible that evolution is true, even if some complex organisms didn’t evolve. Evolution does not claim that all complex organisms must have evolved. Perhaps we will find out how to create new complex organisms through genetic engineering.
3. It is often said that evolution is false because there’s a “missing link.” However, there is no missing link. We have fossil evidence of several transitional species that show how an organism can change over time—including various apes that were similar to human beings. Moreover, it is quite possible for evolution to be true, even if no transitional species were ever discovered. There was no guarantee that we would ever find them because they existed long ago and their remains could have all been destroyed by now.
4. It is often said that evolution must be false because monkeys still exist. “If we evolved from monkeys, then how come monkeys still exist?” However, there is a number of problems with this argument: First, there is no reason to think that the existence of monkeys is incompatible with the truth of evolution. Second, evolution does not state that an ancestral species that evolves into another species must become extinct. There was a point that an isolated group of fish evolved into amphibians, but that doesn’t mean all other fish became extinct. The same could be true of an isolated group of monkeys. Third, the monkeys that exist now are not the same creatures that we evolved from. We have a “common ancestor” with monkeys and apes that exist now.

Can we debunk evolution?

It might be possible to debunk evolution by showing how our observations would be better explained by creationism, but that’s never happened. I do not deny that this is possible, but it would be a very difficult task and biologists would be the best qualified people to make such an argument. At this point in time creationism seems to be incompatible with some of our observations rather than better supported by them than evolution.

Can creationism be proven to be true?

Keep in mind that even if evolution is debunked, that does not mean creationism is true. As stated above, there are other potential explanations for the existence of complex life forms. Creationists need to not only debunk evolution, but show that creationism is “better supported” by the evidence than any other alternative explanation.

Conclusion

In conclusion, creationists rely on poor reasoning that is easily debunked. The premises used by the current secular arguments for creationism are unjustified, and evolution has not yet been refuted.

Update (10/8/2012): I rephrased why creationists reject evolution and discussed the issue in greater detail.

Update (10/28/2012): I slightly rephrased why I reject the first premise of the first argument.

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abstractism – The view that something is necessary insofar as it’s true of every consistent set of statements, and something is possible insofar as it’s true in at least one consistent set of statements. It’s necessary that oxygen is O₂ insofar as it’s true that oxygen is O₂ in every consistent set of statements, and it’s possible for a person to jump over a small rock insofar as at least one consistent set of statements has a person jump over a small rock. Abstractism could be considered to advocate the existence of “abstract entities” insofar as the existence of a consistent set of statements could be considered to be factual for that reason.

accessible world – A world that is relevant to our world when we want to determine if something is necessary or possible. For example, we could say that something is necessary if it’s true of all accessible worlds. Perhaps it’s necessary that contradictions are impossible because it’s true of all accessible worlds. An accessible world is not necessarily a world we can actually go to. They could exist outside our universe or only exist conceptually. See “possible world,” “truth conditions,” and “modality” for more information.

accessibility – (1) The relevant domain used to determine if something is necessary or possible. It is thought that something is necessary if it “has to be true” for all of the relevant domain, and something is possible if it is true of at least one thing within the relevant domain. For example, some philosophers believe that it’s possible for people to exist because they exist in at least one possible world—the one we live in. See “accessible world,” “possible world,” “truth conditions,” and “modality” for more information. (2) In ordinary language, “accessibility” refers to the ability to have contact with something. For example, people in jail have access to food and water; and citizens of the United States have access to move to any city found in the United States.

algorithm – A step-by-step procedure.

analogical reasoning – Reasoning using analogies that can be explicitly described as an “argument from analogy.”

argument – In mathematics and predicate logic, the term ‘argument’ is sometimes used as a synonym for “operands.”

argument place – (1) In logic, it is the number of things that is predicated by a statement. For example, “Gxy” is a statement with two predicated things, so it has two argument places. (In this case “G” can stand for “attacks.” In that case “Gxy” would mean “x attacks y.”) (2) In mathematics, it’s the number of things that are involved with an operation. For example, addition is an operation with two argument places. “2 + 3” has two arguments: “2” and “3.”

arity – (1) In logic, it refers to the number of things that are predicated. The statement “Fx” has an arity of one because there’s only one thing being predicated. For example, “F” can stand for “is tall” and in that case “Fx” means “x is tall.” The statement “Gxy” has an arity of two because there’s two things being predicated. For example, “G” could stand for “loves” and in that case “Gxy” means “x loves y.” (2) In mathematics, arity refers to the number of things that are part of an operation. For example, addition requires two numbers. “1+2” is an operation with the following two variables: “1” and “2.”

argument diagram – A visual representation of an argument that makes it clear how premises are used to support a conclusion. Argument diagrams generally have numbers written in circles, and each number is used to represent a statement. Consider the following argument—“(i) Socrates is a human. (ii) All humans are mammals. (iii) All mammals are mortal. (iv) Therefore, Socrates is mortal.” An example of an argument diagram that can be used to represent this argument is the following:

argument map – A visual representation of an argument that makes it clear how premises are used to support a conclusion. Argument maps are a type of argument diagram, but the premises and conclusions are usually written in boxes. An example of an argument map is the following:

case-based reasoning – Reasoning involving the consideration of similar situations or things. For example, a doctor could consider the symptoms and cause of illness of various patients that were observed in the past in order to decide what is likely the cause of an illness of another patient who has certain symptoms. Case-based reasoning uses the following four steps for computer models: (a) Retrieve – consider similar cases. (b) Reuse – predict how the similar cases relate to the current case. (c) Revise – check to see if the similar cases relate to the current case as was predicted and make a new prediction if necessary. (d) Retain – once a prediction seems to be successful, continue to rely on that prediction until revision is necessary. Case-based reasoning is similar to “analogical reasoning.”

completeness – See “semantic completeness,” “syntactic completeness” or “expressive completeness.”

concretism – The view that possible words exist just like the actual world, and that everyone from a possible world calls their own world the “actual world.” Concretism is an attempt to explain what it means to say that something is necessary or possible—something is necessary insofar as it’s true in every possible world, and something is possible insofar as it is true in at least one possible world. It is necessary that oxygen is O₂ insofar as it’s true that oxygen is O₂ in every possible world, and it’s possible that a person can jump over a small rock insofar as it’s true in at least one possible world. “Concretism” can be contrasted with “abstractism.” See “modality” and “modal realism” for more information.

consistent logical system – A logical system with axioms and rules of inference that can’t possibly be used to prove contradictory statements from true premises.

constant – (1) See “logical constant” or “predicate constant” for more information. (2) In ordinary language, “constant” refers to staying the same.

critical reasoning – A synonym for “critical thinking.”

decidability – A question is decidable if we can determine the answer. For example, logical systems are supposed to be able to determine if arguments are valid. An argument that can’t be determined to be valid by a logical system would be “undecidable” by that logical system. Any logical system that can’t determine if an argument is valid is semantically incomplete. See “semantic completeness” for more information.

deductively complete – See “syntactic completeness.”

derivation – A formal proof of a proposition expressed in formal logic. A derivation can be described as a series of statements that are implied by rules of inference, axioms of a logical system, or other statements that have been derived by those two things. For example, a logical system could have an axiom that states “a or not-a” and have a rule of inference that states “a implies a or b.” In that case the following is a derivation—“a or not-a. Therefore, a or not-a or b.” See “axioms,” “rules of inference,” “logical system,” and “theorem” for more information.

epistemic vigilance – Attributes and mechanisms that help people avoid deception, manipulation, and confusion. For example, we intuitively tend not to trust claims that seem to be “too good to be true” from people who want to sell us something, which helps us stay vigilant against those who want to manipulate us.

existential import – The property of a proposition that implies that something exists. For example, Aristotle thought that the proposition “all animals are mammals” implied that “at least one mammal exists.” However, many logicians now argue that propositions of this type do not have existential import. See the “existential fallacy” for more information.

expressibility – The ability of a logical system to express the meaning of our statements. For example, consider the argument, “all humans are mammals; all mammals are animals; therefore, all humans are animals. According to propositional logic, this argument has the form “A; B; therefore, C” and it would determine this argument to be logically invalid as a result. However, predicate logic is better able to capture the meaning of these statements and it can prove that the argument is logically valid after all. Therefore, predicate logic is expressively superior to propositional logic given this one example. See “valid argument” for more information.

expressive completeness – A logical system is expressively complete if it can state everything it is meant to express. For example, a system of propositional logic with connectives for “and” and “not” is expressively complete insofar as it can state everything any other connective could state. You can restate “A and/or B” as “it’s not the case that both not-A and not-B.” (“Hypatia is a mammal and/or a mortal” means the same thing as “it’s not the case that Hypatia is both a non-mammal and a non-mortal.”) See “expressibility” and “logical connective” for more information.

formal semantics – A domain concerned with the interpretations of formal propositions. For example, “A and B” could be interpreted as “all lizards are reptiles and all dogs are mammals.” (“A” and “B” each represent specific propositions.) See “interpretation,” “translation,” “models,” and “schemes of abbreviation” for more information.

function – A synonym for “operation.”

good argument – An argument that’s rationally persuasive. A popular example of a good argument is “Socrates is a man. All men are mortal. Therefore, Socrates is mortal.” Good arguments give us a good reason to believe a conclusion is true. If an argument is sufficiently good, then we should believe the conclusion is true. Ideally good arguments rationally require us to believe the conclusion is true, but some arguments might only be good enough to assure us that a belief is compatible with rationality. The criteria used to determine when an argument is “good” is studied by logicians and philosophers.

grouping – In logic, grouping is used to make it clear how logical connectives relate to various propositions, and parentheses are often used. For example, “A or (B and C)” groups “B and C” together. In this case “A” can stand for “George Washington was the first president of the United States,” “B” can stand for “George Washington is a mammal,” and “C” can stand for “George Washington is a lizard.” In this case the statement can be interpreted as “George Washington is a lizard, or he is both a mammal and a lizard.” This can be contrasted with the statement “(A or B) and C,” which would be interpreted as “George Washington is either the first president of the United States or a mammal, and he’s a lizard.”

halt – (1) When a mechanical procedure ends. For example, we can use truth tables to know if arguments are valid. The procedure halts as soon as we determine whether or not the argument is valid. See “valid argument” and “truth table” for more information. (2) In ordinary language, ‘halt’ means “stop.”

index of points – A set of points. Some philosophers believe that the truth conditions of necessity and possibility are based on an index of points. Aristotle thought that something was necessary if and only if it is true at all times, and possible if and only if it is true at some time. It’s necessary that “1+1=2” because it’s true at all times, and it’s possible for a person to jump over a small rock because it’s true at some time. In this case the index of points refers to points in time. See “truth conditions” for more information.

inductive validity – A synonym for “strong argument.”

interpretation – (1) To attribute meaning to statements of a formal logical system. Formal logical statements are devoid of content, but we can add content to them in order to transform them into statements of natural language. For example, “A or B” is a statement of a formal logical system, and it can be interpreted as stating, “either evolution is true or creationism is true.” In this case “A” stands for “evolution is true” an “B” stands for “creationism is true.” See “formal semantics,” “models,” and “schemes of abbreviation” for more information. (2) In ordinary language “interpretation” refers to the act of taking statements of one language and translating them into statements of another language.

law of nature – (1) A constant predictable element of nature. For example, the law of gravity states that dropped objects will fall when dropped near the surface of the Earth. (2) A synonym for “natural law.”

lemma – A proven statement used to prove other statements.

letter – (1) A symbol used from an alphabet in symbolic logic. See “predicate letter” and “propositional letter” for more information. (2) A symbol used in an alphabet, such as “A, B, [and] C.” (3) A message written on a piece of paper for the purposes of communication over a distance.

Liebnitz’s law – The view that there can’t be two or more different entities that have the exact same properties. For example, two seemingly identical marbles are both made of different atoms and exist at different places. Two entities that have all the same properties would both have to exist at the same place at the same time. Imagine that we find out that Clark Kent was at the same place and the same time as Superman. That seems like a good reason to think that Clark Kent is Superman because Clark Kent and Superman can’t have all the same properties and be two different people.

logical constant – Symbols used in formal logic that always mean the same thing. Logical connectives and quantifiers are examples of logical constants. For example, “∧” is a logical connective that means “and.” See “logical connective” and “quantifier” for more information. “Logical constants” can be contrasted with “predicate constants.”

logical system – A system with axioms and rules of inference that can be applied to statements in order to determine if propositions are consistent, tautological, or contradictory. Additionally, logical systems are used to determine if arguments are logically valid. See “formal logic,” “axioms,” and “rules of inference” for more information.

main connective – A logical connective that’s inside the least amount of parentheses when put into a formal language. For example, consider the statement “all dogs are mammals or reptiles, and all dogs are animals.” This statement has the propositional form “(A or B) and C.” In this case “and” is the main connective. See “formal logic,” “logical connective,” and “grouping” for more information.

master table – A truth table that defines all logical connectives used by a logical system by stating every combination of truth values, and the truth value of propositions that use the logical connectives. For example, the logical connective “a ∧ b” means “a and b,” so it’s true if and only if both a and b are true. (“Hypatia is a mammal and a person” is true because she is both a mammal and a person. See “logical connective” for more information.) An example of a master table for propositional logic is the following:

maximally complete – See “syntactic completeness.”

metalogic – The study of logical systems. Logic is concerned with using logical systems to determine validity, and metalogic is concerned with determining the properties of entire logical systems. For example, a logical system can be “expressively complete.”

methodological naturalism – See “epistemic naturalism.”

modal antirealism – The view that there are no modal facts—facts concerning necessity and possibility. For example, a modal antirealist would say that it’s not a fact that it’s possible for a person to jump over a small rock.

modal realism – The view that there are modal facts—facts concerning necessity and possibility. For example, it seems like a fact that it’s possible for a person to jump over a small rock; and it seems like a fact that it’s necessary that contradictions don’t exist. See “modality,” “concretism,” and “abstractism” for more information.

moral objectivism – (1) The view that there are moral facts that are mind-independent as opposed to forms of moral realism that depend on subjective states or conventions. This form of moral objectivism requires a rejection of “moral subjectivism” and “constructivism.” See “moral realism” for more information. (2) A synonym for “moral realism.”

NAND – See “Sheffer stroke.”

natural law – (1) A theory of ethics that states that moral standards are determined by facts of nature. Consider the following two examples: One, the fact that human beings need food to live could determine that it’s wrong to prevent people from eating food. Two, the fact that people have a natural desire to care for one another could be a good reason for them to do so. (2) A theory of ethics that states that there are objective moral standards. See “moral objectivism.” (3) A theory of law that states that laws should be created because of moral considerations. For example, murder should be illegal because it’s immoral.

NOR – See “Peirce stroke.”

offensive – A synonym for “suberogation.”

omissible – Not obligatory, but permissible. For example, jumping up and down is generally considered to be permissible and non-obligatory. “Omissible” is not a synonym of “permissible” because all obligatory actions are also taken to be permissible. “Omissible” can be contrasted with “obligatory” and “permissible.”

operands – The input involved with an operation or predicate. For example, “Gxy” is a statement of predicate logic with two operands—two things being predicated. “G” can stand for “jumps over.” In that case “Gxy” means “x jumps over y” and “x” and “y” are each an operands. See “predicate logic” and “operation” for more information.

operation – Something with variables, input, and output. For example, addition is a function with two different numbers as input, and another number that’s the output. You can input 1 and 3 and the output is 4. (1+3=4.) Statements of predicate logic are also said to involve an operation insofar as predicates are taken to be operations. For example, “Fj” is a statement of predicate logic that could also be taken to be an operation. In this case “F” could stand for “is intelligent” and “j” can stand for “Jennifer.” In that case the input is “j” and the output is “Jennifer is intelligent.” See “predicate logic” and “operands” for more information.

Peirce stroke – A symbol used in formal logic to mean “neither this-nor-that” or “not-a and not-b.” (“a” and “b” are any two propositions.) The symbol used is “↓.” For example, “all dogs are lizards ↓ all dogs are fish” means that “it’s not the case that all dogs are lizards, and it’s not the case that all dogs are fish.” See “formal logic” and “logical connective” for more information.

philosophical logic – Logical domains with a strong connection to philosophical issues, such as modal logic, epistemic logic, temporal logic, and deontic logic. “Philosophical logic” can be contrasted with “philosophy of logic.”

philosophy of logic – A philosophical domain concerned with issues of logic. For example, questions involving the role of logic, the nature of logic, and the nature of critical thinking. “Philosophy of logic” can be contrasted with “philosophical logic.”

predicate constants – Constants in predicate logic are specific things that are predicated. The lower case letters “a, b, [and] c” are commonly used. For example, consider the statement, “George Washington is an animal.” In this case we can write this statement in predicate logic as “Ag” where “A” means “is an animal” and “g” stands for “George Washington.” In this case “g” is a constant because it refers to something specific that’s being predicated. Sometimes variables are used instead of constants. For example, “Ax” means “x is an animal” and “x” can be anything. See “predicate logic,” “predicate variables,” and “predicate letters” for more information. “Predicate constants” can be contrasted with “logical constants.”

predicate letters – Capital letters used in predicate logic are used as symbols for predicates, and the letters generally used are F, G,and H. For example, “F” can stand for “is tall.” In that case “Fx” is a statement that means “x is tall.” See “predicate logic,” “relation letters,” “predicate constants,” and “predicate variables” for more information.

predicate variables – Variables in predicate logic are things that are predicated without anything in particular being mentioned. The lower-case letters “x, y, [and] z” are usually used. For example, consider the statement “it is an animal.” In this case “it” is something, but nothing in particular. It could be Lassie the dog, Socrates, or something else. We could write “it is an animal” in predicate logic with a variable as “Fx.” In this case “F” means “is an animal” and “x” is the variable. See “predicate logic,” “predicate letters” and “predicate constants” for more information. “Predicate variables” can be contrasted with “propositional variables.”

propositional letter – Symbols used to stand for specific propositions in propositional logic. Upper-case letters are often used. For example, “A” can stand for “George Washington was the first President of the United States.” “Propositional letters” can be contrasted with “propositional variables.” See “propositional logic” for more information.

propositional variable – Symbols that stand for any possible proposition in propositional logic. Lower-case letters are often used. For example, “a” can stand for any possible proposition. It can stand for “A or B;” or “A and B, or C;” etc. “Propositional variables” can be contrasted with “predicate variables.” See “propositional logic” for more information.

reasoning – The thought process that leads to an inference. For example, a person who knows that all dogs are mammals and that Lassie is a dog can come to the realization that Lassie is a mammal. Reasoning that’s made explicit along with the conclusion are “arguments.” One potential difference between reasoning and arguments is that reasoning does not necessarily include the inference, but arguments must include a conclusion. Everything we say about reasoning or arguments tends to correspond to both. For example, fallacious arguments corresponds to fallacious reasoning of the same type, logically valid arguments has a corresponding logically valid reasoning, etc. Moreover, inductive and deductive types of arguments correspond to inductive and deductive types of reasoning.

reducibility – To be able to express everything from one logical system in another. For example, a propositional logical system with the logical connectives for “and” and “not” can state everything said by other logical connectives. Therefore, a system of propositional logic that has logical connectives for “not,” “and,” “and/or,” “implies,” and “if and only if” can be reduced to a system that only has connectives for “and” and “not.” See “expressibility” and “logical connectives” for more information.

relation letters – Predicate letters used in predicate logic that involve two or more things that are predicated. Capital letters are used to represent predicates, and “F, G, [and] H” are most commonly used. For example, “F” can stand for “attacks.” In that case “Fxy” is a statement that means “x attacks y.” See “predicate logic,” “predicate letters,” “predicate constants,” and “predicate variables” for more information.

rhetoric – (1) Persuasion using ordinary language. In this sense both rational argumentation and fallacious argumentation could be considered to be forms of rhetoric. Rhetoric is the specialization of public speaking, persuasion used by lawyers, and oratory used by politicians. (2) Argumentation used for the purpose of persuasion. This type of rhetoric can involve technical terminology used by specialists. This type of rhetoric is compatible with both public speaking and essays written by philosophers. (3) Persuasion that uses nonrational forms of persuasion through language. Fallacious arguments, propaganda, and various forms of manipulation could be considered to be rhetoric in this sense. This type of rhetoric is thought to be a source of power for sophists, pseudoscience advocates, snake oil salesmen, and cult leaders.

rhetorical arguments – Arguments used for persuasion. Rhetorical arguments are thought to be very important in politics and in the court room. See “rhetoric” for more information.

satisfiability – The ability to interpret a set of statements of formal logic in a way that would make them true. Statements that can all be simultaneously interpreted as true in this way are satisfiable. Consider the statement “a → b.” In this case “a” and “b” stand for any propositions and “→” stands for “implies.” We can interpret “a” as being “all bats are mammals” and “b” as being “all bats are animals.” In that case we can interpret the whole statement as saying “if all bats are mammals, then all bats are animals,” which is a true statement. Therefore, “a → b” is satisfiable. See “formal logic” and “interpretation” for more information.

semantic completeness – A logical system is semantically complete if and only if it can prove everything it is supposed to be able to prove. For example, propositional logic is semantically complete insofar as it can be used to determine whether any possible argument is valid.

semantics – The meaning of words or propositions. “Semantics” can be contrasted with “syntax.”

sentenial – Something relating to sentences. For example, propositional logic is “sentenial.”

sentential logic – A synonym for “propositional logic.”

Sheffer stroke – A symbol used in propositional logic to mean “not both” or “it’s not the case that a and b are both false.” (“a” and “b” are any two propositions.) The symbol used is generally “|” or “↑.” For example, “Dogs are mammals ↑ dogs are lizards” means that “it is not the case that dogs are both mammals and lizards.”

sound logical system – A logical system is sound if every statement it can prove using the axioms and rules of inference is a tautology (a logical truth). See “tautology” for more information.

statement letter – See “propositional letter.”

statement variable – See “propositional variable.”

suberogatory – Actions or beliefs that are inferior to alternatives (or somewhat harmful), but are permissible. Suberogatory beliefs are compatible with rational requirements or normative epistemic constraints; and suberogatory actions are inferior to alternatives (or somewhat bad), but are compatible with moral requirements. For example, being rude is not generally serious enough to be “morally wrong,” but it is suberogatory. “Suberogatory” can be contrasted with “supererogatory.”

subsentential – Something relating to parts of sentences, such as predicate logic.

subsentential logic – A synonym for “predicate logic.”

syllogism – A deductive argument. For example, “Either George Washington was a person or a dog. George Washington was not a dog. Therefore, George Washington was a person.”

syntactic completeness – A logical system is syntactically complete if it has every axiom it needs. Adding an unprovable axiom to a syntactically complete logical system will produce at least one contradiction.

syntax – The arrangement of words or symbols. Syntax can involve rules and symbol manipulation, and formal logic can be considered to mainly involve syntax for that reason. “Syntax” can be contrasted with “semantics.”

temporal interpretation of modality – The view that “necessity” and “possibility” are based on time. For something to be necessary is for it to be true of all times, and for something to be possible is for it to be true at some time. It’s necessary that dogs are mammals because dogs are mammals at all times, and it’s possible for it to rain because there is at least one time that it rains. See “modality” and “truth conditions” for more information. The “temporal interpretation of modality” can be contrasted with the “worlds interpretation of modality.”

theorem – A statement that we can know is true because of the axioms and rules of inference of a logical system. For example, consider a logical system with “a or not-a” as an axiom and the rule of inference that “a implies a or b.” The following is a proof of that system—“A or not-A. Therefore, A or not-A, or B.” In this case “A or not-A, or B” is a theorem. See “derivation,” “axioms,” “rules of inference,” and “logical system” for more information.

trans-world identity – For something to exist in multiple worlds. Some philosophers talk about there being multiple possible worlds. For example, they might say that it was possible for George Washington to become the King of the United States because there’s a possible world where he became the king. In this case we could say that George Washington has trans-world identity because he exists in multiple possible worlds. “Trans-world identity” can be contrasted to “world-bound individuals.” See “modality” and “possible world” for more information.

translation – (1) The restatement of a statement in natural language to a statement of formal logic. For example, “either George Washington was a dog or a mammal” can be translated into propositional logic as “P ∨ Q,” where “P” stands for “George Washington was a dog” and “Q” stands for “George Washington was a mammal” and “∨” is a logical connective meaning “and/or.” See “logical form” for more information. (2) In ordinary language, translation refers to a restatement of a sentence (or set of sentences). For example, the sentence “Il pleut” can be translated from French to English as “It’s raining.”

truth conditions – The conditions that make a statement true. For example, the truth condition of “the cat is on the mat” is a cat on a mat. The statement is true if and only if a cat is on a mat. Sometimes truth conditions are controversial, such as when we say it’s “necessary that people are rational animals.” It could be true if and only if people are rational animals in all times, or in all possible worlds, or perhaps given some other condition.

Turing machine – A symbol manipulation machine that follows rules in order to make movements. Such a machine can do logical and mathematical operations. Turing machines were originally hypothetical devices, but computers could be considered to be Turing machines.

Turing test – A test used to examine the ability of a machine to speak natural language within a conversation. Machines that speak natural language within conversations in exactly the same way as real human beings pass the Turing test. Any machine that passes the Turing test could be said to adequately simulate human behavior in regards to its ability to simulate a conversation in natural language. There could be tests similar to the Turing test that tests a machine’s ability to simulate other types of human behavior.

valid formula – A valid statement. A statement that is true under all interpretations. For example, “A or not-A” is true no matter what proposition “A” stands for. If “A” stands for “nothing exists,” then the statement is “either nothing exists or it’s not the case that nothing exists.” In propositional logic, “valid formula” is a synonym for “tautology.”

valid logical system – A logical system that has valid rules of inference. If a logical system is valid, then it’s impossible for true premises to be used with the rules of inference to prove a false conclusion. See “rules of inference” for more information.

variable – (1) See “propositional variable” or “predicate variable.” (2) A symbol used to represent something else, or a symbol used to represent a range of possible things. For example, “x + 3 = y” has two variables that can represent a range of possible things (“x” and “y.”)

world-bound individuals – For something to only exist in one world. Some philosophers talk about there being multiple possible worlds, but they think each person is world-bound insofar as they can only exist in one world. For example, they might say that it was possible for Thomas Jefferson to become the first President of the United States because there’s a possible world where the person in that world who most resembles Thomas Jefferson became the first President of the United States. That possible world does not contain the actual Thomas Jefferson in it. “Trans-world identity” can be contrasted with “world-bound individuals.” See “modality” and “possible world” for more information.

worlds interpretation of modality – A view of “necessity” and “possibility” based on worlds. For something to be necessary is for it to be true of all worlds, and for something to be possible is for it to be true at some world. It’s necessary that dogs are mammals because dogs are mammals at all words, and it’s possible for it to rain because there’s at least one world that it rains. The “worlds interpretation of modality” can be contrasted with the “temporal interpretation of modality.” See “possible worlds” and “modality” for more information.

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Related

• Comprehensible Philosophy Dictionary
• More Philosophy Definitions Part 1
• More Philosophy Definitions Part 2

What is sophistry?

The sophists were rhetoric teachers in Athens who lived at the same the as Socrates. They were major intellectual figures, and the term “sophist” means “wise person.” We still use the term “sophisticated” to refer to intelligent or crafty people. At that time “sophistry” was roughly equivalent to “rhetoric,” and rhetoric is the art of persuasion using language. However, the term ‘sophistry’ is now generally used to refer to manipulative forms of rhetoric.

Why sophistry was important

The reason that rhetoric was considered to be important at the time was for at least two reasons:

One, public persuasion (i.e. “oratory”) was very important for their democratic form of government—especially when concerning matters of justice. The citizens would side with experts, such as doctors, when making decisions relevant to the domain of the expert, but no one was considered to be an expert of justice. Justice was a “matter of debate.” So, they relied on arguments and debate in order to decide what should be done concerning matters of justice. Many such arguments were manipulative, just as they are now. You can watch television and see political pundits debate moral issues to see what I’m talking about.

Two, people needed to know how to argue well in case they would go to court. The Athenians settled disputes and investigated crimes similar how we do now. They went to court and sued each other. It happened a lot at the time, and people who knew how to argue well were at an advantage.

Sophistry and manipulation

The fact that rhetoric concerns persuasion means that it can be used for any situation and for any purpose. The orator, Gorgias, said that having profound oratory skills allows him to be more persuasive to non-experts than the experts are. He could convince a patient that his medical advice is superior to a doctor’s, even though he personally knows little to nothing about medical science. Sound familiar? Gorgias is referring to the sophist’s ability to be a charlatan—a false expert who people take too seriously. For example, he could trick people into buying a product for the wrong reasons, just like a snake oil salesmen. Snake oil salesmen sell medicine that doesn’t actually work, so no one should buy their product. And yet these guys have been incredibly popular for thousands of years and can make a great deal of money.

Experts can be persuasive, and rhetoric can be used by experts. The problem with rhetoric is that it is so often used for the wrong reasons. For example, by charlatans.

Consider all the unqualified charlatans who pretend to be experts, such as snake oil salesmen, new age gurus, cult leaders, astrologers, fortune tellers, spin doctors, political pundits, and conspiracy theorists. These people can make a lot of money, even though they have no relevant expertise. They are masters of the art of deception and manipulation. False philosophers, false scientists, false doctors, false political experts, and false wise people have been swindling people out of their money for the entirety of human history. There’s more of them than truly qualified philosophers, scientists, doctors, and wise people—and those who are truly qualified rarely make very much money.

Manipulative argument techniques are known as “fallacies” or “sophisms.” For example, to slander one’s opponent often causes an audience to dismiss the arguments of that opponent even though arguments are either good or bad regardless of who makes them.

Socrates saw sophistry (and rhetoric) as being manipulative. He thought we should rely on the best arguments and expertise rather than the nonrational forms of persuasion that rhetoric was often using. The terms ‘sophistry’ and ‘sophist’ are usually defined in the way Socrates saw them—sophistry as manipulation and sophists as manipulative people.

Sophistry and ethics

Many of the sophists traveled the world and realized that each society had somewhat different moral rules. This convinced many of them that morality is relative—there are no moral facts. Instead, there are merely conventional moral beliefs that people in an area will agree with. Perhaps this also reinforced the “democratic spirit” that everyone’s opinion concerning justice was equally good because there are no ethical experts.

Socrates thought that philosophers could become ethical experts, so he rejected the idea that everyone’s ethical opinion was equal. Socrates thought the sophists who used persuasive arguments in political debates were being charlatans—they were not experts, and their opinions were being taken more seriously than the actual experts.

What is philosophy?

Philosophy literally means “love of wisdom” in Greek. It refers to the attitude of those who want to know the truth and be wise rather than dogmatically hold onto their false beliefs. At one point in time philosophy referred to this general attitude being applied to all domains—science, theology, ethics, and logic were all part of philosophy. Moreover, philosophy referred to a method of rational argumentation and debate to be used in order to attain knowledge. We can then try to sincerely consider what beliefs are supported by the best arguments and evidence. We should generally believe whatever is best justified by arguments in this way. That is also how natural science works. Scientists also consider how well justified various beliefs are, and there are arguments they consider for and against scientific beliefs.

Philosophy is often confused with rhetoric, and people often think philosophers are charlatans—false experts who are taken too seriously. However, the truth is that philosophers are the closest thing we often have to experts, and yet good philosophy is rarely taken seriously by anyone. Philosophers know a lot about logic, but people are rarely interested in learning about logic. Formal logic is now used by computers, and informal logic helps us understand and identify fallacious arguments.

The domain of philosophy

The domain of philosophy is now based on what educational institutions teach in philosophy classes. Natural science is no longer considered to be part of the philosophical domain mainly becauseit’s now better taught in classes outside of the philosophy department. Now philosophy includes ethics, metaphysics (the nature of reality), epistemology (the nature of justification, rationality, and knowledge), and logic (the study of consistent, valid, and nonfallacious reasoning). This means that philosophy now mainly involves topics that are more controversial than scientific ones. However, that is not the case with logic, which is less controversial than natural science. Scientists have to rely on our best logic rather than the other way around.

Philosophy as a way of life

The term ‘philosophy’ now mainly refers to the domains of philosophical expertise rather than to general concern for wisdom and knowledge. The interest to attain knowledge is merely assumed to be a goal of philosophy, just like it’s assumed to be the goal of any other person who wants to attain expertise (such in mathematics or science). Such an interest could be considered to be part of a concept of philosophy as being a “way of life.” Additionally, philosophy as a way of life includes an interest to become a better person, to attain happiness, and to live one’s life in accordance with the best ethical expertise available. Those who live a philosophical as a way of life want to know how we should behave, and they try to behave in better ways. They also try to improve themselves by improving their skills of rationality, skills of ethics, and to learn anything that will help them become more ethical.

Are philosophers experts?

Although we know that philosophers want to know something about various things that doesn’t mean they ever do know something about anything at all. However, I think they do know quite a bit. Consider the following:

• Philosophers who later became known as “scientists” seem to clearly know something. We seem to know a lot about the laws of nature and how we can apply our knowledge to technological achievements.
• Philosophers who are experts of logic (also known as “logicians”) seem to clearly know something. We now know how to use formal logic to create computers and knowledge of informal logic is used by scientists in order to know which beliefs are best supported by arguments. For example, we know that an insult against a person is not a good reason to reject the person’s argument.
• We do seem to know something about epistemology. For example, we know that we shouldn’t form beliefs based on fallacious arguments. If we find out that an argument is fallacious, then we should reject the argument. Epistemologists can study hundreds of arguments concerning epistemology in order to have the most informed epistemological beliefs possible.
• We do seem to know something about ethics. For example, we seem to know that killing all the people we can is the wrong thing to do. Ethicists can study hundreds of arguments concerning ethics in order to have the most informed ethical beliefs possible.
• We do seem to know something about metaphysics. For example, we seem to know that other people exist and they have minds of their own. Metaphysicians can study hundreds of arguments concerning metaphysics in order to have the most informed metaphysical beliefs possible.

Finally, having informed beliefs are better than uninformed ones. We often form beliefs for the wrong reasons and being informed gives us an opportunity to realize which beliefs are the best supported by arguments.

Conclusion

Experts can persuade others using language. That means experts can use “rhetoric.” However, rhetoric alone does not actually offer us the expertise we need, and it’s often used in manipulative ways. We can call that use of rhetoric “sophistry.” We have a good reason to think that there are philosophers who are not sophists—philosophers who genuinely want to be experts. Finally, we also have a good reason to think that some philosophers are experts.

Against my better judgment I went ahead and read part 1. As I expected, it was pretty much the same drivel I put up with in Part 2. It was a waste of time. Since he is obviously unable to defend his assertions, continuing the email exchange would also be a waste of time.

I suspect that Neo does not understand what “logically sound” means in this context, and I had a conversation with someone else who had the same problem. Dictionary definitions are not appropriate for this conversation. I defined “logically sound” in Part 1. I will say more about that in what follows.

The conversation continues.

I will now respond to what he said last. Keep in mind that he said all of these things at once before I responded to him here. I will quote several things he says and reply to him below:

James, your reply ignored my question: how could any argument be “meant” to be unsound? And other than you, who made that claim?

My reply: Inductive arguments are not meant to be sound. That is true by definition. You learn that in a logic class. See “Deductive and Inductive Arguments” at the Internet Encyclopedia of Philosophy.

A sound argument is a logically valid argument with true premises. Logically valid arguments have an argument form that couldn’t possibly have true premises and a false conclusion at the same time. For example:

1. All dogs are lizards.
2. If all dogs are lizards, then all dogs are reptiles.
3. Therefore, all dogs are reptiles.

The first premise is false and the conclusion is false. It doesn’t matter—it’s still a valid argument.

This is simply the definition of “valid” and “sound.” You learn this in a logic class. See “Validity and Soundness” at the Internet Encyclopedia of Philosophy.

James, good scientific theories are supported by evidence not arguments.

My reply: Yes, they are supported by arguments. Good scientific theories are conclusions—they should be believed because they are highly justified. Sufficiently good arguments provide us with conclusions that we should accept based on our justifications. An argument is the reasoning process made explicit. There is inductive reasoning and deductive reasoning. Inductive arguments and deductive arguments correspond to these two types of reasoning.

Let’s say a scientist predicts that gravity will cause objects to that are dropped fall at a certain speed at an accelerated rate (in a vacuum while at a certain distance from the center of the Earth). Scientists can then check that prediction using experiments. If objects always fall at that speed in experimental conditions, then the scientists can have the following argument:

1. Dropped objects always fell at a certain speed at an accelerated rate in our experiments.
2. The belief that dropped objects always fall at a certain speed at an accelerated rate is consistent with all observations we have.
3. Therefore, probably all dropped objects will fall at a certain speed at an accelerated rate.

This is an inductive argument. Inductive arguments take observations and make predictions (or generalize) based on those observations. Inductive arguments can only tell us what’s probably true and there’s always a chance they have false conclusions.

For example, at one point many people thought that all swans were white because they only saw white swans. Even so, a black swan was eventually found. That proved the conclusion to be false.

In the first email James had written:

However, not all good deductive arguments are logically sound either.

To which I replied:
“If they aren’t, then I would be even more interested in exactly
how you are defining ‘good’.”

Now James responds:
A good reason to believe something, as stated earlier.

James, why would I have good reason to believe a deductive argument that wasn’t logically sound?

My reply: Because the premises of the deductive argument can be justified without necessarily being true. Sound arguments have to have true premises. For example, arguments can use the highly justified conclusions of science as premises.

Consider the following argument:

1. Objects that are dropped fall at a certain speed at an accelerated rate.
2. If objects that are dropped fall at a certain speed at an accelerated rate, then dropping my glasses will cause them to fall at a certain speed at an accelerated rate.
3. Therefore, dropping my glasses will cause them to fall at a certain speed at an accelerated rate.

This is a deductive argument that uses a conclusion attained by science as a premise—that objects that are dropped fall at a certain speed at an accelerated rate. This premise is probably true, but it could be false. Perhaps no objects will fall in the future at all. That seems unlikely, but it’s perfectly reasonable to believe the glasses will fall as predicted.

Now from James’ first email where he had stated:

There is a sense that deductive arguments would ideally be logically sound, but some deductive arguments have sufficiently justified premises, even if those premises aren’t known to be true for certain.

To which I had replied:
“Only religious nut jobs believe in absolute certainty. Since you are
not one of them, why do you even bother mentioning a concept
that is unknowable (absolute certainty)?”

Now James responds:
Because it has to be said to prove what I want to prove.

My reply: To require that good arguments are sound is to require the premises of good arguments to be known for absolutely certain. Sound arguments have to have true premises. That’s true by definition. It’s what “sound” means in this context.

James, how does resorting to an imaginary concept prove what you want to prove?

[James replied: If we can't know things for certain, then we can't depend on all our arguments being sound.]

James, name one thing that you know “for certain.”

We can know things only to a degree of certainty, as your earlier statement demonstrated when you correctly stated that science is often proved wrong; yet we can depend on our arguments being sound if we rigorously follow the rules of logic.

So I disagree.

My reply: You are wrong. The rules of logic can’t guarantee that our premises are true, but sound arguments have to have true premises.

If you don’t rely on sound arguments to form your beliefs, what do you rely on?

[James replied: We rely on the best information available and we make predictions.]

James, why can’t that “best information available” be a sound argument?

[James replied: Because our best scientific conclusions are not necessarily sound but we still have a good reason to believe them.]

James, name one.

That will require that you produce a scientific conclusion that is either 1) invalid, or 2) contains a false premise.

My reply: A conclusion can’t be invalid or contain a false premise. You mean that the argument must be invalid or contain a false premises. Inductive arguments are technically invalid. The truth of the premises of inductive arguments does not guarantee that the conclusions are true. I already gave examples.

James now quotes me from email #1:

“When your argument has to resort to counterfactuals, you should
realize that your argument is in deep trouble.”
I disagree. The point is we don’t rely on it being sound.

James, why would you rely on something that wasn’t sound?

[James replied: We rely on inductive evidence instead.]

James, evidence is neither inductive nor deductive – reasoning is.

I found one web site where that term was used … incorrectly.

The author defined observations as inductive evidence showing that he doesn’t understand the meaning of induction.

My reply: Yes, I do understand the meaning. If you want to prove otherwise, go ahead. See “Deductive and Inductive Arguments” at the Internet Encyclopedia of Philosophy.

Update (9/9/2012): I made a few clarifications and rephrased some of what I had to say. Our responses were also mixed up a bit and had to be separated.

Update (9/12/2012): I explained that I am responding to something he wrote all at once.

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1. Logic has nothing to do with everyday life.
2. Logic is subjective.
3. Logic can’t help improve our thinking.
4. I am a critical thinker, but other people should learn about it.
5. Arguments are hostile disputes.
6. Some arguments are true, and some are false.
7. All good arguments are logically valid.
8. All good arguments have true premises.
9. If a premise is false, then the conclusion is false.
10. We know about good argumentation from observation.

Once more—each of these assertions are false.

1. Logic has nothing to do with everyday life.

Logic is a philosophical domain concerned with argumentation, reasoning, errors in reasoning, and proper argument form. Many people think logic is too abstract to have any relation to their life. However, logic has a lot to do with everyone’s life.

First, logic as it was developed by Aristotle, the Stoic philosophers, and other philosophers is now being used by computers (and everything else that uses computers, such as cell phones).

Second, understanding logic helps us become more reasonable. In particular, it helps us better understand good argumentation, and what we should believe. One popular argumentative tactic is to insult those who disagree with us in order to take what they say less seriously. We all know why that’s not a good argument. However, there are more subtle ways that arguments can fail us. For example, we should not form beliefs based on logically invalid deductive arguments. Such an invalid deductive argument is the following:

1. If dogs are mammals, then dogs are animals.
2. Dogs are animals.
3. Therefore, dogs are mammals.

The problem is that the first two statements (the premises) are insufficient to support the conclusion. This argument has an invalid argument form—“If A, then B. B. Therefore A.” A counterexample to this argument form is “If dogs are reptiles, then dogs are animals. Dogs are animals. Therefore, dogs are reptiles.”

2. Logic is subjective.

Some people conceive of logic as what makes things “sound good.” Whatever you believe “sounds good” to you, but other beliefs “sounds good” to other people. However, this is a complete misunderstanding about what logic is about. It’s not about what “sounds good.” There is rational criteria involved that determines when an argument is rational, when a conclusion is likely true, and when we should believe something is true.

3. Logic can’t help improve our thinking.

It’s hard to say why anyone would think logic can’t help improve our thinking, but many people do seem to think that. Perhaps they only think that because they think logic is subjective. However, there is data that indicates that logic can improve our critical thinking skills.

First, a study provides evidence that an applied logic class significantly helped improve the critical thinking skills of high school students.¹

Second, a meta-analysis by Claudia María Álvarez Ortiz compares how well several different classes improve the critical thinking of students, and various classes were found to be effective:

1. Critical thinking classes are effective, which generally focus on “informal logic.”
2. Critical thinking classes involving lots of argument mapping practice were the most effective.
3. Other logic classes (that focus more on “formal logic”) were also found to be effective.

(See Does Philosophy Improve Critical Thinking Skills? [2007, PDF] for more information.)

Third, a study by Richard Arum has found that certain college classes were better at improving critical thinking than others. “Students who majored in the traditional liberal arts—including the social sciences, humanities, natural sciences and mathematics—showed significantly greater gains over time than other students in critical thinking, complex reasoning and writing skills.”²

4. I am a critical thinker, but other people should learn about it.

Many people think that logic is important, but they don’t think they need to take a class on logic (or spend much time educating themselves about logic). However, the fact is that even a college education does not guarantee that you know very much about logic or critical thinking. It is important to study these subjects to understand them as well as possible.

First, Richard Arum’s study found that “[f]orty-five percent of students made no significant improvement in their critical thinking, reasoning or writing skills during the first two years of college… After four years, 36 percent showed no significant gains in these so-called “higher order” thinking skills.”³

Second, most teachers know very little about critical thinking, even though they say that they think it’s important. “Though the overwhelming majority (89%) claimed critical thinking to be a primary objective of their instruction, only a small minority (19%) could give a clear explanation of what critical thinking is. Furthermore, according to their answers, only 9% of the respondents were clearly teaching for critical thinking on a typical day in class.”⁴

5. Arguments are hostile disputes.

Perhaps one of the largest barriers to an interest in logic is the view of arguments as hostile disputes. Sometimes we talk about “arguments” when we are talking about shouting matches or some other disrespectful exchange of words. That is not what logical arguments are. Logical arguments are reasons given to believe something involving premises and a conclusion. The premises are supposed to be a reason to believe the conclusion is true. Some arguments are very good reasons to believe the conclusions to be true, and it can be perfectly respectful to tell someone what you think is true and why by using an argument.

The word “debate” is similarly equated with “hostile dispute” when in reality a debate involving logical arguments can be highly educational and perhaps even indispensable to improving ourselves in certain ways. In particular, debates have been shown to help us find out when our reasoning process involves mistakes.

6. Some arguments are true, and some are false.

Arguments can be good, valid, sound, strong, or cogent. But they can’t be true or false. Consider the following good argument:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

In this case the premises and the conclusion are all true. It is unclear why anyone would think the argument itself could be true or false.

7. All good arguments are logically valid.

Logically valid arguments have conclusions that must be true if we assume the premises are true. For example:

1. All lizards are mammals.
2. If all lizards are mammals, then all lizards are animals.
3. Therefore, all lizards are animals.

The premises are false and the conclusion is true, but that’s not the point. We can imagine that the premises are true and that scenario would force us to imagine that the conclusion is also true. The premises are meant to be a reason to believe the conclusion precisely because true premises would be a good reason to think the conclusion is also true.

Invalid deductive arguments don’t give us a reason to believe the conclusion is true. For example:

1. All men are mortal.
2. Socrates is a mortal.
3. Therefore, Socrates is a man.

If we knew the premises are true, that would not mean that we know the conclusion is true. We know all dogs are mortal and Socrates is a mortal, but that doesn’t mean Socrates is a dog.

All good deductive arguments are valid, but not all good inductive arguments are valid. Deductive arguments are supposed to be valid by definition—that’s how we define deductive arguments. However, inductive arguments are not supposed to be valid. That’s how we define them. An example of a good inductive argument is the following:

1. The Sun has risen every day of human history.
2. Therefore, the Sun will probably rise tomorrow.

This argument is technically logically invalid, but it’s not supposed to be logically valid. Inductive arguments make predictions and generalizations based on limited information, and the predictions and generalizations can be wrong. They provide us with hypotheses and theories that can be falsified. Even so, good inductive arguments still give us a good reason to believe the conclusions to be true.

8. All good arguments have true premises.

We should try to have true premises for our arguments, but there is rarely a way to guarantee that any of our premises are true. The best we can hope for is to use premises that are highly justified—premises that are probably true based on our limited information. For example, consider the example of an inductive argument above. We have a very good reason to believe the premise is true, but maybe it’s not. For example, the whole universe might have been created an hour ago and we might all have false memories. In that case the Sun did not rise every day of human history after all.

9. If a premise is false, then the conclusion is false.

Many people refute the argument given by someone else by proving a premise to be false, and seem to assume that the conclusion must also be false as a result. However, an argument can have a false premise and a true conclusion. Consider the following argument:

1. If more than two people exist, then “1+1=3.”
2. 1+1=3
3. Therefore, more than two people exist.

This is a horrible argument. Both premises are false. Even so, the conclusion is true.

10. We know about good argumentation from observation.

Many people think that natural science is the only way to justify our beliefs—they think that we know everything we know because of natural science. However, natural science has never discovered that “1+1=2,” or that contradictions are impossible, or that a good argument must have justified premises. In fact, natural science has never discovered anything about logic. We know a lot about logic, and we know a lot more about logic now than we did two thousand years ago. Even so, we didn’t learn about logic through natural science.

Evidence that Critical Thinking Educators Don’t Understand Logic

Critical thinking traditionally focuses on “informal logic,” so the fact that many critical thinking educators don’t understand logic as well as they should is a problem. I applaud many critical thinking educators who share valuable materials online to give everyone a chance to learn about it, but their false assertions are apparent to experts in the field. The following webpages are created by critical thinking educators that believe in at least one of the myths that were discussed above:

• Good Arguments (Joe Lau & Jonathan Chan falsely state that good arguments must have true premises, and that good deductive arguments must be logically sound. A sound argument must be valid and have true premises. However, good arguments don’t need true premises. They both have PhD’s in philosophy, so this error is more disappointing than usual.)
• Field Guide to Critical Thinking (James Lett falsely asserts that arguments must be logically sound. Arguments need not have true premises and they need not be valid.)
• Purdue University: Using Logic (Ryan Weber & Allen Brizee falsely assert that a syllogism is false. A syllogism is a type of argument, and arguments can’t be false. See Example D.)

Related

Some related blog posts I’ve written:

• Not all Good Arguments Are Logically Sound
• Why Critical Thinking Is Important
• What You Need From Formal Logic
• What Are Good Arguments?

Free introductions to logic:

• Introduction to Critical Thinking & Argument Mapping (A full introduction with a focus on informal logic.)
• Introduction to Formal Logic by Stefan Waner and Steven R. Costenoble (A full introduction to formal logic.)

Notes

A-type proposition – A proposition with the form “all a are b.” For example, “all cats are animals.”

addition – A rule of inference that states that we can use “a” as a premise to validly conclude “a or b.” For example, “Dogs are mammals. Therefore, Dogs are mammals or lizards.”

affirmative categorical proposition – A categorical proposition that has the form “all a are b” or “some a are b.” For example, “all dogs are mammals.”

affirmative premise – A categorical proposition used as a premise that has form “all a are b” or “some a are b.” For example, “some mammals are dogs.”

affirming the disjunct – A fallacy committed by an argument that requires us to mistakenly assume two propositions to be mutually exclusive and reject one proposition just because the other is true. The argument form of an argument that commits this fallacy is “Either a or b. a. Therefore, not-b.” For example, consider the following argument—“Either Dogs are mammals or animals. Dogs are mammals. Therefore, dogs are not animals.”

agent causation – A type of causation that’s neither determined nor random produced from people. Agent causation occurs from an action caused by a person that’s not caused by other events or states of affairs. For example, it’s not caused by the reasoning of the agent. See “prime mover” and “libertarian free will” for more information.

alternate possibilities – Events that could happen in the future or could have happened in the past instead of what actually happened. Alternate possibilities are often mentioned to refer to the ability to do otherwise. For example, some people think free will and moral responsibility require alternate possibilities. Let’s assume that’s the case. If Elizabeth is morally responsible for killing George, then she had an alternate possibility of not killing George. If she was forced to kill George, then she isn’t morally responsible for doing it. Alternate possibilities are often thought to be incompatible with determinism.

appeal to probability – A fallacy committed by an argument that concludes that something will happen just because it might happen. For example, “It’s possible to make a profit by gambling. Therefore, I will eventually make a profit if I keep playing the slot machines.”

appeal to consequences – A type of fallacy committed by arguments that conclude that something is true or false based on the effects the belief will have. For example, “We know it’s true that every poor person can become rich because poor people who believe they can become rich are more likely to become rich.”

appeal to force – A fallacious form of persuasion that is committed when coercion is used to get people to pretend to agree with a conclusion, or in order to suppress opposing viewpoints. The appeal to force can be subtly used in an academic setting when certain views are taboo and could harm a person’s future employment opportunities. However, sometimes people also fear being punished for expressing their “heretical views.” For example, John Adams passed the Sedition Act, which imposed fines and jail penalties to anyone who spoke out against the government. Additionally, there was a time when various heresies (taboo religious beliefs) were punishable by death.

Aristotelian ethics – An ethical system primarily concerned with virtue. Aristotle believes that (a) people have a proper function as political rational animals to help each other and use their ability to reason; (b) happiness is the greatest good worth achieving (c) virtues are generally between two extremes; and (d) virtuous people have character traits that cause them to enjoy doing what’s virtuous and do what’s good thoughtlessly. For example, courage is virtuous because it is neither cowardly nor foolhardy, and courageous people will be willing to risk their life whenever they should do so without a second thought.

Argument from fallacy – See “argumentum ad logicam.”

argument indicator – A term used to help people identify that an argument is being presented. Argument indicators are premise indicators or conclusion indicators. For example, ‘because’ is an argument indicator used to state a premise. See “argument” for more information.

argumentum ad baculum – Latin for “argument from the stick.” See “appeal to force.”

argumentum ad consequentiam – See “appeal to consequences.”

argumentum ad logicam– A type of fallacy committed by an argument that claims that a conclusion of an argument is false or unjustified just because the argument given in support of the conclusion is fallacious. A conclusion can be true and justified, even if people give fallacious arguments for it. For example, Tom could argue that “the Earth exists because Tina is evil.” This argument is clearly fallacious, but the conclusion (that the Earth exists) is both true and justified.

association – A rule of replacement that takes two forms: (a) “a and/or (b and/or c)” means the same thing as “(a and/or b) and/or c.” (b) “a and (b and c)” means the same thing as “(a and b) and c.” (“a,” “b,” and “c” stand for any three propositions.) The parentheses are used to group certain statements together. For example, “dogs are mammals, or they’re fish or reptiles” means the same thing as “dogs are mammals or fish, or they’re reptiles.” The rule of association says that we can replace either of these statements of our argument with the other precisely because they mean the same thing.

association fallacy – A type of fallacy committed by an argument with an unwarranted assumption that two things share a negative quality just because of some irrelevant association. For example, we could argue that eating food is immoral just because Stalin ate food. Also see the “halo effect” and “ad hominem” for more information.

bad company fallacy – See “association fallacy.”

bad reasons fallacy – See “argumentum ad logicam.”

base rate fallacy – A fallacy committed by an argument that makes a statistical error based on information about a state of affairs. For example, we might assume that a test used to detect a disease that’s 99% accurate will correctly detect that more people have a disease than it will falsely claim have the disease. However, if only 0.1% of the population has the disease, then it will falsely detect around ten times as many people as having the disease than actually have it. See “false positive” for more information.

base rate information – Information about a state of affairs that is used for diagnosis or statistical analysis. For example, we might find out that 70% of all people with a cough and runny nose have a cold. A doctor is likely to suspect a patient with a cough and runny nose has a cold in consideration of how common colds are. “Base rate information” can be contrasted with “generic information” concerning the frequency of a state of affairs, such as how common a certain disease is.

biased sample – (1) A sample that is not representative of the group it is meant to represent for the purposes of a study. For example, a poll taken in an area known to mainly vote for republican politicians that proves that the republican presidential candidate is popular with the population at large. It might be the case that the republican candidate is not popular when all other voters are accounted for, and the sample is so biased that we can’t use it to have any idea about whether or not the republican candidate is truly popular with the population at large. Also see “selective evidence” and “hasty generalization” for more information. (2) A fallacy committed by an argument based on a biased sample. For example, to conclude that a republican presidential candidate is popular with the population at large based on a poll taken in a pro-republican area.

bifurcation fallacy – See “false dilemma.”

black or white fallacy – See “false dilemma.”

booby trap – (1) A logical booby trap is a peculiarity of language that makes it likely for people to become confused or to jump to the wrong conclusion. For example, an ambiguous word or statement could make it likely for people to equivocate words in a fallacious way. Some people think all forms of debate are attempts at manipulative persuasion, but there are rational and respectful forms of debate. See “equivocation” for more information. (2) In ordinary language, a booby trap is a hidden mechanism used to cause harm once it is triggered by a certain action or movement. For example, Indiana Jones lifted an artifact from a platform that caused the room to collapse.

borderline case – A state of affairs that can be properly described by a vague term, but it is difficult to say how the vague term can be properly applied. For example, it might not be clear whether or not it’s unhealthy to eat a small bag of potato chips is unhealthy. Even so, we know that eating one potato chip is not unhealthy, and eating a thousand potato chips is unhealthy. See “vague” for more information.

character ethics – See “virtue ethics.”

commutation – A rule of replacement that states that “a and b” and “b and a” both mean the same thing. (“a” and “b” stand for any two propositions.) For example, we know that “all dogs are animals and all cats are animals” means the same thing as “all cats are animals and all dogs are animals.” If we use one of these statements in an argument, then we can replace it with the other statement.

commutation of conditionals – A fallacy committed by arguments that have the logical form “if a, then b; therefore if b, then a.” (“a” and “b” stand for any two propositions.) For example, “If all snakes are reptiles, then all snakes are animals. Therefore, if all snakes are animals, then all snakes are reptiles.”

commutative – To be able to switch symbols without a loss of meaning. “a and b” has the same meaning as “b and a.” For example, “dogs are animals and lizards are reptiles” has the same meaning as “lizards are reptiles and dogs are animals.”

compound proposition – A proposition that can be broken into two or more propositions. For example, “Socrates is a man and he is mortal” can be broken into the following two sentences: (a) Socrates is a man. (b) Socrates is mortal. “Compound propositions” can be contrasted to “non-compound propositions.”

compound sentence – See “compound proposition.”

composition – (1) In logic, ‘composition’ refers to the “fallacy of composition.” (2) When a creditor agrees to accept a partial payment for a debt. (3) The arrangement of elements found in a work of art. (4) Producing a literary work, such as a text or speech.

conclusion indicator – A term used to help people identify that an conclusion is being stated. For example, “therefore” or “thus.” See “conclusion” for more information.

conditional proof – A strategy used in natural deduction used to prove an argument form is logically valid that has an if/then proposition as a conclusion. We know the argument form is valid if we can assume the premises are true and the first part of the conclusion is true in order to deduce the second part of the conclusion. For example, consider the argument “If A, then B. If B, then C. Therefore, if A, then C.” We can use the following conditional proof to know this argument is valid:

1. We can assume the first part of the conclusion (“A”) and prove the second part of the conclusion (“C”).
2. We know “if A, then B” is true and “A” is true, so we know “B” is true. (See “modus ponens.”)
3. We know “if B, then C” is true and “B” is true, so we know “C” is true. (See “modus ponens.”)
4. We have now deduced that the second part of the conclusion is true, so the argument form is logically valid.

conjunction – A rule of inference that states that we can use “a” and “b” as premises to validly conclude “a and b.” (“a” and “b” stand for any two propositions.) For example, “Socrates is a man, and Socrates is mortal. Therefore, Socrates is a man and he is mortal.”

constructive dilemma – A rule of inference that states that we can use the premises “a and/or b,” “if a, then c,” and “if b, then d” to validly conclude “c and/or d.” (“a”, “b,” and “c” stand for any three propositions.) For example, “Either all dogs are mammals and/or all dogs are lizards. If all dogs are mammals, then all dogs are animals. If all dogs are lizards, then all dogs are reptiles. Therefore, all dogs are animals and/or reptiles.”

contradiction – In categorical logic, contradiction is a process of negating a categorical statement and expressing it as a different categorical form. For example, “all men are mortal” can be contradicted as “some men are not mortal.”

contradictory – In categorical logic, a contradictory is the negation of a categorical statement expressed in a different categorical form. For example, “no men are immortal” is the contradictory of “some men are immortal.”

contraposition – (1) To switch the terms of a categorical statement and negate them both. There are two valid types of categorical contraposition: (a) “All a are b” means the same thing as “all non-b are non-a.” (b) “Some a are not b” means the same thing as “some non-b are not non-a.” For example, the following argument is valid—“Some snakes are not mammals. Therefore, some non-mammals are not non-snakes.” (2) To infer a contrapositive from a categorical proposition. See “contrapositive” for more information. (3) In modern logic, it is also known as “transposition.”

contrapositive – A categorical proposition is the contrapositive of another categorical proposition when the terms are negated and switched. For example, the contrapositive of “all mammals are animals” is “All non-animals are non-mammals.” It is valid to infer the contrapositive of two different types of categorical propositions because they both mean the same thing: (a) “All a are b” means the same thing as “all non-b are non-a.” (b) “Some a are not b” means the same thing as “some non-b are not non-a.” For example, “some people are not doctors” means the same thing as “some non-doctors are not non-people.”

contrary propositions – Propositions that are mutually exclusive. For example, “Socrates is a man” and “Socrates is a dog” are contrary propositions.

converse – A categorical proposition or if/then statement with the two parts switched. The converse of “all a are b” is “all b are a.” (“a” and “b” are any two terms.) The converse of “if c, then d” is “if d, then c.” (“c” and “d” are any two propositions.) For example, the converse of “if all fish are animals, then all fish are organisms” is “if all fish are organisms, then all fish are animals.” It is valid to infer the converse of any categorical statement with the form “no a are b” or “some a are b.” See “conversion” for more information.

conversion – To switch the terms of a categorical statement. There are two valid types of conversion: (a) “No a are b” means the same thing as “no b are a.” (b) “Some a are b” means the same thing as “some b are a.” For example, the following is a valid argument—“No birds are dogs. Therefore, no dogs are birds.”

definiendum – The term that is defined by a definition. Consider the definition of “argument” as “one or more premises that supports a conclusion.” In this case the definiendum is “argument.” “Definiendum” can be contrasted with “definiens.”

definiens – The definition of a term. Consider the definition of “premise” as “a proposition used to give us reason to believe a conclusion.” In this case the definiens is “a proposition used to give us reason to believe a conclusion.” “Definiens” can be contrasted to “definiendum.”

DeMorgan’s laws – A rule of replacement that takes two forms: (a) “It’s not the case that both a-and-b” means the same thing as “not-a and/or not-b.” (b) “It’s not the case that a and/or b” means the same thing as “not-a and not-b.” (“a” and “b” stand for any two propositions.) For example, “it’s not the case that dogs are either cats or lizards” means the same thing as “no dogs are cats, and no dogs are lizards.”

denying a conjunct – A logical fallacy committed by arguments with the following form—“It’s not the case that both a-and-b. Not-a. Therefore, b.” This argument form is logically invalid. For example, “Socrates isn’t both a dog and a person. Socrates isn’t a dog. Therefore, Socrates isn’t a person.”

distribution – A rule of replacement that takes two forms: (a) “a and (b and/or c)” means the same thing as “(a and/or b) and (a and/or c).” (b) “a and/or (b and c)” means the same thing as “(a and b) and/or (a and c).” (“a”, “b,” and “c” stand for any three propositions.) For example, “all lizards are reptiles, and all lizards are either animals or living organisms” means the same thing as “either all lizards are reptiles or animals, and either all lizards are reptiles or living organisms.”

division – (1) See “fallacy of division.” (2) A mathematical operation based on a ratio or fraction. For example “4 ÷ 2 = 2.” (3) To split objects into smaller parts.

doctrine of the maturity of chances – (1) The false assumption that the past results of a random game will influence the future results of the game. For example, a person who loses at black jack five times in a row might think that she is more likely to win if she plays another game. (2) See the “gambler’s fallacy.”

double negation – (1) A rule of replacement that states that “a” and “not-not-a” both mean the same thing.(“a” stands for any proposition.) For example, “Socrates is a man” means the same thing as “it’s not the case that Socrates isn’t a man.” (2) A “double negative.” When it’s said that something isn’t the case twice. For example, “it’s not the case that Mike didn’t turn the TV on means the same thing as “Mike turned the TV on.”

E-type proposition – A proposition with the form “no a are are b.” For example, “no cats are reptiles.”

enthymeme – (1) A categorical syllogism with an unstated premise. For example, “all acts of abortion are immoral because all fetuses are persons.” In this case the missing premise could be “all acts of killing people are immoral.” (2) Any argument with an unstated premise or conclusion. For example, “all fetuses are people and all acts of killing people are immoral” has the unstated conclusion “all acts of abortion are immoral.”

epistemic modality – The distinction between what is believed and what is known. Moreover, epistemic modality can involve the degree of confidence a belief warrants. For example, we know that more than three people exist and we are highly confident that this belief is true. We communicate epistemic modality through terms and phrases, such as “probably true,” “rational to believe,” “certain that,” “doubt that,” etc.

epistemic randomness – When something happens that is not reliably predictable. For example, when we roll a six-sided die, we don’t know what number will come up. We say that dice are good for attaining random results for this reason.

equivalence – A rule of replacement that takes two forms: (a) “a if and only if b” means the same thing as “if a, then b; and if b, then a.” (b) “a if and only if b” means the same thing as “a and b” and/or “not-a and not-b.” (“a” and “b” stand for any two propositions.) For example, “Socrates is a rational animal if and only if Socrates is a person” means the same thing as “Socrates is a rational animal and a person, or Socrates is not a rational animal and not a person.”

etymological fallacy – A fallacy committed by an argument when a word is equivocated with another word it’s historically derived from. For example, “logic” is derived from “logos,” which literally meant “word.” It would be fallacious to argue that “logic” is the study of words just because it is historically based on “logos.”

exclusive premises – A fallacy committed by categorical syllogisms that have two negative premises. There are no logically valid categorical syllogisms with two negative premises. For example, “No dogs are fish. Some fish are not lizards. Therefore, no dogs are lizards.”

existential quantifier – A term or symbol used to say that something exists. For example, “some” or “not all” are existential quantifiers in ordinary language. “Some horses are mammals” means that at least one horse exists and “not all horses are male” means that there is at least one horse that is not a male. The existential quantifier in symbolic logic is “∃.” See “quantifier” for more information.

exportation – A rule of replacement that states that “if a and/or b, then c” means the same thing as “if a, then it’s the case that if b, then c.” (“a”, “b,” and “c” stand for any three propositions.) For example, “if Socrates is either a mammal or an animal, then Socrates is a living organism” means the same thing as “if Socrates is a mammal, then it’s the case that if Socrates is an animal, then Socrates is a living organism.”

false conversion – See “illicit conversion.”

fallacy fallacy – (1) See “argumentum ad logicam.” (2) A type of fallacy committed by an argument that falsely claims another argument commits a certain fallacy. For example, Lisa could argue that “Sam is an idiot for thinking that only two people exist. We have met many more people than that.” Sam could then respond, “You have committed the ad hominem fallacy. My belief should not be dismissed, even if I am an idiot.” In this case Lisa’s argument does not require us to believe that Sam is an idiot. It is an insult, but it can be separated from her actual argument.

fallacy of composition – A fallacy committed by an argument that falsely assumes that a whole will have the same property as a part. For example, “molecules are invisible to the naked eye. We are made of molecules. Therefore, we are invisible to the naked eye.” The “fallacy of composition” is often contrasted with the “fallacy of division.”

fallacy of the consequent – See “affirming the consequent.”

fallacy of division – A fallacy committed by an argument that falsely assumes that a property that a whole has will also be a property of the parts. For example, “We can see humans with the naked eye. Humans are made of molecules. Therefore, we can see molecules with the naked eye.”

false positive – A positive result that gives misleading information. For example, to test positive for having a disease when you don’t have a disease. Let’s assume that 1 of 1000 people have Disease A. If a test is used to detect Disease A and it’s 99% accurate, then it will probably detect that the one person has the disease, but it will also probably have around ten false-positive results—it will probably state that ten people have Disease A that don’t actually have it.

false precision – See “overprecision.”

formal language – Languages that are devoid of semantics, such as the languages used for formal logic. See “formal logic” and “formal system” for more information. “Formal language” can be contrasted with “natural language.”

gambler’s fallacy – Fallacious reasoning based on the assumption that the past results of a random game will influence the future results of the game. For example, if you toss a coin and get heads twice in a row and conclude that you are more likely to get tails if you keep playing. Gamblers who lose a lot of money often have this assumption when they make the mistake in thinking that they will more likely start winning if they keep playing the same game.

golden mean – Aristotle’s concept of virtues as being somewhere between two extremes. For example, moderation is the character trait of wanting the right amount of each thing, and it’s between the extremes of gluttony and an extreme lack of concern for attaining pleasure.

golden rule – A moral rule that states that we ought to treat other people how we want to be treated. For example, we generally shouldn’t punch other people just because they make us angry insofar as we wouldn’t want them to do it either.

grandfather’s axe – A thought experiment of axe with all the parts that have been replaced. The question is whether or not it’s the same axe.

guilt by association – See “association fallacy.”

hidden assumption – An assumption of an argument that is not explicitly stated, but is implied or required for the argument to be rationally persuasive. For example, consider the argument “the death penalty is immoral because it kills people.” This argument requires a hidden assumption—perhaps that “it’s always immoral to kill people.”

hot hand fallacy – An argument commits this fallacy when it requires the false assumption that good or bad luck will last a while. For example, a gambler who wins several games of poker in a row is likely to think she’s on a “winning streak” and is more likely than usual to keep winning as a result.

hypothetical syllogism – A rule of inference that states that we can use “if a, then b” and “if b, then c” to validly conclude “if a, then c.” (“a” and “b” stand for any two propositions.) For example, “if all dogs are mammals, then all dogs are animals. If all dogs are animals, then all dogs are living organisms. Therefore, if all dogs are mammals, then all dogs are living organisms.”

I-type proposition – A proposition with the form “some a are b.” For example, “some cats are female.”

illicit affirmative – The fallacy committed by categorical syllogisms that have positive premises and a negative conclusion. All categorical syllogisms with this form are logically invalid. For example, “Some dogs are mammals. All mammals are animals. Therefore, some dogs are not animals.”

illicit contraposition – In categorical logic, illicit contraposition refers to a fallacy committed by an invalid argument that switches the terms of a categorical statement and negates them both. There are two types of illicit contraposition: (a) No a are b. Therefore, no non-b are non-a. (b) Some a are b. Therefore, some non-b are non-a. For example, “Some horses are non-unicorns. Therefore, some unicorns are non-horses.”

illicit conversion – Invalid forms of conversion—invalid ways to switch the terms of a categorical statement. There are two types of illicit conversion: (a) All a are b. Therefore, all b are a. (b) Some a are not b. Therefore, some b are not a. For example, the following is an invalid argument—“Some mammals are not dogs. Therefore, some dogs are not mammals.”

illicit process – A fallacy committed by categorical syllogisms that have a term distributed in the conclusion without being distributed in a premise. All categorical syllogisms that commit this fallacy are logically invalid. For example, “All lizards are reptiles. Some reptiles are lizards. Therefore, all reptiles are lizards.” See “distribution,” “illicit major” and “illicit minor” for more information.

illicit negative – The fallacy committed by categorical syllogisms that have one or two negative premises and a positive conclusion. All categorical syllogisms with that form are logically invalid. For example, “No fish are mammals. Some mammals are dogs. Therefore, some fish are dogs.”

illicit transposition – See “improper transposition.”

implication – A rule of replacement that states that “if a, then b” and “not-a and/or b” both mean the same thing. (“a” and “b” stand for any two propositions.) For example, “if dogs are lizards, then dogs are reptiles” means the same thing as “dogs are not lizards, and/or dogs are reptiles.”

improper transposition – A logically invalid argument with the form “If a, then b. Therefore, if not-a, then not-b.” For example, “If all lizards are mammals, then all lizards are animals. Therefore, if not all lizards are mammals, then not all lizards are animals.” See “transposition” for more information.

interchange – In categorical logic, interchange is to switch the first and second term of a categorical statement. For example, the interchange of “all men are mortal things” is “all mortal things are men.” See “conversion” for more information.

indirect proof – A strategy used in natural deduction used to prove an argument form is logically valid consisting of assuming the premises of an argument are true, but the conclusion is false. If this assumption leads to a contradiction, then the argument form has been proven to be logically valid. For example, consider the argument form “If A, then B. A. Therefore, B.” (“A,” “B,” and “C” are specific propositions.) An indirect proof of this argument is the following:

1. Assume the premises are true and the conclusion is false (not-B is true).
2. We know that “if A, then B” is true, and B is false, so A must be false. (See “modus tollens.”)
3. Now we know that A is true and false.
4. But that’s a contradiction, so the original argument form is logically valid.

intentional objects – The object that our thoughts or experiences refer to. For example, seeing another person involves an intentional object outside of our mind—another person. Some intentional objects are thought to be abstract entities, such as numbers or logical concepts.

inversion – To infer an if/then proposition from another if/then proposition. See “inverse” for more information.

inverse – An if/then proposition that is inferred from another if/then proposition. It is valid to conclude that one if/then proposition can be inferred from another whenever they both mean the same thing. It is valid to conclude from any proposition with the form “if a, then b” that “if not-b, then not-a.” For example, we can infer that “if it is false that all dogs are animals, then it is false that all dogs are mammals” from the fact that “if all dogs are mammals, then all dogs are animals.” See “transposition” for more information.

jargon – Technical terminology as used by specialists or experts. Jargon is not defined in terms of common usage—how people generally use a term in everyday life. Instead, they are defined in ways that are convenient for specialists. For example, logicians, philosophers, and other specialists define “valid argument” in terms of an argument form that can’t possibly have true premises and a false conclusion at the same time, but people use the term “valid argument” as a synonym for “good argument.” See “stipulative definition” for more information. “Jargon” can be contrasted with “ordinary language.”

logical equivalence – Two sentences that mean the same thing. For example, “no dogs are lizards” is logically equivalent to “no lizards are dogs.”

missing conclusion – A conclusion of an argument that is not explicitly stated, but is implied. For example, consider the argument “the death penalty kills people and it’s immoral to kill people.” This argument implies the unstated conclusion—that the death penalty is immoral.

monadic predicate – A predicate that only applies to one thing. For example, “x is mortal” could be stated as “Mx.” (“M” stands for “is mortal,” and “x” stands for anything.)

monadic predicate logic – A system of predicate logic that can express monadic predicates, but can’t express polyadic predicates. See “monadic predicate” and “predicate logic” for more information.

Monte Carlo fallacy – See “gambler’s fallacy.”

natural language – Language as it is spoken. Natural language includes both specialized language used by experts and ordinary language. “Natural language” can be contrasted with “formal systems.”

natural deduction – A method used to prove deductive argument forms to be valid. Natural deduction uses rules of inference and rules of equivalence. For example, consider the argument form “A and (B and C). Therefore, A.” (“A,” “B,” and “C” are three specific propositions.) The rule of implication known as “simplification” says we can take a premise with the form “a and b” to conclude “a.” (“a” and “b” stand for any two propositions.) We can use this rule to take “A and (B and C)” to conclude “A.” Therefore, that argument is logically valid.

negative categorical proposition – A categorical proposition that has the form “not all a are b” or “some a are not b.” For example, “some dogs are not mammals.”

negative conclusion – A categorical proposition used as a conclusion with the form “no a are b” or “some a are not b.” For example, “no dogs are reptiles.”

negative premise – A categorical proposition used as a premise with the form “no a are b” or “some a are not b.” For example, “some animals are not mammals.”

non-compound proposition – A sentence that can’t be broken into two or more propositions. For example, “Socrates is a man.” “Non-compound propositions” can be contrasted to “compound propositions.”

non-compound sentence – See “non-compound proposition.”

O-type proposition – A proposition with the form “some a are not-b.” For example, “some cats are not female.”

obverse – A categorical proposition is the obverse of another categorical proposition when it has a certain different quantification and a negated second term. There are four different forms of obversion: (a) The obverse of “all a are b” is “no a are non-b.” (b) The obverse of “no a are b” is “all a are non-b.” (c) The obverse of “some a are b” is “some a are not non-b.” (d) The obverse of “some a are not b” is “some a are non-b.” It is always valid to infer the obverse of a categorical propositions because the two propositions mean the same thing.

obversion – To infer the obverse of a categorical proposition. See “obverse” for more information.

ontological randomness – When something happens that could not possibly be reliably predicted because it could have happened otherwise. If anything ontologically random happens, then determinism is false—there are events that occur that are not sufficiently caused to happen due to the laws of nature and state of affairs. Ontological randomness can be contrasted with “determined” events and the acts of “free will.” It is generally thought that acts of free will are not random (and perhaps they’re not determined either). Imagine that you time travel to the past without changing anything, and all people make the same decisions, but a different person won the lottery as a result. That would indicate that there are elements of randomness that effects reality.

ordinary language – Language as it is used by people in everyday life. Words in ordinary language are generally defined in terms of “common usage” (i.e. how people tend to use the word). “Ordinary language” can be contrasted with “formal language” and “jargon.”

overprecision – A fallacy committed by an argument that requires precise information for the premises in order to reach the conclusion, and it uses misleadingly precise premises in order to do so. For example, a person was told that a frozen mammoth was five thousand years old five years ago, so she might insist that the frozen mammoth is now “5,005 years old.”

polyadic predicates – A predicate that applies to two or more things. For example, “John is taller than Jen” could be expressed as “Tab.” (In this case “T” stands for “is taller,” “a” stands for “John” and “b” stands for “Jen.”

positive categorical proposition – A categorical proposition with the form “all a are b” or “some a are b.” For example, “some mortals are men.”

positive conclusion – A categorical proposition used as a conclusion with the form “all a are b” or “some a are b.” For example, “some animals are mammals.”

positive premise – A categorical proposition used as a premise with the form “all a are b” or “some a are b.” For example, “all mammals are animals.”

predicate term – See “major term.”

premise indicator – A term used to help people identify that an premise is being stated. For example, “because” or “considering that.” See “premise” for more information.

proof by absurdity – See “indirect proof.”

proposition type – Different logical forms categorical propositions can take. There are four proposition types: A, I, O, and E. Each of these refers to a different logical form: (A) all a are b, (I) some a are b, (O) some a are not b, and (E) no a are b.

propositional connectives – See “logical connectives.”

propositional variables – Symbols used in propositional logic to represent propositions. Capital letters tend to stand for specific propositions. For example, “A” could stand for “Socrates is mortal.” Lower-case letters or Greek letters tend to stand for any possible proposition. For example, “a” could stand for any possible proposition.

public reason – John Rawls’s concept of reason as it should exist to justify laws and public policy. Ideally, everyone should be able to rationally accept the laws and policies no matter what their worldview is, so laws and public policies should be justified in secular ways that don’t require acceptance of controversial beliefs. Public reason does not require controversial religious beliefs or a comprehensive worldview precisely so it can help assure us that every reasonable person would find the laws and public policies to be justified—even if they have differing worldviews.

quantificational logic – See “predicate logic.”

randomness – See “epistemic randomness” or “ontological randomness.”

reasonable pluralism – Disagreement among people who have reasonable yet incompatible beliefs. A plausible example is of a person who believes that intelligent life exists on another planet and another person who doesn’t think life exists on another planet. John Rawls coined this phrase because he believed that society should fully embrace cultural diversity involving various worldviews and religious beliefs insofar as such religious beliefs and worldviews can be reasonably believed—the evidence we have for many of our beliefs is inconclusive, but it can be reasonable to have the beliefs until they are falsified (or some other standards of reason are violated).

reduction – To conclude or speculate that the parts of something are identical to the whole. For example, water is H₂O, and diamonds are carbon molecules with a certain configuration.

relational predicate logic – A system of predicate logic that can express both monadic and polyadic predicates.

rules of replacement – Rules that tell us when two propositions mean the same thing. We can replace a proposition in an argument with any equivalent proposition. For example, we know that “all dogs are animals and all cats are animals” means the same thing as “all cats are animals and all dogs are animals” because of the rule known as commutation—“a and b” and “b and a” both mean the same thing. (“a” and “b” stand for any two propositions.)

schema – See “scheme of abbreviation.”

scheme of abbreviation – A guide used to explain what various symbols refer to for a set of symbolic logical propositions, which can be used to translate a proposition of symbolic logic into natural language. For example, consider the logical proposition, “A ∧ B.” A scheme of abbreviation for this proposition is “A: The President of the USA is a man; B: The President of the USA is a woman.” “∧” is used to mean “and/or.” We can then use this scheme of abbreviation to state the following proposition in natural language—“The President of the USA is a man or a woman.”

set – (a) A group of things that all share some characteristic. For example, the set of cats includes every single cat that exists. (b) In Egyptian mythology, Set is the god of deserts, storms, and foreigners. Set has the head of an animal similar to a jackal, and he is known as “Sēth” in Ancient Greek.

Ship of Theseus – A ship used as part of a thought experiment. Imagine a ship is slowly restored and all the parts are eventually replaced. This encourages us to ask the question—Is it the same ship?

simplification – A rule of inference that states that we can use “a and b” as premises to validly conclude “a.” (“a” and “b” stand for any two propositions.) For example, “Socrates is a man and Socrates is mortal; therefore, Socrates is a man.”

social construction – The ability of collective attitudes and actions to create something. For example, our collective attitudes and actions create money, language, and the Presidency of the USA. These things would stop existing if we no longer believed in them.

social reality – Reality that exists because of the collective attitudes and actions of many people. For example, money, language, and the Presidency of the USA only exist because of the attitudes and actions of people. These things would stop existing if our attitudes and actions were changed in certain ways. See “institutional fact” for more information.

spurious accuracy – See “overprecision.”

state of affairs – A situation or state of reality. For example, the state of affairs of dropping an object while standing on the Earth will lead to a state of affairs consisting of the object falling to the ground.

subject term – See “minor term.”

symbolic logic – A formal logical system devoid of content. Symbols are used to replace content and logical connectives. For example, “if all men are mortal, then Socrates is mortal” could be written as “A → B.” In this case “A” stands for “all men are mortal, “B” stands for “Socrates is mortal” and “→” stands for “implies.” See “formal logic” and “logical connectives” for more information.

tautology – A rule of replacement that has two forms: (a) “a” and “a and a” both mean the same thing. (b) “a” and “a and/or a” both mean the same thing. (“a” stands for any propositions) For example, “Socrates is a man” means the same thing as “Socrates is either a man or a man.”

temporal modality – What makes a proposition true or false based on whether it is being applied to the past, present, or future. For example, dinosaurs existed in the past, but they do not presently exist.

transposition – A rule of replacement that states that “if a, then b” means the same thing as “if not-b, then not-a.” (“a” and “b” stand for any two propositions.) For example, “if Socrates is a dog, then Socrates is a mammal” means the same thing as “if Socrates is not a mammal, then Socrates is not a dog.”

Trigger’s Broom – A broom used in a thought experiment in which all the parts of the broom have been replaced. This encourages us to ask the question, “Is it still the same broom?”

truth preservation – The property of reasoning that can’t have true premises and false conclusions. See “valid” for more information.

universal quantifier – A term or symbol used to say something about an entire class. For example, “all” and “every” are universal quantifiers used in ordinary language. “All horses are mammals” means that if a horse exist, then it is a mammal.” This statement does not say that any horses actually exist. The universal quantifier in symbolic logic is “∀.” See “quantifier” for more information.

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Related

• Comprehensible Philosophy Dictionary
• More Philosophy Definitions Part 1
• More Philosophy Definitions Part 3

1. We can give uncontroversial examples of good arguments.
2. Good arguments must not be fallacious.
3. Good arguments are rationally persuasive.
4. The premises of good arguments are sufficiently justified.
5. The conclusion of good arguments are likely true given the assumption that the premises are true.

1. We can give uncontroversial examples of good arguments.

Here are two uncontroversial examples of a good arguments:

Argument 1

1. All dogs that have been studied by scientists were discovered to be mammals.
2. Therefore, all dogs are probably mammals.

This is an inductive argument. Inductive arguments generalize using limited information to attempt to provide us with a conclusion that is likely true given the premises. Inductive reasoning is used when we try to make plausible predictions or use statistics.

Argument 2

1. All dogs are mammals.
2. If all dogs are mammals, then all dogs are animals.
3. Therefore, all dogs are animals.

This is a deductive argument. Deductive arguments are meant to prove that a conclusion must be true as long as we assume the premises are true.

2. Good arguments must not be fallacious.

Fallacies are errors in reasoning. Good arguments must not be fallacious because such argument don’t give us a good reason to believe something. Consider the following three types of fallacies:

1. Straw man argument – To mischaracterize an argument or belief in order to make it seem less reasonable than it really is. For example, imagine that Samantha argues that “we should continue to use the death penalty because some people will decide not to commit murder knowing they could be executed.” I would then give a straw man argument if I responded by saying, “Samantha thinks that we should continue to use the death penalty because no one will choose to commit murder knowing they could be executed for it, but obviously murderers are willing to take that chance in places that still use the death penalty. They are not deterred knowing about the harsh punishment.” In this case I attempted to refute an argument, but it is a notably worse argument than the one given by Samantha. She never claimed that the death penalty would deter everyone from committing murder, but only that it could deter some people from committing murder.
2. False dilemma – Also known as “black and white thinking.” An argument that requires us to only consider so many options when there is at least one reasonable option that’s ignored. For example, a person might argue that “the President of the United States is either a dog or a cat. He’s not a dog, so he’s a cat.” There is at least one other option—the President is a human being.
3. One-sidedness – Also known as “suppressed evidence.” An argument commits the one-sidedness fallacy when it gives considerations in favor of a conclusion but it leaves out important considerations against the conclusion. For example, a person might argue that we should vote for the Democratic candidate for president because the Republican candidate supported the bank bailout when both candidates actually supported the bailout.

3. Good arguments are rationally persuasive.

Good arguments give us a sufficient reason to believe something is true insofar as the reason is compatible with rationality. What exactly is “rationality?” At the very least rationality requires that we don’t contradict ourselves or use fallacies. However, rationality in a broad sense involves what we should believe:

1. Rationally required – These are propositions that we should believe, and it would be wrong not to believe them. We would be irrational not to believe these things. For example, we are rationally required to believe that at least three people exist based on our experiences of our family, friends, and strangers. A belief is only rationally required if we have a very strong reason to believe it’s true.
2. Rationally permissible – These are beliefs that are optional. We need not believe them, but there’s nothing wrong with believing them. For example, it seems perfectly reasonable to believe that computers will be greatly improved within the next ten years considering how much they have been improved over time in the past. A belief is only rationally permissible if we have a good enough reason to believe it’s true.
3. Rationally impermissible – These are propositions that we should not believe—it would be wrong to believe them. We could say it would be irrational to have these beliefs. For example, it is rationally impermissible to believe that we can jump to the moon based on our experiences of our human limitations and the laws of nature. A belief is rationally impermissible only if we have a very good reason to think it’s false.

How do we know exactly if a belief is rationally required or permissible? That is a controversial issue in epistemology (the philosophical study of knowledge, rationality, and justification). However, this issue is particularly related to the concept of “justified beliefs,” which is discussed below.

4. The premises of good arguments are sufficiently justified.

What exactly does it mean for a premise to be sufficiently justified? It means that it’s rational to believe the premise is true because we have a good enough reason to believe it—there is enough evidence to warrant the belief. Some beliefs have such a strong justification that we have to believe them, some beliefs are justified enough to be compatible with rationality, and we have such strong justification against certain beliefs that the beliefs are incompatible with rationality.

Keep in mind that the classic definition of “knowledge” is of a “justified true belief.” If no beliefs are justified, then it would be impossible to know anything. Yet we know lots of things. We know “1+1=2,” we know that more than three people exist, and we know that dogs are mammals.

If we have a good argument that leads to a belief in a true conclusion, then it is reasonable to say that we “know the conclusion is true.” Even so, what qualifies as a “sufficient justification” is not always easy to understand. A permissive view is that a person who has rationally permissible premises only needs premises that are not yet refuted similar to a scientific hypothesis that is assumed to be true until proven otherwise.

Additionally, it is not always obvious what kind of a justification makes our beliefs likely true or rationally required. For example, some philosophers argue that the only justifications that matter involving the world are those involving observation, but other philosophers believe that intuition (a form of justification that is more difficult to understand) should count for something as well. Intuitive beliefs are those that we think seem true when we have a hard time explaining how we can know they are true. For example, it can be argued that we find it intuitive that “2+2=4” and yet we have a hard time explaining how we can know it to be true. It would also be “counterintuitive” to find out that “2+2=5.”

Another example of an intuitive belief is that at least one of two contradictory statements are always false. Two statements are contradictory when one statement is incompatible with the other. For example, “Socrates is a man” contradicts “Socrates is not a man.” It would be counterintuitive to find out that Socrates is both a man and not a man.

5. The conclusion of good arguments are likely true given the assumption that the premises are true.

Premises of good arguments need to be good reasons to believe that the conclusions are true, which means that if the premises of a good argument are true, then the conclusion is likely true as a result. It is unlikely for the premises of a good argument to be true and the conclusion to be false at the same time. In other words, good deductive arguments are logically valid, and good inductive arguments are logically strong.

Logically valid arguments can’t possibly have true premises and a false conclusion at the same time because they have a logically valid argument form. For example, “A. If A, then B. Therefore, B” has a logically valid argument form. An argument with this form is “All pit bulls are dogs. All dogs are reptiles. Therefore, all pit bulls are reptiles” is a valid argument because it’s impossible for the premises to be true and the conclusion to be false at the same time. Even so, one of the premises is false, and the conclusion is false.

Logically valid arguments with true premises always have true conclusions. These arguments are logically sound. For example, “All dogs are mammals. If all dogs are mammals, then all dogs are animals. Therefore, all dogs are animals.” The premises and conclusions of logically sound arguments are always true.

Logically strong inductive arguments are unlikely to have true premises and a false conclusion at the same time. If we assume the premises are true, then the conclusion is also likely true. For example, “All dogs we have ever seen were reptiles. Therefore, all dogs are probably reptiles.” This is an inductively strong argument, but the premise is false.

An argument is logically cogent if and only if it’s logically strong arguments and has true premises. For example, “All dogs we have ever seen were mammals. Therefore, all dogs are probably mammals.”

All good deductive arguments are valid and all good inductive arguments are strong, but not all good arguments have true premises because we don’t need to know that the premises are true. Instead, the premises merely need to be sufficiently justified. For example, consider Argument 2 from above:

1. All dogs are mammals.
2. If all dogs are mammals, then all dogs are animals.
3. Therefore, all dogs are mammals.

Both premises are confirmed by our best science, but it is possible that the first premise false. We don’t know it’s true for absolute certain. Perhaps one day we will find out that dogs are a species that merely greatly resembles mammals. Even so, this is still a perfectly good argument. It is perfectly reasonable to use the results of our best science as premises for our arguments, even though the conclusions of our best science is occasionally discovered to be false.

Conclusion

Understanding good argumentation requires us to better understand rationality and justification. Although philosophers do not agree unanimously about what makes a good argument, there is a great deal of agreement. There are uncontroversial examples of good arguments, there are uncontroversial examples of fallacies, good arguments are meant to help us understand what is true, and good arguments are meant to help us understand what we should believe. Finally, good arguments are central to our understanding of knowledge—to know anything implies that we had a sufficiently good reason to believe it. If we have a good argument for a conclusion and the conclusion is true, then we know the conclusion is true.

Update (12/3/2012): Various corrections and clarifications were made.

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One reason that not all good arguments are logically sound is because good arguments used in science are inductive, and inductive arguments are not meant to be logically sound. However, not all good deductive arguments are logically sound either. There is a sense that deductive arguments would ideally be logically sound, but some deductive arguments have sufficiently justified premises, even if those premises aren’t known to be true for certain. A good deductive argument must be logically valid, and it must have sufficiently justified premises. Even so, not all good deductive arguments are logically sound.

A good way to know if good deductive arguments can fail to be logically sound is to consider a good deductive argument that can fail to be logically sound. An uncontroversial example of a good argument is the following:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

Is this argument logically sound? Probably, but maybe not. The premises are sufficiently justified, but that doesn’t mean we know they are true for certain. If the premises are true, then it’s sound. If the premises are false, then it is not sound. We need to know why we would believe the conclusion is true. The reason that we should believe the conclusion is true is because the premises are well-justified, not because we know the argument is sound.

Consider the first premise in particular. Perhaps there is an immortal man who has kept his immortality a secret. If we found out a man is immortal, then the argument will no longer be a good argument. However, it would not be reasonable to require all good deductive arguments to have premises we know are true for absolute certain. Right now the premise is sufficiently justified and will remain so until we have significant counter-evidence against it. Imagine that all good deductive arguments had to use premises that we know are true for certain. In that case scientific conclusions could never be used for the premises of deductive arguments. Our best science concludes that all men are mortal, but sometimes scientific conclusions are discovered to be wrong. It would be absurd to say that no good deductive argument could use scientific conclusions as premises. Our best scientific conclusions are sufficiently justified and uncontroversial good deductive arguments can use our best scientific conclusions as premises.

Examples of scientific conclusions are “if the laws of nature will still exist tomorrow, then the law of gravity will still help us make predictions tomorrow” and “the laws of nature will still exist tomorrow.” A perfectly good deductive argument that uses these conclusions as premises is the following:

1. The laws of nature will still exist tomorrow.
2. If the laws of nature will still exist tomorrow, then the law of gravity will still help us make predictions tomorrow.
3. Therefore, the law of gravity will help us make predictions tomorrow.

We don’t know that these premises are true for absolutely certain. Sometimes the conclusions of science are proven wrong at some later point. The laws of nature might not exist tomorrow. Even so, it is perfectly reasonable to assume they will. This is a good argument, but it might not be logically sound.

We can summarize my argument that not all deductive arguments are logically sound as the following:

1. We know that the above deductive arguments are good arguments, but we don’t know for certain that they are logically sound. They are good arguments whether or not they are logically sound.
2. If they are good arguments whether or not they are logically sound, then not all good deductive arguments are logically sound.
3. Therefore, not all good deductive arguments are logically sound.

In conclusion, not all good deductive arguments are logically sound. We hope our good deductive arguments are logically sound, but sometimes they aren’t. Instead, I suggest that we define good deductive arguments as those that are logically valid with sufficiently justified premises.

Update (8/23/2012): I added the following clarification concerning why the first argument is good and why it doesn’t need to be logically sound — “We need to know why we would believe the conclusion is true. The reason that we should believe the conclusion is true is because the premises are well-justified, not because we know the argument is sound.” Additionally, I added a summary of my argument. Finally, someone replied with a lengthy email who seems to misunderstand this essay because he/she didn’t see part 1. If you strongly disagree or feel confused, make sure to read part 1.

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Related:

• Good Arguments (I disagree with the author that good arguments must have true premises, and that good deductive arguments must be logically sound.)
• Not All Good Arguments Are Logically Sound (Part 1)

What are good arguments?

For our purposes here a good argument is one that is rationally persuasive and does not make use of informal fallacies (informal errors in reasoning). Good arguments give us a sufficient reason to rationally agree with a conclusion. I will not discuss “informal fallacies” in detail here.

However, it is not entirely clear what a “good argument” is because it is not entirely clear when an argument is rationally persuasive. What exactly “rationality” consists of is a controversial topic that is studied by epistemologists (philosophers who study reasoning, justification, and knowledge).

Even so, there are uncontroversial examples of good arguments. A common example of a good argument is the following:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

(“Socrates” refers to a real historical figure discussed by Plato and other ancient philosophers and Socrates is said to have died by drinking hemlock.)

This argument gives us a sufficient reason to rationally agree with the conclusion that “Socrates is mortal” because the premises are highly justified, and anyone who believes the premises are true has no choice but to think that the conclusion is true. A person would not be irrational to believe the conclusion and it might even be rationally required for us to believe the conclusion—perhaps anyone who knows about this argument yet believes that Socrates is immortal (or is even undecided about it) is irrational considering how highly rationally persuasive this argument is.

What does it mean for an argument to be logically sound?

“Logical soundness” requires that an argument is both logically valid and that all the premises are true.

What’s a valid argument?

Logically valid arguments have a form that guarantees that the argument can’t have true premises and a false conclusion at the same time. For example, consider the following valid argument:

1. If all dogs are mammals, then all dogs are animals.
2. If all dogs are animals, then all dogs are flowers.
3. Therefore, if all dogs are mammals, then all dogs are flowers.

This argument is logically valid because we can’t imagine that the premises are true and the conclusion is false at the same time. If we imagine that both the premises are true, then the conclusion must also be true.

To better understand why an argument is logically valid, it can be a good idea to consider the logical form. In this case the logical form is the following:

1. If a, then b.
2. If b, then c.
3. Therefore, if a, the c.

All arguments with this logical form are logically valid. Each variable (a, b, and c) can stand for any proposition. Keep in mind that valid arguments can have false premises or conclusions. However, if an argument with this form has true premises, then it’s logically sound—and we are guaranteed that the conclusion is also true. Why? Because valid arguments can’t have true premises and a false conclusion at the same time.

It can also be a good idea to consider an invalid argument to see how it differs from a valid one. An example of an invalid argument is the following:

1. If all dogs are mammals, then all dogs are animals.
2. All dogs are animals.
3. Therefore, all dogs are mammals.

In this case we can imagine that the premises are true but the conclusion is false insofar as both premises could be true even if not all dogs are mammals.

The logical form of this invalid argument is the following:

1. If a, then b.
2. b.
3. Therefore, a.

Any argument with this form is invalid. We can replace the variables with new propositions to show that an argument with this form can have true premises and a false conclusion at the same time. Let’s replace a with “all cats are reptiles” and b with “all cats are animals.” In that we get the following invalid argument:

1. If all cats are reptiles, then all cats are animals.
2. All cats are animals.
3. Therefore, all cats are reptiles.

Now both premises are true, but the conclusion is false. The problem with invalid arguments of this kind is not that a premise or conclusion is false. The problem is that the premises do not give us a sufficiently good reason to think the conclusion is true—even if the premises are true, the conclusion can still be false.

What’s a sound argument?

An example of a sound argument is plausibly the following:

1. If all dogs are mammals, then all dogs are animals.
2. If all dogs are animals, then all dogs have DNA.
3. Therefore, if all dogs are mammals, then all dogs have DNA.

One problem with just about any example of a “sound argument” is that there’s some uncertainty involved. Our best science tells us that the premises are true, but there’s a chance that the science is wrong. We can say that this argument is “probably sound” but we can’t say we know it is sound for absolute certain. It is possible that one of the premises are false and that the conclusion is false as a result.

If it were true that all good arguments are logically sound, that would imply that we almost never know for sure if an argument is good. The best we could do is say that it’s probably a good argument.

What’s the difference between inductive and deductive arguments?

Inductive arguments are meant to give us a conclusion that’s probably true based on limited data, but deductive arguments are meant to guarantee that the conclusion is true. Deductive arguments are meant to be valid, but inductive arguments are not meant to be valid. The above arguments were all deductive, but not all good arguments are deductive. For example, consider the following good inductive argument:

1. The laws of nature existed throughout human history.
2. Therefore, the laws of nature will probably exist tomorrow.

This argument could be considered to be logically invalid, but it’s not meant to be logically valid. It’s only meant to tell us what is probably true based on limited information. This is how scientific arguments for theories work. Science makes predictions based on limited data. The predictions could always have a chance of being false. For example, it is possible that the laws of nature will not exist tomorrow. We predict they will, but we can’t prove they will for absolute certain.

The fact that we can have good inductive arguments is potential proof that not all good arguments are logically sound. If all good arguments are logically sound, then the above argument about the laws of nature would fail to be a good argument—and all scientific arguments for theories would be also fail to be good arguments. And yet many of the most persuasive rational forms of reasoning to believe anything is based on science (and inductive arguments).

Proof that not all arguments are logically sound

An argument against the belief that all good arguments are logically sound is the following:

1. At least some good scientific theories were proven to be false.
2. If at least some good scientific theories could proven to be false, then not all good arguments are logically sound.
3. Therefore, not all good argument are logically sound.

Premise 1 – Is it true that “at least some good scientific theories were proven to be false?” Yes. For example, I think Newton’s theory of physics is a good example. It was believed that Newton’s theory of physics was complete and could predict any physical motion, but it failed to predict the motion of Mercury. However, Einstein’s theory of physics was able to predict the motion of Mercury and is now considered to be a better (and more complete) theory of physics.

Premise 2 – Is it true that “if at least some good scientific theories could proven to be false, then not all good arguments are logically sound?” Yes. Scientists give good arguments in favor of good scientific theories. If all good scientific theories are proven to be true by sound arguments, then they can’t be proven to be false. Sound arguments would guarantee the theories are true because the premises would be true and the arguments for the theories would be valid—valid arguments can’t have true premises and false conclusions at the same time.

Given these two premises, the conclusion must be true (that not all good arguments are logically sound). Why? Because this argument uses a valid logical form. The logical form is “a. If a, then b. Therefore, b.” This valid logical form is well-known to be valid and is called “modus ponens.”

Conclusion

I am all for good arguments, and I think people should know more about what good arguments are. I want them to know more about what it means to give people a sufficient reason to rationally agree with a conclusion. Saying that not all good arguments are logically sound doesn’t mean we shouldn’t try to present sound arguments now and then. However, it does mean that we can’t condemn all arguments that fail to be sound.

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Related:

• Not All Good Arguments Are Logically Sound Part 2
• Field Guide to Critical Thinking (James Lett wrongly asserts that arguments must be sound because soundness is part of the scientific method.)
• Informal Fallacies Part 1

The Absolute – A term for “God” or “the Good.”

absurdism – The view that it is absurd for people to try to find the meaning of life because it’s impossible to do so.

æon – Latin for “life,” “age,” or “for eternity.” Plato used this term to refer to the eternal world of the Forms.

agent-neutral reasons – A reason for action that is not dependent on the person involved. For example, everyone could be said to have a reason to find a cure for cancer insofar as it would save lives. The assumption is that the reason to cure cancer does not depend on unique motivations or duties of an individual and saving lives is good for its own sake. Classical utilitarianism is an agent-neutral ethical theory insofar as it claims that all ethical reasons to act concern whatever has the most valuable consequences. “Agent-neutral reasons” are often contrasted with “agent-relative reasons.”

agent-relative reason – A reason for action that is dependent on the person involved. For example, a person has a reason to give money to a friend in need insofar as she cares for the friend. Ethical egoism is an agent-relative theory that claims that the only reasons to act are agent-relative. Agent-relative reasons are often taken to only describe ethical reasons for action as opposed to instrumental reasons for action. “Agent-relative reasons” are often contrasted with “agent-neutral reasons.”

The All – Another term for “the absolute,” “God,” or “the Good.”

amoral – Lacking an interest in morality. For example, an amoral person doesn’t care about what’s morally right or wrong, and a person acts amorally when she doesn’t care about morality at that moment in time. Many people think that babies and nonhuman animals act amorally because they have no concept of right or wrong. “Amoral” can be contrasted with “nonmoral.”

Bayesian epistemology – An epistemic theory featuring a formal apparatus for induction based on deduction and probability calculus. The formal apparatus is used to better understand probabilistic coherence, probabilistic confirmation, and probabilistic inference.

class conflict – The struggle between social classes to attain greater power or to maintain their current level of advantage. For example, the working class could fight for a higher minimum wage, and the wealthy could fight to keep receiving corporate welfare.

class warfare – See “class conflict.”

cognition – A mental process. For example “inferential reasoning” is a form of cognition.

conditionalization – Concerning how we ought to update our beliefs and degrees of confidence when we attain new information. For example, a person who believes all swans are white ought to reject that belief once she sees a black swan.

continuant – (1) A persisting thing. For example, we often think people persist through time and continue to exist from one moment to the next. (2) A persisting thing that “endures.”

convention – What is true based on agreement. For example, it’s a convention that people drive on the right side of the road in the United States (on two lane roads), so it would be generally wrong to drive on the left side of the road in the United States.

credence function – A comparison between the actual state of the world and the credence (subjective degree of confidence) a person has of the world being that way. Ideally people will have a strong credence towards factual statements. For example, people should be very confident that more than five people exist considering that society couldn’t function without thousands of people existing.

credence – A subjective degree of confidence concerning the odds we believe that something could be true. See “psychological certainty.”

cultural evolution – See “sociocultural evolution.”

desire-dependent reason – A reason for an action that depends on a desire. For example, a person who yearns to eat chocolate has a reason to eat chocolate. “Desire-dependent reasons” can be contrasted with “desire-independent reasons.”

desire-independent reason – A reason for action other than a desire. For example, John Searle argues that promises are desire-independent reasons. If you promise to do something, then you have a reason to do it, even if you don’t desire to do it. “Desire-independent reasons” can be contrasted with “desire-dependent reasons.”

Demiurge – (1) A godlike being theorized by Plato that is thought to be similar to an artisan who crafts and maintains the physical universe. Plato did not describe the Demiurge as the creator of the entire physical universe, and Platonists often thought that the entire physical universe was created or dependent on a greater being called “the Good.” (2) According to Neoplatonists, the Demiurge is “Nous” (the mind or intellect of the Good).

destiny – (1) A fated course of events, which is generally thought to be fated due to a person having a certain purpose. For example, King Arthur could have been said to be destined to become a king insofar as he was meant to be a king and would become a king no matter what choices he made. (2) A probable future event involving a person’s purpose that could be willfully achieved, but could be avoided given resistance. Perhaps King Arthur was destined to become king and could make choices to become the king, but could have fought against his destiny and become a blacksmith instead.

deus – Latin for “god” or “divinity.”

discursive – (1) Requiring “inferential reasoning.” (2) Rambling or discussing a wide range of topics.

discursive concept – According to Immanuel Kant, discursive concepts are general concepts known through inferential reasoning or experience rather than concepts known from a “pure intuition” (without depending on experience or generalization). For example, the concept of the person is a discursive concept because it requires various experiences and generalizations. “Discursive concepts” can be contrasted with “non-discursive concepts.”

discursive reasoning – See “inferential reasoning.”

divine providence – The view that everything that happens in the universe is guided and controlled by a divinity. It is generally believed that the divinity controls the universe to make sure that better things happen than would happen otherwise. Sometimes it is believed that the divinity assures us that everything that happens is predestined and “for the best” (or at least “everything happens for a good reason”) It is often thought that divine providence is a logical consequence of the assumption that God is all-good, all-knowing, and all-powerful; and it is often thought to conflict with our experiences of evil in the world.

divinity – A god or godlike being. See “God,” “Demiurge,” “Monad,” “the Good,” or “Universal Reason.”

emanation – How lower levels of existence, such as physical reality, flows from and depends on an ultimate eternal being. Those who believe in emanation tend to think that the ultimate reality is God or “the Good.” Emanation is the idea that creation is ongoing and eternal rather than out of nothing. In that sense the physical universe has always existed.

emanationism – The view that reality as we know it exists from emanation—all of existence as we know it depends on and constantly flows from an ultimate eternal being. See “emanation” for more information.

endurance theory – See “enduratism.”

endurantism – The view of persistence and identity that states that a persisting thing is entirely present at every moment of its existence. Endurantists believe that things can undergo change and still be the same thing. For example, a single apple can be green and then turn red at a later time. Endurantists believe that persisting things have spatial parts, but they don’t have temporal parts. See “temporal parts” for more information. “Endurantism” is often contrasted with “perdurantism.”

endure – (1) For a single thing to fully exist at any given moment in time, and to continue to exist at different moments in time despite the fact that it is likely to undergo various changes. See “endurantism” for more information. (2) To survive adversity or to continue to exist despite taking damage. (3) To tolerate an attack or insult.

eon – See “æon.”

eudaimonism – Ethical theories concerned with happiness or flourishing. Eudaimonist theories of ethics tend to be types of “virtue ethics.” Eudaimonist theories tend to argue that we should seek our happiness or flourishing, and that virtue is a necessary condition of being truly happy or flourishing. Socrates, Aristotle, Epicurus, and the Stoics are all examples of “eudaimonists.”

extension – What a term refers to. For example, the “morning star” and “evening star” both have the same extension. “Extension” is often contrasted with “intension.”

extensionality – Exensionality is concerned with the reference of words. For example, “the morning star” and “the evening star” both refer to Venus, so they both have the same extensionality. “Extensionality” is often contrasted with “intensionality.” Also see “sense” and “reference” for more information.

fate – (1) A fated event is an inevitable event that will occur no matter what we do. For example, every choice we make will lead to our death; so it’s plausible to think we are all fated to die. Sometimes it’s thought that a fate is inevitable because of a divine influence. Fate is often thought to be a separate concept from “determinism” in that a determinist does not necessarily think that everything that happens will happen no matter what choices we make. (2) “Fate” is another term the Stoics used for the concept of “Universal Reason.” (3) In ordinary language, “fate” is often synonymous with “destiny.”

foundational – The starting point or building blocks that everything else depends on. For example, a foundational belief can be justified without inferential reasoning or argumentation. The axioms of logic are a plausible example of foundational beliefs.

The Good – Plato’s term for the Form of all Forms. It is the ultimate being that all other types of reality depend on for their existence, and it is the ultimate ideal that determines how everything should exist. It’s also known by Neoplatonists as the “One” or the “Monad.” See “Plato’s Forms” for more information.

gunk – Any whole or type of stuff that can be indefinitely split into smaller pieces. Gunk can be made of smaller parts without an indivisible or indestructible “smallest part” (i.e. atom). “Gunk” is often contrasted with “atoms.”

gunky time – The view of time as being infinitely divisible. If time is gunky, then there is no such thing as a shortest moment of time. See “gunk” for more information.

inner sense – Our ability to experience states of the mind as opposed to the external world. “Inner sense” can be contrasted with “outer sense.”

institutional fact – Facts that exist because of collective attitudes or acceptance. For example, the value of money is an institutional fact and money would have no value if people didn’t agree that it has value. Institutions, such as the police force, government, and corporations all depend on institutional facts (and can only exist due to collective attitudes and acceptance).

intension – What a term means or how a word refers to things, which is often given in terms of a description. For example, the intension of “the morning star” is “the last star that can be seen in the morning” and the intension of “the evening star” is “the first star we can see at night.” Therefore, they both have a different intension, even though they both refer to Venus. “Intension” is often contrasted with “extension.”

maximize expected utility – (1) The position of decision theory that states that a person ought to make the decision based on whatever will probably lead to the greatest utility (the most valued or desired state). See “utility theory” and “stochastic dominance” for more information. (2) To make a decision that will probably lead to the most preferable outcome considering all possible outcomes of all possible decisions.

merology – The philosophical study of parts and wholes. Mereology concerns what the parts are of various things and how various parts and wholes relate. One merological question is whether there are atoms (smallest indivisible parts) of all objects, or whether all objects are ultimately gunky (can be split into smaller pieces indefinitely). Another merological question is whether or not an object is the same object if we replace all of its parts with functionally equivalent parts, such as if we replaced all the parts of a pirate ship with new but nearly identical parts.

meronomy – A type of hierarchy dealing with part-whole relationships. For example, protons are parts of molecules.

metalangauge – Language or symbols used to discuss language. Formal logical systems are metalanguages. See “formal logic” for more information.

metavariables – A symbol or variable that represents something within another language. For example, a logical system could have various either/or statements. “A or B” and “A and B, or C” are two different either/or statements within a logical system. We could then use metavariables to talk about all either/or statements. For example, “a or b” would represent all either/or statements of our logical language assuming that the lower-case letters are metavariables.

metalinguistic variable – See “metavariable.”

moral atomism – See “moral generalism.”

moral generalism – The view that there are abstract moral criteria (rules, duties, or values) that can be applied in every relevant situation to determine what we ought to do. Moral generalists often believe that analogies can be used to discover what makes an action right or wrong. For example, kicking and punching are both analogous insofar as we could do either to try to hurt people, and they both tend to be wrong insofar as hurting people is bad. “Moral generalism” is often contrasted with “moral particularism.”

moral holism – See “moral particularism.”

moral particularism – The view that there are no abstract moral criteria (rules, duties, or values) that can be applied in every relevant situation to determine what we ought to do. Instead, what we ought to do depends on the circumstance we are in without being determined by such things. Moral particularists sometimes agree that rules of thumb and analogies can be useful, but they don’t think we can discover objective criteria that determines what we ought to do in every situation. For example, kicking and punching are both analogous insofar as we could do either to try to hurt people. “Moral particularism” is often contrasted with “moral generalism.”

moral rationalization – Arguments used in an attempt to justify, excuse, or downplay the importance of immoral behavior. Moral rationalizations may superficially appear to be genuinely good arguments, but they fail on close examination. For example, many people deny that they are responsible for the harm they cause when they were one person out of many who were needed to cause harm, such as certain corporate employees. They are likely to say they are like a “cog in a machine” or “just doing my job.” See “rationalization” for more information.

moral responsibility – See “responsibility.”

noëtic structure – Everything a person believes and the relationship between all of the beliefs. Also, noëtic structure involves how confident the person is that various statements could be true and the strength in which each belief influences other beliefs. For example, finding out that there is no external reality would have a dramatic effect on our noëtic structure insofar as we are very confident that an external reality exists and many of our beliefs depend on that belief. Perhaps hurting “other human beings” would no longer be immoral insofar as they don’t really exist anyway. See “worldview” for more information.

non-discursive concept – According to Immanuel Kant, it’s a concept known from “pure intuition” (known a priori without depending on experience). For example, space and time. According to Kant, we couldn’t even have experiences without these concepts. “Non-discursive concepts” can be contrasted with “discursive concepts.”

non-discursive reasoning – See “non-inferential reasoning.”

nonmoral – Something that is neither morally right nor morally wrong. For example, mathematics is nonmoral, and a person who scratches an itch is acting nonmorally. “Nonmoral” can be contrasted with “amoral.”

The One – A Neoplatonist term for “the Good.”

outer sense – Sense perception used to experience the external world, such as the five senses (touch, taste, sound, smell, and sight). See “perception” for more information. “Outer sense” can be contrasted with “inner sense.”

objective reason – See “agent-neutral reason.”

partonomy – See “merenomy.”

perdurance theory – See “perdurantism.”

perdurantism – The view of persistence and identity that states that a persisting thing only partly exists at any given moment, and it’s entire existence must be understood in terms of its existence at every single moment that it exists. Perdurantism states that each persisting thing has distinct temporal parts throughout its existence in addition to having spatial parts. See “temporal parts” for more information. “Perdurantism” is often contrasted with “endurantism.”

perdure – For a single thing to only partly exist at any given moment in time, and for its full existence to require a description of it at every single moment in time that it exists. See “perdurantism” for more information.

philodoxer – “A lover of opinion.” Philodoxers love their own opinion more than the truth. They are contrasted with “philosophers” who love the truth more than their own opinion. Philodoxers are more close-minded than philosophers.

philosopher – (1) “A lover of wisdom.” Used as a contrast to “sophists” who claim they are wise and “philodoxers” who love their own opinion more than the truth. (2) A lover of learning. Someone who spends a great deal of time to learn and correct her beliefs. (3) A professional who is highly competent regarding philosophy, and spends a lot of time teaching philosophy or creating philosophical works.

post-hoc justification – A justification given for a belief that we already have. Although we often have a hard time explaining why our beliefs are justified, even if we know they clearly are, post-hoc justifications do not explain why we actually have a belief. As a result, they often exist to persuade or even manipulate others into sharing our belief. Post-hoc justifications are often motivated by bias rather than a genuine interest in the truth, and they are often rationalizations rather than genuinely good arguments. For example, people have been shown to be generally repulsed by consensual incest and they have an intuition that consensual incest is wrong, but most of the arguments they give against consensual incest are superficial. See “rationalization” for more information.

probability calculus – Mathematical rules that determine the odds of various propositions being true. For example, the probability of a tautology being true is 100%, and the probability of a contradiction being true is 0%. Also, in general, the odds of two propositions being true is lower than merely one of the two being true.

predestination – (1) The view that a deity determines everything that happens, usually thought to be “for the best.” Predestination in this sense is thought to logically imply that determinism is true, and it has inspired debates over “free will” for that reason. See “divine providence” for more information. (2) A synonym for “fate” or “destiny.”

predetermination – See “predestination.”

probabilism – The view that the degrees of confidence we have for various beliefs ought to be based on probability calculus. Probabilism states that we often lack certainty, but we should still try to believe whatever is likely true. For example, we ought to be confident that we won’t roll a six when we roll a six-sided-die, but we should not be confident that we will roll a two.

problem of evil – The question about how a divinity could exist if evil exists. It is sometimes thought that a divinity exists that’s all-powerful, all-knowing, and all-good, but that would imply that the divinity would assure us that less evil exists than actually exists. For example, it is plausible that the divinity would not make lead such a convenient yet poisonous metal that would take us thousands of years to discover to be poisonous.

providence – See “divine providence.”

pure intuition – An a priori cognition. According to Immanuel Kant, a pure intuition is the way we know about “non-discursive concepts,” such as space and time.

rationalization – Persuasive nonrational arguments given to believe something without a genuine concern for what’s true. Rationalizations are meant to superficially appear to be genuinely good arguments, but they fail on close examination. For example, a person who believes that the Earth is flat and is told that we have pictures of the Earth from space and we can see that it’s round could rationalize that the pictures are probably fake. A great deal of philosophical writing could be closer to rationalization than to genuinely good argumentation, but rationalization plagues everyday thought and can be difficult to avoid. See “moral rationalization” and “post-hoc justification” for more information.

redistribution of wealth – To take wealth away from some people and give it to others. It is sometimes thought that it is morally justified to tax the wealthy to provide certain services for the poor. For example, many people insist that Robin Hood is a hero because he risks his well being to take from the rich to give to the poor (who would otherwise suffer from an unjust system).

redistributionism – The view that we should have “redistribution of wealth” (perhaps to take from the wealthy to help the poor).

responsibility – (1) Being in control of one’s moral decisions. A person who is morally responsible can be legitimately praised or blamed for her moral actions. Moral responsibility requires a certain level of sanity, competence, and perhaps free will. It is plausible that small children and nonhuman animals lack responsibility because they might lack the competence required. Additionally, there are excuses that can temporarily invalidate a person’s moral responsibility, such as when people are harmed on accident or when a person is coerced into harming others. (2) To have a duty or to be morally required to act a certain way. For example, parents are responsible for caring for their children.

scientific method – The way science makes discoveries, which involves hypotheses, observations experiments, and mathematical models. It is often thought to follow the “hypothetico-deductive method.”

sensible intuition – According to Immanel Kant, it’s the concepts required for experience. For example, space and time. Without those concepts it would be impossible to experience the phenomenal world.

sentential – The property of being related to sentences or propositions. For example, sentenial logic is a synonym for “propositional logic.”

social convention – See “convention.”

social construct – Something that exists from collective attitudes and agreement. For example, money or the Presidency of the United States. These things wouldn’t exist if people unanimously decided that they don’t exist.

social progress – For a culture to be improved through changes in political institutions, economic systems, education, technology, or some other cultural factor. Technological improvement is perhaps the least controversial form of social progress.

sociocultural evolution – The view that people continue to find ways to adapt to their environment using technology, political systems, laws, improved education, and other cultural factors. See “social progress” for more information.

sophist – (1) “Wise person.” (2) A rhetoric teacher from ancient Greece. Some of those teachers traveled to other countries, and questioned the taboos and cultural beliefs of the Greeks because those taboos and cultural beliefs were not shared by everyone in other countries. (3) Someone who is willing to use fallacious reasoning to manipulate the beliefs of other people. This sense of “sophist” is often contrasted with “philosopher.”

spatial parts – Physical parts of an object, such as molecules, hairs, or teeth. “Spatial parts” can be contrasted with “temporal parts.”

statewise dominance – The property of a decision that can be said to be “superior” to another based on the decision-maker’s preferences and the fact that the outcomes of the decision are preferable. Every possible outcome of a statewise dominant decision is just as preferable as the other except at least one outcome must be more preferable. See “stochastic dominance” for more information.

state-by-state dominance – See “statewise dominance.”

stochastic – Regarding the probability calculus. Stochastic systems have predictable and unpredictable elements that can be taken to be part of a probability distribution.

subjective reason – See “agent-relative reason.”

syntactical variable – See “metavariable.”

temporal parts – Time-dependent parts of a persisting thing often thought of as time-slices based on the assumption that a persisting thing only exists in part at any given time-slice. We can talk about the temporal parts of a person in terms of the person yesterday, the person today, and the person tomorrow; and the person is thought to only exist in her entirety given every moment of her existence. We can talk about the person in any given time slice (such as August 3, 10:30 am). One reason some philosophers believe in temporal parts is because it can explain how an object can have two conflicting properties, such as how a single apple can be both green (while growing) and red (when ripe). If it has temporal parts, then we can say it is green in an earlier time-slice, and red in a later time-slice.

Universal Reason – The mental or intelligent element of the universe conceived as a deity by the Stoics. The Stoics saw the entire universe as a god—matter is the body and Universal Reason is the mind. They believed that Universal Reason has a divine plan and determines that everything that happens in the universe happens for a good reason.

world of ideas – The realm of the Forms. See “Plato’s Forms” for more information.

worm theory – See “perdurantism.”

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We all know that dictionaries can be valuable when we need to  know what a term means while reading a book, but the philosophical dictionary isn’t just to help us understand the arcane language spoken by elitist intellectuals. The terminology used by philosophers often refer to concepts that are central to rational thought or that help us make important distinctions. Go here to take a look for yourself.

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Related:

• 20 Important Philosophical Definitions
• Critical Thinking Concepts Everyone Should Know About

ad hominem – A Latin phrase that literally means “to the person.” It refers to insults and usually to fallacious forms of reasoning that make use of insults or disparaging remarks. For example, we could respond to the a doctor’s claim that “smoking is unhealthy” by saying the doctor who made the argument drinks too much alcohol.

anecdotal evidence – To attempt to persuade people to agree to a conclusion based on the experiences of an individual or even many individuals. Anecdotal evidence is often a fallacious type of argumentation. For example, many individuals could have experiences of winning sports games while wearing a four-leaf clover, but that doesn’t prove that four-leaf clovers actually give sports players luck. No fallacy is committed when the experiences of people are sufficient to give evidence for a causal relation and mere correlation can be ruled out. Fallacious appeals to anecdotal evidence could be considered to be a form of the “hasty generalization” fallacy. Also relevant is the “cum hoc ergo propter hoc” fallacy.

appeal to authority – (1) An argument that gives evidence for a belief by referencing expert opinion. Appeals to authority are not fallacious as long as it actually appeals to the unanimous opinion of experts of the relevant kind. (2) A fallacious argument that appeals to the supposed expert opinion of others when the opinion referred to is not unanimous or uncontroversial among the experts or when the supposed expert that is appealed to is not an expert of the relevant kind.

appeal to ignorance – A fallacious argument that concludes something on the basis of what we don’t know. For example, to claim that “we should agree that extraterrestrials don’t exist because we can’t yet prove they exist” is fallacious because there are other reasons we might expect extraterrestrials to exist, such as the vastness of the universe.

argument – (1) To provide statements and evidence in an attempt to lead to the plausibility of a particular conclusion. For example, “Punching people is generally wrong because hurting people is generally wrong” is an argument. (2) A verbal battle. (3) The discussion that concerns a disagreement.

begging the question– A logical fallacy that is used by an argument that uses a premise to prove a conclusion when a controversial premise trivially implies that the conclusion is true. For example, “The death penalty is murder, so the death penalty is wrong” requires a controversial premise (that the death penalty is murder) to prove something else controversial (that the death penalty is wrong). “Begging the question” is similar to “circular reasoning.”

charity – (1) The virtue in a disagreement or debate to describe other people’s beliefs and arguments accurately rather than to misrepresent them as being less reasonable than they really are. If we are not charitable in this way, then we will create a fallacious “straw man” argument. (2) The virtue concerned with helping others who are in need. For example, giving money to the poor is often charitable in this sense. (3) An organization or institution that exists to try to help others who are in need.

circular argument – An argument with a premise that’s identical to the conclusion. For example, “All dogs are animals because all dogs are animals.” The logical form of a circular argument is “a; therefore a.” Circular arguments are similar to the “begging the question” fallacy. Also see “circular reasoning.”

circular reasoning – (1) Reasoning involving the justification of beliefs that require us to accept other beliefs that aren’t justified unless we assume the belief that we want to justify in the first place. A simple form of circular reasoning is the following: A is justified because B is justified; B is justified because C is justified; and C is justified because A is justified. For example, “We should agree that stealing is wrong because it should be illegal; we should agree that stealing should be illegal because we shouldn’t want people stealing from us; and we shouldn’t want people stealing from us because it’s wrong.” (2) A “circular argument.”

conclusion – A statement that is meant to be proven or made plausible in consideration of other statements. “Conclusions” are often contrasted with “premises.”

continuum fallacy – A fallacy that is committed by an argument that appeals to the vagueness of a term to unreasonably conclude something (perhaps based on the fact that we can’t draw the line). For example, we might not know where to draw the line concerning how many hairs can be on a person’s head before that person is no longer bald, but we would commit the continuum fallacy to conclude from that fact that no one is bald.

contradiction – When two propositions cannot both be true due to their logical form. “Socrates was a man” and “Socrates was not a man” are two statements that can’t both be true because the logical form is “a” and “not-a.” (“a” is any proposition.)

counterexample – (1) An object or state of affairs that disproves a belief. For example, a white raven disproves the belief that “all ravens are black.” (2) An argument meant to prove another argument to be logically invalid by using the same argument form as the other argument, but the counterargument must have obviously true premises and an obviously false conclusion. Consider the invalid argument, “If dogs are lizards, then dogs are reptiles. Dogs are not lizards. Therefore, dogs are not reptiles.” A counterexample would be, “If dogs are reptiles, then dogs are animals. Dogs are not reptiles. Therefore, dogs are not animals.”

criticism – (1) An argument that is meant to persuade us to reject a belief of another argument. See “objection.” (2) Disparaging remarks, fault-finding, or judging something as falling short of certain requirements or standards.

cum hoc ergo propter hoc – Latin for “with this, therefore because of this.” A logical fallacy committed when an argument concludes that something causes something else to happen due to a correlation. For example, the fact that a person takes a sugar pill before recovering from an illness doesn’t prove that she recovered from the sugar pill. She might have recovered for some other reason. This fallacy is a version of the “false cause” fallacy.

debate – A prolonged discussion concerning a disagreement that is characterized by two or more sides that (a) try to give reasons to believe differing incompatible conclusions, (b) try to explain why the conclusions of the opposing side should be rejected, and (c) try to explain why the arguments given by the opposing side should be rejected. Debates need not be between two people and they need not exist in a face-to-face presentation. A single philosophical essay can be considered to be part of a debate that’s been going on for hundreds or thousands of years by philosophers in different time periods reading various arguments and responding to them.

deduction – Reasoning or argumentation characterized by the fact that the truth of the premises are meant to guarantee the truth of the conclusion. For example, “All men are mortal. Socrates is a man. Therefore, Socrates is a mortal.” Deduction is often contrasted with “induction.”

epistemic certainty – The degree of justified confidence we have in our beliefs. To be certain that something is true could mean (a) that we have a maximal degree of justification for that belief, (b) that we can’t doubt that it’s true, or (c) that it’s impossible for the belief to be false. To be absolutely certain that something is true is to have no chance of being wrong. For example, we are plausibly absolutely certain that “1+1=2.”

fact – A state of affairs, relation, or part of reality that makes a statement true. For example, it’s true that objects fall and will continue to fall because it’s a fact that “the law of gravity exists” or accurately describes relations that exist in reality.

fallacy – An error in reasoning. Formal fallacies are committed by invalid arguments and informal fallacies are committed by errors in reasoning of some other kind.

false dilemma – A fallacious argument that requires us to accept fewer possibilities than there plausibly are. For example, we could argue the following—“All animals are mammals or lizards; sharks are not mammals; therefore, sharks are lizards.” False dilemmas are related to the “one-sidedness” fallacy.

hasty generalization – A fallacious argument that concludes something because of insufficient evidence. Hasty generalizations conclude that something is true based on various observations when the observations are not actually a sufficient reason to believe the conclusion is true. For example, to conclude that all birds use their wings to fly based on seeing crows and swans would be a hasty generalization. Not all generalizations are fallacious. See “induction” for more information.

induction – To generalize based on a sample. For example, the view that the future will resemble the past in order to arrive at conclusions. To see only white swans could lead to the conclusion that all swans are white. To see that bread has always been nutritious could lead to the conclusion that similar bread will still be nutritious tomorrow. Induction is often contrasted with “deduction.” Not all inductive reasoning is well-reasoned. See “hasty generalization” for more information.

invalid – (1) An argument form that can have true premises and a false conclusion at the same time. An example of an invalid argument is the following—“Socrates is either a man or a mortal. Socrates is a man. Therefore, Socrates is not a mortal.” “Invalid” is often contrasted with “valid.” See “logical form” for more information. (2) Unreasonable. (3) Inappropriate or failing to meet specified requirements. (4) Someone who is chronically ill.

Justification – (1) Evidence or reasons to believe something. Observation is one of the strongest forms of justification; but self-evidence, intuition, and appeals to authority could also be legitimate forms of justification. For example, people can justify their belief that they can feel pain by having actual pain experiences. (2) The supporting premises of an argument.

justified belief – Some philosophers believe that justified beliefs are those that are given a sufficiently good justification, but it is possible that justified beliefs are defensible beliefs that one has no sufficient reason to reject. For example, a typical uncontroversial example of a justified belief is the belief that “1+1=2” but few to no people know how to properly justify this belief using argumentation.

knowledge – Classically defined as “justified true belief,” but many argue that it must be “justified in the right way” or that there might be a fourth factor. An eyewitness who sees a murderer commit the act knows who the murderer is because the belief is justified through observation and the belief is true. However, consider a situation where Sally believes that cows are on the hillside because she mistakes cardboard cutouts of cows as the real thing, and some real cows are on the hillside hiding behind some trees. The belief is justified and true, but some philosophers argue that Sally doesn’t actually know that cows are on the hillside.

logical form – The logical form of an argument consists in the truth claims devoid of content. “The sky is blue or red” has the same logical form as “the act of murder is right or wrong.” In both cases we have the form, “a or b.” (“a” and “b” are statements.) In this case the truth claim is that one thing is true and/or another thing is true.

nonrational persuasion – Fallacious and manipulative forms of persuasion. Nonrational persuasion does not always take the form of an argument, and it often appeals to our biases. For example, the news could continually have stories about how our enemies harm innocent people to give us the impression that our enemies are evil. This is similar to the “one-sidedness fallacy,” but no actual argument needs to be presented. People are likely to jump to conclusions on their own.

one-sidedness: (1) A fallacy committed by an argument that present reasons to believe something while ignoring or marginalizing the reasons against believing it. For example, a person selling a vacuum cleaner could tell us how it can pick metal objects off the floor, but decide not to mention that it tends to break after being used a few times. “One-sidedness” is also known as “selective evidence” and highly related to “cherry picking” and “quoting out of context.” (2) To be incapable or unwilling to see things from more than one reasonable point of view.

philosophy – (1) Literally means “love of wisdom.” The quest to attain knowledge and improve ourselves. It generally refers to various domains of study that involve systematic attempts to greater understanding while attempting to be reasonable other than those domains that have been designated to mathematicians or scientists. Arguments and theories concerning the proper domain of philosophy is known as meta-philosophy. (2) Opinions regarding what’s important in life or how one should conduct oneself. (3) A declaration of principles, values, or goals of an institution.

psychological certainty – The feeling of some degree of confidence about a belief. To be psychologically certain that something is true is to feel highly confident that it’s true. For example, a person might feel absolutely confident that trees really exist and later find out that our entire world takes place within a dream.

proposition – A truth claim or the conceptual meaning behind an assertion. The statement “Socrates is a man and he is mortal” contains two propositions. (a) Socrates is a man and (b) Socrates is mortal. Propositions are not statements because there can be multiple statements that refer to the same proposition. For example, there are many languages that offer us different ways to say, “Socrates is a man and he is mortal.”

red herring – A fallacious kind of argument that is meant to distract people from arguments and questions made by the opposing side. These kinds of arguments are meant to derail the conversation or change the subject. For example, a politician might be asked if we should end our wars, and she might reply, “What’s really important right now is that we improve the economy and create jobs. We should do that by lowering taxes.”

reductio ad absurdum – Latin for “reduction to the absurd.” Also known as the “argument from absurdity.” It’s a form of argument that justifies why an argument or claim should be rejected insofar as it would have absurd consequences. For example, consider the following argument—“Stars exist; the Sun is a star; therefore Stars don’t exist.” This argument leads to an absurd consequence in the form of a logical contradiction (i.e. that something exists and doesn’t exist.)

slippery slope – (1) An argument that requires us to believe that incremental causal changes will likely happen given that we make certain decisions. For example, having violence on television might desensitize people to violence and lead to even greater violence on television in the future by an ever-increasing demand for more thrilling forms of entertainment. (2) An informal fallacy committed by arguments that require us to believe that some decision will likely lead to incremental changes for the worse without sufficient evidence for us to accept that the changes are likely to actually happen. For example, some people argue that we shouldn’t legalize same-sex marriage because that would likely lead to marriages between brothers and sisters, and eventually it would lead to marriages between humans and nonhuman animals.

sound argument – An argument that’s valid and has true premises. For example, consider the following sound argument—“If all dogs are mammals, then all dogs are animals. All dogs are mammals. Therefore, all dogs are animals.”

statement – Classically defied as a sentence that’s true or false. However, some philosophers argue that a statement could have some other truth value, such as neither true nor false. For example, “This sentence is false” might be neither true nor false.

straw man – A fallacious form of reasoning consisting of misrepresenting another person’s arguments or beliefs in order to convince people that the arguments or beliefs are less reasonable than they really are. For example, Andrea might claim that “stealing is generally wrong,” and Charles might then reply, “No. Andrea wants us to believe that stealing is always wrong, but sometimes stealing might be necessary for survival.” The opposite of straw man is being charitable to another person’s arguments and beliefs—to present them as rationally defensible as they really are.

true – The property that propositions have that makes them based on reality. According to Aristotle, a statement is true if it corresponds with reality. For example, “Socrates was a man” is true. However, there might be other uses of the word true, such as, “The pawn can move two spaces forward when it is first moved in a game of Chess.” Many such “truths” are based on agreements or human constructions and are not factual in the usual sense of the word. “True” is often contrasted with “false.”

valid – An argument is valid when it has a logical form that assures us that true premises guarantee the truth of the conclusion. It is impossible for a valid argument to have true premises and a false conclusion at the same time. For example, consider the following valid argument—“If Socrates is a dog, then Socrates is a mammal. Socrates is a dog. Therefore, Socrates is a mammal.” “Valid” is the opposite of “invalid.” See “logical form” for more information.

• Related: 20 Important Philosophical Definitions

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Why are philosophical terms important?

Language helps us think. Without language it would be a lot more difficult (or perhaps impossible) to understand what subatomic particles are, to be able to consider what the ultimate form of reality is, or to consider what theory of justification is best. Words and terms are some of the most important parts of language and can be very helpful tools for improving our thought.

Philosophical language (or jargon) has a lot to offer. Philosophers spend a lot of time making new distinctions and they find it both convenient and sometimes indispensable to use their specialized jargon. For example, David Hume discussed “matters of fact” (facts about the world known through observation/empirical evidence) and “relations of ideas” (things true by definition and known through understanding a language that do not help us understand the world itself). Hume thought all knowledge must consist of these two categories. Later Immanuel Kant realized that Hume assumed all knowledge was analytic a priori (nonempirical and true by definition) and synthetic a posteriori (empirical and not true by definition), but that made room for one more category. Kant thought there was a type of knowledge that Hume missed—the synthetic a priori (nonempirical knowledge that’s not true by definition). For example, Kant thought the truths of geometry could be known from a synthetic a priori justification.

Twenty important philosophical terms

a posteriori – Latin for “from the later.” A posteriori statements or beliefs are justified entirely by observation. (e.g. “Human beings are mammals.”) The opposite of a priori.

a priori – Latin for “from the earlier.” A priori statements or beliefs are justified (at least in part) by something other than observation. Many philosophers agree that things that are true by definition have an a priori justification. (e.g. “All bachelors are unmarried.”) The opposite of a posteriori.

analytic – Analytic statements or beliefs that are true because of their meaning. (e.g. “All bachelors are unmarried.”) The opposite of “synthetic.”

synthetic – Statements that cannot be true by definition. Instead, they can be true because of how they relate to something other than their meaning, such as how they relate to the world. For example, “humans are mammals” is synthetic and can be justified through empirical science. “Synthetic” is the opposite of “analytic.”

categorical imperative – An imperative is a command or requirement. Categorical imperatives are overriding commands or requirements that don’t depend on our desires, and are rational even if we’d rather do something else. For example, it is plausible that we have a categorical imperative not to run around punching everyone in the face just for entertainment. The mere fact that someone might want to do it does not make it morally acceptable. Categorical imperatives are often contrasted with “hypothetical imperatives.”

hypothetical imperative – Imperatives are commands or requirements. Hypothetical imperatives are those we are required to do in order to fulfill our desires or goals. For example, if you are hungry, then you have a hypothetical imperative to get some food to eat. “Hypothetical imperatives” are often contrasted with “categorical imperatives.”

coherentism – The view that we start with various assumptions and such assumptions are justified as long as they are part of a coherent world view (mutually supported by other beliefs). It is often claimed that an assumption is justified through coherence if it is useful as part of an explanation. Observation itself is meaningless without assumptions, and observation appears to confirm our assumptions as long as our observations are consistent with them. For example, my assumption that a table exists can be confirmed by touching the table. Some philosophers argue that coherentism should be rejected because it legitimizes “circular reasoning,” which we ordinarily recognize as being a fallacious form of justification.

foundationalism – The view that there are privileged or axiomatic foundational beliefs that need not be proven. The source of privileged beliefs could be from self-evidence, non-inferential reasoning, non-empirical intuitive evidence, or perhaps even beliefs based on certain experiences. Foundationalism is one possible solution to the problem of justification requiring an infinite regress or circular reasoning. If everything we know needs to be justified from an argument, then we need to prove our beliefs using arguments on and on forever, or we need to be able to justify beliefs with other beliefs in a circular mutually supportive fashion; but foundationalism requires us to reject that everything we know must be justified with an argument (or argument-like reasoning).

emergence – (1) Epistemic emergence refers to our inability to know how to reduce one phenomenon into another. For example, chemistry is epistemically emergent insofar as we don’t know how to reduce it to physics—the laws of physics seem insufficient to predict the behavior of all chemical reactions. (2) Metaphysical emergence refers to when something is “greater than the sum of its parts” or the irreducible existence of a phenomenon that exists because of an underlying state of affairs. For example, some scientists and philosophers think that the mind is an emergent phenomena that exists because of brain activity, but the mind is not the same thing as brain activity.

reductionism – (1) Relating to identity theories or identity relations. For example, scientists think that water is identical with H₂O. (2) The view that something is nothing but than the sum of its parts parts. Some philosophers think that particles and energy (the reality described by physics) is the only real part of the universe and everything else is actually “nothing but” physical reality as described by physicists. Moral reductionists think that moral reality is actually nothing but non-moral facts of some other sort.

instrumental value – The usefulness of something. For example, knives have instrumental value for cutting food.

intrinsic value – Something with value just for existing. We might say happiness is “good for its own sake” to reflect that it is good without merely being useful to help us attain some other goal. If something is intrinsically good, then it is something we should try to promote. For example, if human life is intrinsically good, then all things equal, saving lives would plausibly be (a) rational, (b) a good thing to do, and (c) the right thing to do.

logical possibility – (1) The status of a proposition or series of propositions concerning the rules of formal logic—logically contingent (non-impossible) statements could be true, logically necessary statements have to be true (are tautologies), and logically impossible statements have to be false (because they form a contradiction). For example, it is logically contingent that the Earth exists. (2) A synonym for “logical contingence.” We might say that “it’s logically possible that the Earth exists” rather than that it’s “logically contingent.”

metaphysical possibility – (1) A range of modal categories concerning reality as it exists assuming that the laws of nature could have been different. The range includes metaphysical contingence, possibility (non-impossibility), necessity, and impossibility. Metaphysical possibility can be described as the status of a statement or series of statements considering all possible worlds—A statement is metaphysically contingent if it’s true in some possible worlds and false in others, possible if is true in some possible worlds, metaphysically necessary if it is true in all possible worlds, and metaphysically impossible if it’s false in all possible worlds. For example, some philosophers argue that “water is H₂O” is a metaphysically necessary statement. Assuming they are right, if we found a world with something exactly like water (tastes the same, boils at the same temperature, and nourishes the body) but it is made of some other chemical, then it would not really be water. (2) The status of a statement being metaphysically possible (non-impossible) as opposed to a range of modal categories. This status of possibility refers to what could be true or necessarily true about reality assuming that the laws of nature could have been different. A statement is metaphysically possible if it is “true in at least one possible world.” For example, it is metaphysically contingent that the H₂O exists because there is at least one possible world where it exists—the one we exist in.

matters of fact – Empirical statements concerning the physical world. They can be known to be true or false from observation. For example, “All dogs are mammals” is a matter of fact. David Hume believed the only statements that could be justified were “matters of fact” and “relations of ideas.”

relations of ideas – Statements that can be justified by (or true in virtue of) understanding the definitions of words. For example, “All bachelors are unmarried” is a relation of idea, and we can justify the fact that it’s true by appealing to the definitions of words. David Hume thought the only statements that could be justified are “relations of ideas” and “matters of fact.”

objective ought – Preferable behavior of a person while considering few (or no) constraints on the person’s knowledge. What we objectively ought to do is often thought to be based on the actual effects our behavior has. For example, utilitarians often say that we ought to do whatever maximizes happiness, even if we have no idea what that is. A person might try to help others by sharing food and accidentally give others food poisoning, and utilitarians might say that the person objectively ought not to have done so, even though the person might have done what was likely to help others from her point of view. “Objective ought” is often contrasted with “subjective ought.”

subjective ought – What we ought to do with consideration of the knowledge of the person who will make a moral decision. What we subjectively ought to do is based on what is reasonable for us to do given our limited understanding of what will happen. For example, some utilitarians say we ought to do whatever we have reason to think will likely maximize happiness. We might say that a person who gives food to a charity is doing what she ought to do as long as it was very likely to help people and very unlikely to harm them, even if many of the people who eat the food have an unexpected allergic reaction. “Subjective ought” is often contrasted with “objective ought.”

reference – (1) The objects that terms refer to. The terms “morning star” and “evening star” have different meanings, but they both have the same reference (Venus). Gottlob Frege contrasted “reference” with “sense.” (2) A source of information used for citations. (3) Someone who can vouch for your qualifications.

sense – (1) What Gottlob Frege called “sinn” to refer to the meaning or description of a word. For example, “the morning star” and “the evening star” both have different senses, but refer to the same thing. Gottlob Frege contrasted “sense” with “reference.” (2) The ability to understand. For example, we might talk about someone’s good sense. (3) To perceive. For example, we might say that we sense people in the room when we can see them. (4) An ability of perception; such as sight, sound, touch, taste, and smell.

Update (6/23/12): I updated the definitions for “logical possibility” and “metaphysical possibility.”

Related:

• The Wikipedia Philosophy Glossary
• Critical Thinking Concepts Everyone Should Know About

 You can follow Ethical Realism on Facebook or Twitter.

The following are an introduction to philosophical thinking:

1. Why Philosophy is Important
2. Intellectual Virtues, Dogmatism, Fanaticism, & Terrorism
3. How to Become a Philosopher
4. Eight Essential Philosophy Books
5. 10 Myths About Beliefs
6. What You Need From Formal Logic
7. Proving an Argument Is Logically Valid (optional)
8. Examples of Valid & Invalid Logical Reasoning
9. Four Requirements for Good Arguments
10. Four Sorts of Justification
11. Four Terrible Ways to Argue
12. Four Argument Strategies
13. Examples of Errors in Reasoning
14. Is Knowledge Impossible?
15. Three Forms of Evidence
16. Knowledge, Justification, and Theoretical Virtues
17. Arguments for Intuition
18. Three Kinds of Intuitive Arguments
19. Writing Philosophical Arguments
20. Philosophical Thought & An Illustration of An Objection
21. Philosophical Thought & an Illustration of a Positive Argument
22. How to Get an A in a Philosophy Class (optional)
23. How to Take Notes in Philosophy Class (optional)
24. How to Teach Philosophy (optional)
25. A Proposal for a Philosophical Community (optional)

These blog posts are also found in my ebook, How To Become a Philosopher.

 You can follow Ethical Realism on Facebook or Twitter.

There are things we can do to be more unreasonable and biased, and there are things we can do to be more reasonable and unbiased. I will discuss various forms of nonrational persuasion, cognitive bias, manifestations of unreasonable thought, why it matters, and what we should do about it. Simply knowing about forms of nonrational persuasion can help us identify them and stay vigilant. However, I will also discuss other concrete suggestions that can help us be more reasonable. For example, we should try to understand why intelligent people disagree with us and we should generally try to avoid marginalizing people from “other groups.”

Nonrational forms of persuasion

The main concern here are nonrational forms of persuasion that are used to influence what people think and believe. When intended, these forms of persuasion are forms of “manipulation.” Keep in mind that the nonrational forms of persuasion I will discuss might have legitimate and perfectly reasonable uses. Nonrational forms of persuasion include the following:

1. Informal fallacies – Philosophers specialize in understanding errors in reasoning (i.e. informal fallacies), which are often used in nonrational forms of persuasion. These forms of persuasion are used in argumentation and should be familiar to people who watch political debates. Cherry picking, anecdotal evidence, and slandering opponents are common. However, not all forms of manipulation require “arguments.”

2. False balance – The news media often gives us two sides of a story and makes them both seem reasonable when there is actually only one reasonable side to the story. For example, a fringe scientist could be found in order to make it seem like there is a debate over the plausibility of evolution among the experts, when the actual experts do not actually debate over that issue. (Keep in mind that the tactics I discuss are not limited to the “news media.” There are other people who might use the same nonrational forms of persuasion.)

3. One-sidedness – The news media often gives us one side of a story without giving the other side. This is a form of cherry-picking (i.e. selective evidence), but no actual argument is necessary. Instead, the person who reads or watches the news is likely to “draw their own conclusions” based on the one-sided information presented—and are likely to draw the wrong conclusions. For example, the news often has stories that discuss the evils done by our “enemies” rather than stories about the good things they do in order to give us the message that our enemies are inferior or evil.

4. Repetition – The news media can repeat the same story, similar stories, or message in order to convince the audience of something. For example, repeating stories about the crimes and immoral acts done by Muslims could give the impression that Muslims are generally immoral, or inferior than other groups. Many liberals have been known for repeating the message that “all opinions are equal” and many conservatives have been known for repeating the message that wealthy people are “job creators.”

5. Sensationalism – The news media often exaggerates stories. It often blows things out of proportion. This is to be expected in order to increase viewership, but the media is also likely to minimize the importance of stories that conflict with the ideology of the author. For example, a pro-Democrat group could publish news that exaggerates stories that illustrate the weaknesses of Republicans (and minimize or omit stories relevant to the weaknesses of Democrats).

6. Proof of sincerity – Many people will emphasize their sincerity (or the sincerity of their authority figures) rather than properly argue that their beliefs are true. Being willing to endure pain or death could be evidence of sincerity, but it doesn’t prove any belief in particular is true.

7. Moral virtues – Many people will emphasize the moral virtues of their authorities to encourage us to agree with the authorities. Character witnesses can be important in a court room to assure us that someone is generally honest, but it doesn’t prove that person’s beliefs are rational.

8. Marginalization – By ignoring, insulting, dehumanizing, or demonizing “others” we can attempt to convince people that we are superior. They can’t be trusted, but we can. Marginalization is effective at influencing what we think because people we don’t respect are much less likely to be taken seriously or understood by us. We are much more likely to dismiss and misunderstand everything said by those we disrespect. This is highly related to the “ad hominem” fallacy—insulting people is often used to distract us from the actual arguments and make us biased against their arguments.

9. Peer pressure – Peer pressure can exist in the form of praise and blame. People can offer us friendship and community. Their approval is important to us in order to maintain friendship and attain various social benefits. We are likely to at least pretend to agree with those in our community if disagreement is likely to lead to negative social consequences. The suppression of disagreement might actually make it more likely that we learn to agree with others in our group.

10. Isolation – Some groups will require that we stay away from “outsiders” who might have differing viewpoints. We are more likely to have the same biases and beliefs as those we spend our time with, and we are unlikely to change our mind if no one criticizes our beliefs. People on Facebook who block everyone who disagrees are creating a group they interact with in which criticism is suppressed, and they are less likely to change their mind about their beliefs as a result.

11. No questioning – Some groups will require that we refrain from “negative thinking,” “disagreement,” or “questioning the beliefs of the group.” Asking people to suppress their questions and critical thought might actually make it likely that we will learn to agree with the beliefs of the group.

Forms of Cognitive Bias

One reason that nonrational forms of persuasion can be so convincing is because we commonly suffer from certain rational shortcomings (i.e. forms of cognitive bias). All people appear to fall victim to cognitive biases no matter how rational they are. Our ability to reason is much more restricted than we think. I will discuss various forms of cognitive bias:

1. Confirmation bias – If we find some belief to be initially plausible, then we are likely to rationalize that belief to others using any positive evidence we can come up with while simultaneously ignoring and/or dismissing counter-evidence. The confirmation bias is probably the most important bias we commonly suffer from.

2. Belief bias – People who think a conclusion is true are much more likely to think an argument for that conclusion is reasonable. This is related to the confirmation bias.

3. Selective perception – People’s expectations effect their experiences and observations. This is related to theory-laden observation and the confirmation bias.

4. Halo effect – We tend to think that we can generalize from certain positive characteristics of a person or group to other characteristics of that person or group (or negative characteristics of a person or group to other characteristics). This often takes the form of thinking a person who believes something we agree with is reasonable, or someone who believes something we disagree with is unreasonable; and arguments given by those who believe other things we agree with are more likely reasonable than arguments given by those who believe other things we disagree with. For example, if we think a group of people (or a person) is inferior, stupid, or evil, then we are less likely to understand their arguments, and we are more likely to dismiss their arguments out of hand. This is one reason why ad hominem fallacies (insults during arguments) are effective. (Jonathan Haidt discusses this bias as a tendency to demonize other groups in his new book, The Righteous Mind, and in this video interview.)

5. Dunning–Kruger effect – Unskilled people tend to think they are much more skilled than they really are. (However, keep in mind that we have some reason to think even the most reasonable people overestimate how reasonable they are. Perhaps no one is reasonable enough to realize they are often unreasonable.) This bias gives many people an unwarranted sense of certainty and is often caused by the inability of unskilled people to realize when they make mistakes.

6. The Downing effect – People with below average IQ often overestimate their IQ, but people with above average IQ often underestimate their IQ.

7. Overconfidence effect – People systematically have a false sense of certainty. For example, many people might sense that they are 99% certain that they are right about answers they give on quizzes when they are actually only 40% certain.

8. Illusory superiority – People systematically think they have positive qualities in all areas—that they are at least “above average” at just about everything. This is similar to the overconfidence effect in that people think they are better than they really are.

9. Self-serving bias – People tend to falsely attribute their successes to skill and positive personal attributes, and falsely attribute failures to external and situational factors. For example, a person might think she did well on a multiple choice exam because she was prepared, but that she didn’t do well on another exam because the test questions were too hard.

Manifestations of nonrational thought

There are many unreasonable forces, ideas, thoughts, and groups many people fall victim to. There’s charlatans, conspiracy theorists, cults, new age sophistry, pseudoscience, anti-science paranoia, bad science, manipulation in the media, and nonrational political arguments:

1. Charlatans – Charlatans are people who claim to have a product that does something it doesn’t really do, or expertise that they don’t actually possess. Swindlers, snake-oil salesmen, quacks, cult leaders, false prophets, false psychics, unqualified scientists, and unqualified philosophers are all examples of charlatans. They generally want to make money and hope that tricking people is a good way to do so.

2. Conspiracy theorists – Examples of conspiracy theorists are those who believe the Moon landing was a hoax and those who believe aliens landed at Roswell, New Mexico. Although conspiracies do exist, the evidence required to justifiably believe that a conspiracy exists tends to be quite high. Many conspiracy theorists have a false sense of certainty and require insufficient evidence for their theories.

3. Cults – Although there might be cults that aren’t harmful, most people in cults are far too trusting of the cult leaders, who tend to be charlatans. Cult leaders tend not to have the expertise they claim to have—perhaps none of them actually have it. Additionally, harmful cults engage in forms of “brainwashing” demonize “outsiders,” require members to keep a distance from family and friends, and keep questions to themselves in order to give members a false sense of certainty (and suppress critical thinking).

4. New age sophistry – The word “philosophy” is supposed to refer to an honest and reasonable attempt to understand reality, but that’s not how everyone uses the word. Many people (and bookstores) use the category “philosophy” to include the opposite of philosophy—sophistry (unreasonable assertions, arguments, and worldviews). Perhaps what we call “new age spirituality” is one of the most common offenders. Not all new age spiritual writings are necessarily sophistry, but a lot of it is. Moreover, almost no new age writings would qualify as “philosophy” according to those who are properly trained in the field.

5. Pseudoscience – Pseudoscience is material treated as science by many people that should not be treated that way. Homeopathy and astrology are not considered to be science by those who are properly trained in the field, so they should not be considered to be science. One reason that pseudoscience is a problem is because, like other forms of sophistry, it is not properly proven or justified and yet many people arrogantly feel they are certain that their beliefs in them are justified anyway. (To emphasize the impact pseudoscience has on society, consider how just about every newspaper has an astrology section, but they don’t have an Astronomy section.)

6. Anti-science paranoia – There’s a great deal of distrust for good science. Evolution, global warming, and the the fact we need certain vaccines are not debated among the experts of the appropriate fields and they consider these positions to be the most justified available. Non-experts almost never have a good reason to denounce the most authoritative opinions of scientists, yet that is exactly what many anti-science advocates do.

7. Bad science – According to at least one study, most “scientific” findings are false. Scientific studies generally should not be taken seriously until they are properly proven (e.g. through repeated results in other studies). People should not blindly believe what “science says.” That doesn’t mean that we shouldn’t trust scientists at all. It means that science is a process and good science generally takes time. What is not controversial in science among the proper experts should be taken seriously, but the fact that a single scientist believes something controversial among his or her peers is not sufficient evidence that we should agree.

8. Manipulation in the media – There are various ways the media manipulates us (whether intentional or not). Outright lies are not needed. Instead, the media can present us with selective facts, exaggerate, present only one side, present two sides of non-issues, and so on.

9. Nonrational political arguments – Nonrational arguments can be very persuasive, and political debate might be the most common source of nonrational arguments. In general, politics is a wretched hive of scum and villainy. In addition, people who devote their loyalty to a political group commonly marginalize “outsiders” and fail to properly investigate why many intelligent people see things differently.

Why does it matter?

Unreasonable beliefs and nonrational thought is probably harmless for the most part, but not always. Reasons to want to avoid being unreasonable include the following:

1. Unreasonable thought leads to false beliefs – The more unreasonable we are, the more likely our beliefs will be false. We will be more likely to form new false beliefs and continue to hold false beliefs. Learning to be more reasonable will help us form beliefs that are more likely true.

2. Unreasonable thought leads to manipulation – People want us to believe certain things because it benefits them in one way or another, and being unreasonable makes us easier to persuade (and more gullible in general). People in power want us to approve of the status quo to avoid a power struggle and avoid losing power, people who want money want to convince us to give them our money, and so on. We can waste our money and votes if we allow people to manipulate us in unreasonable ways.

3. Unreasonable thoughts leads to mistakes – Many unreasonable people are sincere, but their beliefs can lead to harmful mistakes. People who believe evolution is false are more likely to want to keep children from learning it in schools, people who believe in homeopathy are more likely to waste their money on homeopathy, people who believe vaccines are harmful are more likely to refuse the best medical advice available, people who believe certain groups are evil are more likely to be willing to harm those groups, people who think “all opinions are equal” are less likely to challenge unreasonable thought, and people who believe the wealthy are “job creators” (or in trickle down economics) are more likely to endorse corporate welfare when they shouldn’t.

4. Unreasonable thoughts can waste our time – Believing in the wrong things often causes people to waste their time learning more about those unreasonable things (rather than good science and philosophy).

What we should do

Although we suffer from cognitive biases and are bombarded by nonrational forms of persuasion, there are measures we can take to increase our ability to be reasonable. For example:

1. Educate ourselves – We can study informal fallacies (errors in reasoning) and nonrational forms of persuasion. Knowing what they are can help us look out for them.

2. Learn critical thinking and logic – There’s more to being reasonable than knowing how to identify errors in reasoning. Critical thinking education should help us understand what good reasoning consists of in addition to including critical thinking activities that can improve our critical thinking skills through practice. Logic classes also have a great deal to say about what makes a good argument.

3. Peer review – We can present our beliefs and arguments to others who can double-check them. We are more likely to know when we have errors in reasoning if many other people know how we reason and criticize us when we make mistakes. This is one reason that science uses “peer review” and is so successful as a result. Sometimes scientific findings are wrong, but additional scientific research by other scientists can often help us know when that happens.

4. Understand both sides – We can seek out counter-evidence to our beliefs. We can try to understand why intelligent people disagree with our beliefs. We can also criticize our family and friends when we believe they have an unreasonable belief or argument.

5. We can try not to marginalize “outsiders” – We are less likely to understand others if we think they are inferiors.

Conclusion

Everyone suffers from cognitive bias and are bombarded by nonrational forms of persuasion. Perhaps all of us fall victim to manipulation and bias, but we can still try to be more reasonable and avoid becoming less reasonable. Being unreasonable can not only cause us to have more false beliefs, but it can also cause us to make mistakes that can thwart human flourishing. We should try to avoid being unreasonable and we should try to help others avoid being unreasonable. Being willing to criticize others and be criticized can help, but learning more about critical thinking and logic can also be a good idea.

Update (5/31/2012): I changed the term “unreasonable forms of persuasion” to “nonrational forms of persuasion.”

Update (12/18/2012): I made more clarifications and corrections.

Related:

• Ethical Realism: Ethics and rationalization
• Ethical Realism: Introduction to Critical Thinking & Argument Mapping
• Ethical Realism: Informal Fallacies Part 1
• Costly Cognitive Biases
• 5 Persuasion Techniques
• Interview with Jonathan Haidt concerning social psychology’s findings concerning morality and politics
• A video interview with Jonathan Haidt concerning social psychology’s findings concerning morality and politics
• Wikipedia: Cognitive Bias
• Why Most Published Research Findings are False
• Characteristics Associated with Cultic Groups

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1. What are facts?

The word “fact” has at least one meaning within our language and other languages could have equivalent words that refer to the same concept. We can better understand the concept of “facts” by studying language, considering how the word should be used, and realizing that there are common confusions involving the word.

The meaning of the word “fact” that I’m interested in refers to the reality, state of affairs, or relations that make statements true. What exactly facts refer to is controversial among philosophers, but it involves reality, states of affairs, and/or relations. The statement “the law of gravity exists” is true because it’s a fact that the law of gravity exists—there is a reality, state of affairs, or relation that makes the statement true. Dropping objects causes them to fall and so forth.

It can be useful to consider examples of facts (and examples of how people use the word “fact”). People might say that “it’s a fact that the cat is on the mat,” “it’s a fact that 1+1=2,” or “it’s a fact that John has a headache.” These are all correct ways to use the word. If the statements are true, then corresponding facts exist that make them true.

It can also be useful to consider false statements that claim something is a fact. I can say, “It’s a fact that the Moon is made of cheese” or “it’s a fact that the law of gravity doesn’t exist.” These statements about facts are false precisely because they don’t refer to facts that exist—they don’t properly describe reality.

Some people talk about facts that don’t exist, but there are no facts that don’t exist. If something is a fact, then it exists. There are false claims that something is a fact that isn’t a fact. For example, the claim that “it’s a fact that the Moon is made of cheese” is false precisely because it does not describe a fact at all. (The statement does not refer to a non-existent fact.)

2. Are facts supposed to refer to something that exists?

One important issue is whether or not the word “fact” is meant to refer to something that exists as we use the word in our language (or whether or not the concept refers to something that is supposed to exist). Some people might claim that the word “fact” isn’t even meant to refer to anything that exists. Perhaps saying that “it’s a fact that a cat is on the mat” isn’t supposed to describe something that exists. However, I think saying “it’s a fact that the cat is on the mat” is meant to describe something that exists. The statement could be false—there might not be such a fact. However, the possibility of false statements concerning facts is not the issue here. The question is whether stating something is a fact is a claim about whether a fact exists.

Saying that facts are supposed to exist (based on how we use the word) should not be surprising. Consider the statement that “it’s a fact that the law of gravity exists.” This statement is saying that there is a fact and that fact is that the law of gravity exists. Saying there is something means that something exists. In other words, the statement makes the claim that there is a fact that exists and that fact is that the law of gravity exists. If we said that this fact doesn’t exist, then the statement “the law of gravity exists” would be false.

I think we can know the word “fact” is meant to refer to something that exists as we use it in our language by considering proper and improper ways to use the word (given the definition of “fact” that I am concerned with) . Particularly relevant are two ways many people commonly use the word “fact” that are wrong:

One, some people talk about facts being true. However, facts are not true. Facts are what make statements true. The word “true” refers to statements that describe facts. This is one reason it is appropriate to say that facts exist and statements are true or false. Facts can’t be true or false.

Two, some people think that “facts” differ from “opinions” in that facts have been proven to be true. However, facts do not only refer to things we prove to be true. Consider how many people say that “evolution is just a theory” and others say “evolution is a fact.” These are not mutually exclusive because theories can describe facts.

There are facts whether or not anyone can prove they’re true. Facts do not generally depend on our beliefs or desires. They exist whether or not anyone thinks about them, proves they’re true, and so on. The law of gravity exists whether or not anyone knows about it, thinks about it, proves it, or believes it. Assuming it is true that “the cat is on the mat,” “1+1=2,” and “John has a headache,” these statements are true whether or not anyone knows about them, thinks about them, proves them, or believes them because they would describe facts that exist.

There are exceptions—some facts do depend on thoughts, beliefs, and proofs (to some extent). Those are facts about what people think, believe, or prove. It is a fact that many people believe that murder is immoral, it is a fact that many people have proven something to be true, and so on. However, these facts do not depend on any beliefs, thoughts, or arguments other than the ones that relate directly to the facts in question. For example, many people think that murder is immoral, even if I don’t know that they think that; and scientists have proven that the law of gravity exists, even if we have not proven that they did so.

Once we realize that facts do not depend on our beliefs, desires, or arguments, we can realize how facts (if they exist at all) can be discovered—and how the discovery of facts indicates that they exist as part of reality outside of ourselves. Assuming fact exist, they don’t refer to something “made up” or delusional. Scientific facts seem like a good example. We seem to know it’s a fact that the law of gravity exists and we know the fact doesn’t just exist because we made it up or have some sort of delusion.

3. Do any facts exist?

Perhaps one more reason to think facts are supposed to exist (as we use the word in our language) is that we have some good reasons for thinking there are at least some facts that exist—some forms of realism are plausible. However, my main concern in this section is not that facts are supposed to refer to things that exist given how we talk about them. My concern is that I think it is plausible that at least some facts really do exist. Forms of realism include physical realism, scientific realism, mathematical realism, epistemic realism, and moral realism. I will discuss various forms of realism and explain why I think it’s plausible that physical facts exist in particular.

Physical realism – Physical realism states that there are physical facts. For example, it’s a fact that the law of gravity exists, atoms exist, and an external world exists. There are physical anti-realists who deny that there are physical facts. They don’t think the law of gravity really exists, that atoms really exist, or that an external world really exists. Some physical anti-realists are idealists who think that the world as we know it only exists in our minds—it’s something like a shared dream.

Scientific realism – Scientific realism is not concerned with physical realism—both scientific realists and anti-realists generally agree that it’s a fact that rocks exist. Scientific realism is about whether or not there are facts that certain unobservable entities exist, such as electrons or quarks. The effects of various phenomena are observed in science and scientific theories postulate various entities to explain those effects (such as electrons and quarks). The question is whether or not such unobservable entities really exist or not. Perhaps they are merely convenient fictions.

Mathematical realism – Mathematical realism states that there are mathematical facts, mathematics is not merely a “human invention,” and mathematical truths are not “true by definition.” Mathematics seems to defy language and require intuitive thought. Mathematical facts seem to exist no matter what we believe or prove. Most mathematical realists in contemporary philosophy are called “mathematical platonists” and they believe that mathematical facts are “abstract entities” (similar to Plato’s forms)—they are timeless, nonphysical, and exist outside space and time.

Epistemic realism – Epistemic realism states that there are facts concerning justification and rationality. According to Epistemic realists, it’s a fact that believing that “1+1=3 is true just because your parents told you it’s true is irrational;” and it’s a fact that “it’s rational to believe that the law of gravity exists because of our experience with falling objects.” Moreover, they agree that it’s a fact that “the belief that all men are mortal is justified” and that “the belief that fairies exist is unjustified.”

Moral realism – Moral realists agree that there are facts concerning what’s good or bad, what’s right or wrong, and what we ought or ought not do. They agree that it’s a fact that torturing children is morally wrong, that helping people is often morally right, and that we morally ought not kill other people every second of the day. Moral realists do not think morality depends on our beliefs and desires. Moral facts can be discovered and exist before we know about them.

There are many philosophers who endorse one or more type of realism, which is evidence that many philosophers agree that some facts exist, and I think all of the sorts of realism mentioned here are endorsed by some philosophers precisely because they are at least somewhat plausible given our understanding of the world. Additionally, I think that physical realism is plausible in particular.

Why physical realism is plausible.

Physical realism is plausible for at least two reasons:

First, we have some reason to think science as we ordinarily think of it requires the assumption that physical realism is true. The world is not just part of my dream—it exists independently to my beliefs and desires. The laws of nature exist before I know about them and even if no one thinks about them at all. Scientists want to discover how the world is with the assumption that there are facts that exist whether or not we know about them. If there are no physical facts, then it is unclear what the heck scientists are doing and why they seem to be discovering things.

Second, the assumption of physical facts, such as the fact that an external world exists, seems necessary to explain our experiences of discovery and of a world of regularities that exists in a certain predictable way no matter what we believe or desire. If there is no external world, then we have no idea why we have the experiences that we do.

4. The objection against facts.

An objection to the belief that facts exist (and the assertion that no facts exist whatsoever) is that it violates Occam’s razor, but I think this objection is self-defeating. I will argue that any argument against the existence of facts is self-defeating, and that’s just one more reason for us to believe at least some facts exist.

The objection against facts that I will discuss is that the existence of facts violates Occam’s razor—it multiplies entities beyond necessity. If it’s a fact that a cat is on the mat, then there are even more things that exist. One, a cat. Two, a mat. Three, the fact that a cat and a mat exists and relate in a certain way. Perhaps it would be simpler (and more reasonable) to dispense with facts altogether and just say there’s a cat on a mat.

My reply

First, in order to know if the existence of facts violates Occam’s razor, we need to know what it means for them to be “beyond necessity.” If two theories (or explanations) are equal in all respects except one theory is more complex, then we should prefer the simpler one. Consider the theory that “the law of gravity exists” and a competing theory that “gravity exists and fairies exist.” In this case both theories equally explain my experiences of falling objects, but the theory that “gravity exists and fairies exist” should be rejected because it’s needlessly multiplies entities for absolutely no reason—it’s exactly like the simpler theory except it says fairies exist for no reason.

Even so, it is usually not that easy to decide when Occam’s razor is violated because there can be good things about a theory that might still violate Occam’s razor. For example, many versions of string theory were popular and some postulated more dimensions than others, but it wasn’t clear which version of string theory was best.

Moreover, some philosophers think Occam’s razor is a good reason to reject certain theories even if it’s the best possible explanation available. For example, some philosophers might prefer to have no theory concerning what mathematics is about rather than accept mathematical platonism because they don’t think it’s plausible that “abstract entities” exist.

The question is whether or not the belief that “facts exist” is more complex than an alternative perspective without a good reason for being more complex. Is there a simpler theory that explains our experiences just as well? Consider the theory that we can talk about reality without any facts existing. This theory would confuse what it means to say “facts exist” because the existence of facts is the existence of some sort of reality. Without facts, there is no reality being referred to. Therefore, it doesn’t make sense to say that “there is a cat on a mat, but it’s not a fact that a cat is on the mat.”

It is far from obvious that the belief that “facts exist” violates Occam’s razor because there is no plausible alternative it competes with (that we currently know about). We don’t know of a simpler and more reasonable alternative to the belief that “facts exist.” There might be an alternative like that, but if so, it’s an alternative that no one seems to know about at this point of time.

Second, I think there is a good reason to think that the belief that “facts exist” doesn’t multiply entities in the way stated above at all. To say the true statement, “gravity exists,” is to automatically describe a fact. To say that “it’s a fact that gravity exists” means something like “the statement ‘gravity exists’ is true because of some relation to reality.” Facts are part of reality—whatever it is. Facts are not separate from reality. Therefore, facts seem to exist in a reductive way similar to saying that “sandwiches exist.” The word “sandwiches” refers to various potential configurations of particles and nothing more. If those configurations of particles exist, then sandwiches exist. The fact that sandwiches are configuration of particles does not mean that sandwiches don’t really exist and only configurations of particles exist. Additionally saying sandwiches exist doesn’t mean two different things exist for every sandwich—both a sandwich and a configuration of particles.

In a similar way I suggest that when we say that “it’s a fact that a cat is on the mat,” we’re not saying “it’s a fact that a cat is on the mat and the cat is on the mat.” We are merely saying one thing about reality and there are multiple ways to describe that reality.

A similar issue is that there are so many different “facts” concerning reality. We might say that “it’s a fact that a cat is on the mat” or “it’s a fact that a brown cat is on the mat” or “it’s a fact that a cat is on the red mat.” This seems to imply that there are several different facts just because we can describe reality in so many ways. I don’t think it violates Occam’s razor precisely because we are merely describing reality in many ways. Saying several facts exist doesn’t actually mean there are more entities that exist because our talk of facts is reducible to reality (whatever that is).

Third, the objection is self-defeating because any good argument will require potentially true premises, but no statements can be true if facts don’t exist (perhaps other than the statement that facts don’t exist). Good arguments require us to have plausible premises and a plausible conclusion—we need to believe that the premises are true in order to believe that the conclusion is true. The objection requires us to believe that it’s true that “believing ‘facts exist’ violates Occam’s razor” and it requires us to believe the conclusion is true (that facts don’t exist). Perhaps the conclusion would be true if facts don’t exist, but the premise can’t be. Therefore, the objection is self-defeating.

Finally, any objection to the existence of facts would be self-defeating in this way—they can’t be good arguments if facts don’t exist. The fact that all such arguments are self-defeating gives us a good reason to agree that at least some facts do exist. At least it could be consistent with our ability to produce good arguments. An anti-realist who rejects all facts could reject argumentation entirely, but such a person would no longer have the ability to be reasonable.

Conclusion

The word “fact” is meant to refer to something that exists. Moreover, it is not only reasonable to believe that facts exist, but arguments that facts exist seem much more reasonable than arguments that conclude that no facts exist. Finally, there might be an alternative view to the view that at least some facts exist, but it’s not clear what that view is. Until then we can’t consistently argue that no facts exist, and it seems like we have to believe that facts exist to consistently believe that it’s possible to give good arguments (for any belief whatsoever).

Updates (5/23/12, 6/20/2012): I made some minor corrections and clarifications.

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Philosophers don’t generally talk about default positions or burdens of proof. Those terms might be more suitable for determining good debate than help us better understand knowledge or rationality. We might wonder if these concepts are useful at all when considering what we should actually believe.

An illustration of a default position.

Some entities seem so ordinary that they should be assumed to exist without significant evidence. I have two sisters, and you are likely to believe me even if you don’t know me personally. I think it’s perfectly rational to believe me. These are two entities in the universe that make a difference in the world. They eat food, raise children, and do lots of interesting things. It could be argued that you know you can generally trust people who make claims like this and that the fact that I tell you that I have two sisters is genuine evidence because you know the claim is ordinary.

It could be further argued that it would be irrational to believe that I have two sisters out of nowhere if you had no evidence whatsoever. In that case the “default position” prior to evidence is that my sisters don’t exist, but it is perfectly rational to believe that I have two sisters considering the evidence that is presented.

The default position concerning atheism.

To understand what the term means, we should consider a context in which it exists. Perhaps the most popular argument involving the “default position” is that atheism is the default position (at least in the sense of not believing in gods)¹ and therefore theists have the “burden of proof.” This is the sort of context that I am interested in.

The argument that atheism is the default position.

I believe that the concept of a “default position” was popularized by Antony Flew who introduced us to the argument that a skeptical form of atheism should be presumed prior to debate in his essay “The Presumption of Atheism,” which can be found in his book from 1984, God Freedom and Immorality: A Critical Analysis, (or on the website Positive Atheism). Flew argues that a skeptical form of atheism (not believing in gods) is the default position and those who say gods exist (or don’t exist) have the burden of proof.

Flew doesn’t use the term “default position” but there does seem to be a concept of a “default position” at work—the burden of proof is only against theism prior to the consideration of evidence and arguments.

Flew’s essay seems to present something like the following two arguments:

Argument 1

1. An onus of proof against theism will help us avoid believing in gods without knowledge.
2. We want to avoid having beliefs without knowledge.
3. Therefore, we should require an onus of proof against theism while having debates.

Argument 2

1. If we have an onus of proof against theism (prior to argument), then (prior to argument) there will be no reason to believe in gods.
2. If we have no reason to believe in gods, then it’s irrational to believe in gods.
3. We should not be irrational.
4. Therefore, we should have an onus of proof against theism (prior to argument).

An argument against believing gods don’t exist.

Note that Flew does not seem to give us a strong reason to treat theism as more problematic than the denial that gods exist. We could rephrase his arguments in the following way:

Argument A

1. An onus of proof against the belief that gods don’t exist will help us avoid believing gods don’t exist without knowledge.
2. We want to avoid having beliefs without knowledge.
3. Therefore, we should require an onus of proof against the belief that gods don’t exist while having debates.

Argument B

1. If we have an onus of proof against the belief that gods don’t exist (prior to argument), then (prior to argument) there will be no reason to believe gods don’t exist.
2. If we have no reason to believe gods don’t exist, then it’s irrational to believe that gods don’t exist.
3. We should not be irrational.
4. Therefore, we should have an onus of proof against the belief that gods don’t exist (prior to argument).

However, many people want to say that the default position should be against those who disagree with them. Theists sometimes claim that the burden of proof is on atheists, and those who believe gods don’t exist sometimes claim that the burden of proof is on theists.

The meaning of the term “default position.”

In consideration of the arguments given by Flew, the “default position” seems to refer to either of the following:

1. The proper starting point of debate.
2. Positions that are rationally required prior to debate and without one’s background knowledge being considered.

Flew does not want to say that we know that believing that gods exist is irrational precisely because he is only talking about believing in gods prior to debate and without considering our background knowledge. Default positions (as he understands them) do not generally tell us what to believe. They only tell us what we can rationally believe given absolutely no evidence for or against a belief. According to Flew, we can’t rationally believe anything without evidence, so it’s trivially true that a skeptical position (what he calls atheism in the theist debate) is the starting point of any debate given his principles.

When is something a default position?

Not everyone agrees about what should count as the “default position” because people have different attitudes concerning debate and often disagree bout what principles of justification are true. I will discuss three views concerning what default positions we should have. One, we might think that various far-fetched entities should be rejected within the default position. Two, we might think that we should assume things don’t exist as a default position. Three, we might think that we should reject beliefs as a default position when there are two or more alternatives to the belief. I do not personally endorse any of these views and I will touch upon the reasons why I do not endorse them.

1. Shouldn’t assertions for far-fetched entities be rejected as a default position?

Perhaps a case can be made that the default position is against those who believe in unicorns, dragons, goblins, and fairies—that it’s rational to believe these things don’t exist prior to debate and without considering our background knowledge. However, I find it plausible that those who believe in such entities have the burden of proof precisely because such entities conflict with our background knowledge. These entities have never been scientifically proven to exist and we find it more likely that people made them up. The “default position” should not include our background knowledge, and I don’t think we can generally know which beliefs are “far fetched” without considering our background knowledge.

2. Shouldn’t we assume things don’t exist as a default position?

It has been suggested that anyone who says something exists has a special burden of proof and the non-existence of that thing should be the default position. This principle could seem useful to explain why we should believe that unicorns, dragons, goblins, and fairies don’t exist. If we accept this principle, then the default position is that unicorns, dragons, goblins, and fairies don’t exist. We are likely to struggle to prove they don’t exist, but we are still rational in thinking they don’t exist anyway.

I do not currently agree that claims of existence should be rejected as a default position. Consider the following two beliefs:

1. The past resembles the future. – We justifiably believe that induction is reliable, which means we justifiably believe that we can generalize facts about the future based on facts about the past. For example, we justifiably believe that the same laws of physics will probably exist in the future. This seems to imply that we believe that we live in a universe with unchanging laws of physics—that a universe exists with unchanging laws of physics. We ultimately can’t prove anything like this without fallacious reasoning. If we should assume something exists without proof, then the principle that default positions deny the existence of things is false.
2. The external world exists. – We all live in a shared world of tables, chairs, animals, and so on. The world does not only exist in a dream. It isn’t clear what kind of evidence we have for believing such a thing exists, and many philosophers struggle to prove such a thing. At the same time the assumption that the external world exists is rational, even if we don’t know how to prove it’s true. If such an assumption is rational, then we should deny the principle that default positions deny the existence of things.

Rather than accepting that claims of nonexistence should be the default positions, we could accept something like Karl Popper’s epistemology. We could invent hypotheses and assume they are true until we have a good reason to reject them. We could be rational in holding assumptions until there are better alternative beliefs available.

3. Shouldn’t it be a default position to reject beliefs when there are two or more alternatives?

For example, if I roll a six-sided die, then the odds of rolling a 2 are 1:6. I will only roll a 2 (on average) once out of six rolls. It seems like the default position (prior to knowing if the die is weighted and so on) is to reject the belief that I will roll a 1 because (as far as I can tell) there are five other possibilities and I have no way to say one possibility is more likely than the rest. The default position would require me to withhold judgment in this case because there’s no reason to think any number in particular will be rolled. In a similar way it has been suggested that it can be irrational to believe anything else as a default position when there are two or more alternatives because as far as we know each belief in particular is “probably false.” (As far as we know when we have no evidence for any of the possibilities.)

Another example is that the belief that only one god exists and it’s Zeus seems irrational because the god could also be Jehovah, Isis, or Thor. Given these four gods, there’s only a 1:4 chance that one god exists and the god is Zeus (as far as we know when in a default position). Assuming we should believe whatever is likely true (and reject likely false beliefs), we should believe that it is false that one god exists and it is Zeus because there’s at least a 3:4 chance of such a belief to be true (prior to debate and without considering our background knowledge).

I am not convinced that the default position is to reject beliefs when there are two or more alternatives.

First, the alternatives we consider can be arbitrary and can lead to strange default positions as a result. We might wonder if fairies exist and decides that (1) fairies might not exist, or (2) fairies might exist and be blue, or (3) fairies might exist and be green. When considering these options, the default position would be to reject all three possible beliefs. However, the default position would seem to endorse the view that fairies exist because two possible beliefs are mentioned and only one possible belief is mentioned in which fairies do not exist. In that case we might think fairies probably exist—as far as we know there’s a 2:3 chance that fairies exist.

Second, I can imagine that there are at least two alternatives to the view that induction is reliable because the laws of physics will probably be the same in the future. It’s also possible that (1) induction will not be reliable because the laws of physics will be radically different in the future, (2) induction will not be reliable because laws of physics won’t exist in the future at all, or (3) induction will not be reliable because the laws of physics will only be the same in the next 100 years. If the default position is the reject beliefs when two or more alternatives are available, then (given these options) the default position is to reject the belief that induction is reliable. In fact, three of four possibilities are that induction will not be reliable in the future. Even so, I think the default position (assuming there are default positions) should be that induction will be reliable in the future. For some reason the principle in question seems to be giving us the wrong answer in this situation.

Are there any default positions?

Philosophers almost never talk about “default positions” and we might wonder if such a concept is useful or if it refers to anything at all. The problem with default positions is it requires us to analyze what beliefs are rational prior to evidence and without considering our background knowledge. However, almost nothing in philosophy is ever considered in such abstract isolation and we might wonder if it’s even possible.

Does giving people who make assertions a burden of proof help us achieve knowledge?

Flew seems to suggest that we should give a burden of proof to everyone making an assertion because it will help us avoid having beliefs without having knowledge, so we should give theists (and those who believe gods don’t exist) a burden of proof during debates for that reason. This is an interesting argument, but the conclusion seems either trivial or too extreme.

Is it trivial? – First, I agree that anything controversial in a debate should at some point be challenged and the challenge should be responded to. Second, I agree that all debates—such as the debate over theism—should start with arguments on “both sides.” And I know of no one who would disagree with Flew if that’s what he wants to say.

Is it too extreme? I think we should make the clarification that not all assertions made in debates should be challenged. There are limits to what the “burden of proof” should cover and sometimes premises of arguments should not be questioned precisely because adequately rational people will already agree with them. Questioning every premise would become a waste of time at some point.

Are beliefs irrational unless we have a reason to believe them?

Flew tells us that all beliefs are irrational unless we have a reason to believe them and Argument 2 trivially followed from that belief. If all beliefs are irrational until they are justified by an argument, then of course all assertions should be given justifications (and those who make assertions will have a burden of proof until they give the justifications).

However, I do not agree that all beliefs are irrational unless we have a reason to agree with them. I have many beliefs that I don’t know how to give evidence for, but I think I know they are true anyway. I know “1+1=2” even though I can’t prove it. I know the induction is reliable, even though I can’t prove it (and I realize it’s possible that I am wrong).

Additionally, few to no people can prove that “all beliefs are irrational unless we have a reason to agree with them.” Anyone who can’t prove it and believes it anyway is being irrational. The principle itself seems self-defeating.

Finally, imagine that you can prove that “all beliefs are irrational unless we have a reason to agree with them.” In that case you will need an argument containing evidence. But that argument containing evidence will have premises that must also be proven to be true. We will need an argument to prove all of our premises are true, and that will go on forever (and clearly we are incapable of doing that)—or we will end up reusing certain premises and rely on a circular argument, which is fallacious.

If is possible that the assertion that “all beliefs are irrational unless we have a reason to agree with them” is self-evident, but I see no reason to think it is.

Perhaps one reason to believe in “default positions” is if we believe that some beliefs are irrational prior to debate and evidence. Perhaps a default position of that kind exists, but we need to know what criteria determines certain beliefs to be irrational within default positions, and I don’t agree with the criteria given by Flew.

Conclusion

There is at least one trivial concept of a default position that I have no problem with (concerning who has a burden of proof prior to debate), but it is less clear that a stronger sort of default position exists that concerns what beliefs are irrational prior to debate or evidence.

I find Argument A to be plausible and agree that theists should have a burden of proof prior to debate (at least when that’s the topic of the debate), but Flew’s argument was a bit deceptive because people who believe gods don’t exist should also have the burden of proof for the exact same reason.

Notes

One idea behind the burden of proof is that someone has to prove something or we have no reason to agree with the claim being made. We could then say that person has a “burden of proof.” The person should prove (or argue for) some assertion they have made. If I tell you that fairies exist, then you could argue that I have the “burden of proof” because the existence of fairies is far-fetched or you simply know of no reason to agree with me.

Imagine that I say that fairies exist, and you disagree. I might then say that you have no evidence that fairies don’t exist, so you should believe in them. My reply would seem to irrationally reverse the burden of proof. It seems like I should give you a reason to believe in fairies and that you have no burden to give evidence that they don’t exist.

The burden of proof is related to Carl Sagan’s saying that “extraordinary claims require extraordinary evidence.” If we find out that fairies exist, then our worldview would be significantly changed, so it is seen as an “extraordinary claim.” We aren’t going to believe in fairies unless there is better reason to believe in fairies than there is to continue to maintain our current understanding of the world.

A principle of debate

In a debate, people are required to give us reasons to agree with their controversial claims. It is a fact that those in a debate (or those in the audience who watch a debate) probably think fairies don’t exist, so anyone in a debate who asserts (or requires us to believe) that fairies exist has a burden of proof.

Both sides in a debate start with a burden of proof.

People in a debate disagree about some fact (such as the existence of God). One person in the debate is then required to argue why the fact exists, and the opponent is required to argue why the fact doesn’t exist. The burden of proof is not specifically against only one side of a debate because both sides need to prove that their claim is true.

Arguments require premises—certain beliefs must be agreed-upon or we will not agree with the conclusions. Consider someone who argues that “Hypatia is a woman; all women are mortal; therefore, Hypatia is mortal.” In this case everyone is likely to already agree with the premises, so it’s a good argument in a debate.

However, imagine that someone argues that, “Hypatia is a fairy, all fairies are mortal; therefore, Hypatia is mortal.” In this case no one is likely to agree with the premises, so the debater has the burden of proof to tell us why we should agree with them.

The burden of proof in a debate can shift.

Although both sides start with a burden of proof in a debate, the person who has the burden of proof can shift. When two sides debate and they both argue for their conclusion, we might not know which side’s conclusion is true (and which arguments are unsound). At that point both sides will have a burden of proof to explain why the opposing argument is unjustified or inferior. Any side that fails to meet this burden of proof will continue to have one. For example, one person might argue that abortion should be legal because women have a right to choose and another can argue that abortion should be illegal because the fetus is a person. At this point we might not have any idea which argument is reasonable (if either of them are), and we will not be able to know which conclusion is true as a result.

For this reason, each side will not only argue for their conclusion, but they will object to the arguments given by the opposing side. The opposing side then has a burden to reply to the objections (and they will continue to have a burden of proof until they reply to the objections). Perhaps the pro-life advocate argues that women don’t have a right to choose because human life has value and the fetus is a human life. At that point the pro-choice advocate would have the burden of proof to explain why this argument is unconvincing, and that burden of proof will continue to exist until the explanation is given. (We could say that the pro-life advocate has the burden of proof to prove that abortion should be legal at that point.)

There are some exceptions to the general rules discussed here. Some arguments might be too unreasonable to be worth responding to. If an argument is obviously unreasonable, then it will not shift the burden of proof.

Do all assertions we make in a debate require the burden of proof?

Not all assertions we make in a debate necessarily have a burden of proof because those we debate with (and the audience) will agree with certain points. People already agree that murder is morally wrong, that other people have minds, that there is an external world, that we can legitimately generalize certain data, and so on. These beliefs do not need to be proven in a debate.

If we did have to prove all our premises to be true in a debate, then it would lead to a circular argument or an infinite regress. Imagine that Samantha argues that “Hypatia is a woman; all women are mortal; therefore, she’s a mortal.” I think this argument is perfectly good, but Johnny could then object, “How do you know all women are mortal?” Hypatia could then respond, “All women we know about died in the past before reaching two-hundred years of age.” Johnny could then object, “But maybe immortal women keep it a secret.”

This line of questioning can go on and on forever. For that reason people in a debate should agree not to question such plausible premises and they should provide as much evidence for their own position as they demand the opposing side to provide.

What about the supernatural?

Some people argue that assertions of the supernatural are “extraordinary” and require extraordinary evidence for that reason. This is only true within a debate when people don’t already agree with the supernatural assertions and demand those assertions to be justified.

A principle of rationality

Some people argue that it really is irrational to believe in fairies and rational to disbelieve in them because of the “burden of proof” (in or out of a debate context). This concerns how much we should be justifying our beliefs on our own, and what beliefs we are rationally required to have. Many people want to say that we should believe that fairies don’t exist and that fairy-believers have a burden of proof to justify their belief. If there’s no reason to believe in fairies, then it is supposedly irrational to believe in them.

The rational principle of the burden of proof is that some beliefs are initially irrational or less plausible than the alternatives prior to debate or evidence. Some beliefs should be rejected unless we have sufficient reason to agree with them. We would have a “burden of proof” for any belief that’s irrational without a needed argument.

The rational principle of the “burden of proof” is not entirely separable from the “burden of proof” in debate because people who debate also care about rational standards and often conclude that people should agree with them, or that people are irrational for disagreeing with their conclusions.

When are beliefs rational or irrational?

Rational beliefs (i.e. rationally permissible beliefs) are generally said to be beliefs we can have without being irrational. When is a belief rational? This is a controversial subject, but a relatively permissive view would say that beliefs are rational as long as we have absolutely no reason to reject them. This belief seems to be shared with Karl Popper, who argued that scientists (and people generally) invent hypotheses that are consistent with their understanding of the world and they believe those hypotheses are true until we have a sufficient reason to reject them. (Or, at least we know they should reject their hypotheses once we have a sufficient reason to reject them.)

The relatively permissive view does not say that we need to prove all our beliefs to be true, and that view seems to be implausible. For example, it would seem to lead to circularity or an infinite regress, just like when someone who questions us in a debate over and over forever. See “Beliefs are Innocent Until Proven Guilty” for more information.

A belief is irrational when we have a sufficient reason to reject it. We have a sufficient reason to believe that “1+1=3” is false, so it is irrational.

If a belief has a “burden of proof” (as a principle of rationality), then the belief is irrational unless the burden of proof can be met. (There should be a good argument as to why the belief is not irrational.) No belief has a burden of proof (or is irrational) in absolute isolation. However, beliefs can have a burden of proof (or be irrational) in consideration of our background knowledge (our current understanding of the world). At one point the belief that the Earth revolves around the Sun could have seemed irrational and perhaps we would need evidence in support of such a belief in order to justify it. That is no longer the case—now it would be shocking to find out that the Earth doesn’t revolve around the Sun.

Should we believe in fairies?

No, we shouldn’t believe in fairies. It is rational to believe fairies don’t exist, and it is irrational to believe fairies do exist.

Why shouldn’t we believe in fairies?

There are at least two main factors that determine the fact that we shouldn’t believe in fairies (and that those who believe in fairies have a “burden of proof”): (1) Occam’s razor and (2) our background knowledge. Beliefs come with a burden of proof whenever the belief requires us to believe something that conflicts with our current understanding of the world for no good reason. I will describe how Occam’s razor and our background knowledge determine that those who believe in fairies have a burden of proof.

Occam’s razor – Occam’s razor is a principle of justification that helps us determine if a belief is sufficiently rational or not. There are at least two important formulations of Occam’s razor. One, “don’t multiply entities beyond necessity.” (i.e. Don’t posit the existence of entities unless it’s necessary.) Two, “if two explanations are equally good in every other respect, then the simplest explanation is better than the other one.” Occam’s razor should encourage us to question whether the existence of fairies would be multiplying entities beyond necessity or if they are used as an explanation when simpler explanations are available.

People generally believe in fairies based on various experiences—stories they’ve heard and objects that seem to move around on their own. The problem with these experiences is that there are simpler explanations as to why such stories exist and why objects seem to move on their own other than the existence of fairies. For example, people could have just made the stories up, we might have faulty memories concerning where we leave objects, and humans might have a tendency to anthopomorphize (imagine that people-like entities exist where they don’t exist).

If everything we know about the world would remain unchanged without assuming fairies exist, then we have no reason to believe in them, and we have at least one reason to reject them (i.e. Occam’s razor). That should be the case for everyone. We should know how to fully understand the world without positing the existence of fairies.

Our background knowledge – Occam’s razor doesn’t help us in isolation. We need to know whether or not we are multiplying entities beyond necessity (or whether or not the simplest explanation is just as good as the alternative explanations in other respects). Consider fairies. They significantly conflict with our background knowledge (because, for example, (a) we would have expected to know about them by now, (b) they require us to believe in magic, and (c) we thought people just made them up). And if they don’t exist, then nothing we know about the world is changed. Therefore, fairies seem entirely extraneous and it doesn’t seem necessary to believe in them.

On the other hand the existence of pain is perfectly compatible and perhaps necessary based on our experiences, and the positing the existence of a neighbor being home seems reasonable when a light is turned on next door. In these cases we are positing the existence of substantial entities, but they seem compatible and perhaps even necessary for us in order to adequately explain and/or understand the world around us.

What about extraordinary claims?

There is a sort of “extraordinary claim” that is relevant to the burden of proof as a rational principle. In this case a claim is “extraordinary” when it significantly conflicts with our background knowledge. The claim that “extraordinary claims require extraordinary evidence” is relevant because the more a belief conflicts with our background knowledge, the more evidence will be needed before we can rationally accept the belief. We often say these claims are “ambitious” or “far-fetched.”

What about supernatural entities?

There is a question concerning whether or not anything supernatural exists whatsoever. Many people claim that every supernatural claim is “extraordinary” and conflicts with our worldview. It is true that supernatural claims do conflict with certain people’s worldview, but not necessarily everyone’s. For example, some people might think that psychological phenomena is supernatural, but there’s nothing extraordinary about psychological phenomena.

Even so, people should be consistent about how they treat supernatural entities. People reject supernatural entities in general. There is no religion I know about that agrees that all supernatural entities exist. Do fairies, vampires, werewolves, zombies, ghosts, demons, demigods, angels, or gods exist? Does Thor, Zeus, or Isis exist? Given any particular supernatural entity, it seems reasonable to expect people to be skeptical. Anyone who wants to reject all supernatural entities except a few of their favorites could be inconsistent in how they apply the burden of proof or principles of rationality in general.

Conclusion

Although people who discuss burdens of proof seem to be talking about the burden of proof as it exists within debate, I think what we are more interested in are burdens of proof as they exist in rationality. We want to know what we should believe, what beliefs are rationally permissible, what beliefs are rationally required, and what beliefs are irrational.

Related:

• Beliefs are Innocent Until Proven Guilty
• Five Tips To Better Debates
• Is Knowledge Impossible?

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I have made a comprehensive introduction to argument mapping and critical thinking in the form of an ebook. There is evidence that argument mapping is the most effective way to learn critical thinking. Argument maps are visual representations of arguments that help us understand how arguments work, and what it takes to make a good argument.

This ebook is no longer available for free on the website because I am considering publishing it. If you would like to use it for free to teach a class, you can send me a message with information about who you are, the school you work at, and how you plan on using it in the class. I would also require that you give me feedback. You can send me a message using the form below.

Abstract

This ebook is an introduction to argumentation, argument form, informal fallacies, and more by using illustrative argument maps.

Contact me:

Claudia María Álvarez Ortiz completed an in-depth analysis regarding the most effective forms of critical thinking education in 2007. Her MA thesis was Does Philosophy Improve Critical Thinking Skills? It can be downloaded for free right here in PDF format. She wanted to know how effective philosophy classes are at teaching critical thinking compared to other classes. Her study provides evidence for the following conclusions:

1. Philosophy classes aren’t significantly more effective than many other classes at teaching critical thinking (when the classes aren’t focused on primarily teaching critical thinking) (86).
2. Critical thinking classes are effective at teaching critical thinking (ibid.).
3. Critical thinking classes taught within the analytic philosophy tradition are generally more effective than the alternatives (ibid.).
4. Critical thinking classes that have lots of argument mapping practice are by far the most effective at teaching critical thinking (87-88, 100).

The meta-analysis suggests that the most effective class for critical thinking meets the following criteria:

1. It’s taught in the philosophy department.
2. It’s focused on critical thinking.
3. It has lots of argument mapping practice.

It should be mentioned that the meta-analysis equates “critical thinking” with “informal logic” (11). I don’t know everything that is measured by informal logic tests, but I know that informal fallacies would be involved.

There is a “broader conception” of critical thinking that includes formal logic, creative thinking, and so on. I think formal logic (at the very least) should be included in the definition and I’m not sure why it’s usually excluded from critical thinking classes. Formal logic skills should be even easier to test than informal logic skills.

Additionally, the actual benefits various classes have on us as human beings was not mentioned in this study. At the very least, there is evidence that philosophy for children helps children in many ways (see “Philosophy for Kids”) and there’s evidence that philosophy for adults helps them do better on standardized tests (see “Philosophers Excel on Standardized Tests”).

Finally, I am confident that there are benefits for attaining informal logic skills as well, but I do not know of any studies that tell us exactly what those benefits are. Even so, I would expect the following benefits to be included:

1. Better ability to persuade others.
2. Better ability to correct one’s beliefs that rely on poor reasoning.
3. Identifying deception, such as the manipulation commonly used by politicians, quacks, and cults.

Related:

• Introduction to Critical Thinking & Argument Mapping
• A Study Finds That Formal Logic Can Help High School Students

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