What You Need From Formal Logic

It can be extremely helpful to know some formal logic in order to visualize how arguments work. Formal logic shows how the form of an argument can be valid or invalid. Assuming that the premises of an argument are true, a valid argument form guarantees that the conclusion is true. An invalid argument doesn’t. In other words, a valid argument can give us a complete reason to agree to a conclusion and an invalid argument can’t. Knowledge about argument form helps us achieve clear thinking and relevant arguments.

I will discuss the following:

1. Argument form
2. Validity
3. Two common logic mistakes

Argument form

Argument form is the truth claim structure of an argument. The logical form can be revealed by erasing the content of the argument. In propositional logic, the content of assertions (propositions) are erased and all that is left are truth claims about those propositions.

For example, “If Socrates is a dog, then he is a mammal” and “if Socrates is a man, then he is mortal” both have the same form, “If A, then B.”

‘A’ and ‘B’ represent propositions. ‘A’ could be “Socrates is a dog,” “Socrates is a man,” or any other proposition. ‘B’ could be “Socrates is a mammal,” “Socrates is mortal” or any other proposition.

“If A, then B” gives us a truth claim: If ‘A’ is true, then ‘B’ is true. However, if ‘A’ is true and ‘B’ is false, then “If A, then B” would be false. For example, “If Socrates is a man, then he is a mortal” is true because then both ‘A’ and ‘B’ are true. He is a man and a mortal. However, “If Socrates is a man, then he is a reptile” is false. ‘A’ (Socrates is a man) is true and ‘B’ (Socrates is a reptile) is false, so it is false that “If Socrates is a man, then he is a reptile.”

Validity

A valid argument can’t have true premises and a false conclusion, but an invalid argument can. Valid arguments are not guaranteed to be good arguments, but they have an appropriate type of logical form.

An example of a valid argument is the following:

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

The argument form is the following:

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

Any argument using this form is valid, such as, “If Socrates is a lizard, then he is a reptile. He is a lizard. Therefore, he is a reptile.” Even though he isn’t a reptile, the truth of the premises guarantees the truth of the conclusion. The only problem is that one premise is false. Socrates is a famous philosopher, so he isn’t a lizard.

Why aren’t valid arguments automatically good? For one thing the fact that the above argument is valid didn’t make the premises or conclusion true. Socrates is not a lizard, so we have no reason to agree that he is a reptile. Good arguments need very plausible premises.

Why do we care? Because a valid argument with true premises also have true conclusions, and invalid arguments don’t give us a good reason to agree with the conclusions. Imagine that someone gives you an argument with true premises and a false conclusion:

1. If Socrates is a dog, then he is a mammal.
2. Socrates is a mammal.
3. Therefore, Socrates is a dog.

Socrates is a famous philosopher, so he is a human, not a dog. The premises are true and the conclusion is false, so it is invalid. The premises didn’t prove the conclusion to be true. In fact, the argument fails to convince us of the conclusion because it could be false, even though the premises are true. Any argument with this form will be invalid. The argument form is the following:

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

Some arguments with this form could look convincing, such as:

1. If Socrates is a man, then he isn’t a woman.
2. Socrates is not a woman.
3. Therefore, Socrates is a man.

This argument in and of itself doesn’t prove the conclusion to be true until we realize that there needs to be an additional premise—That Socrates can’t be both a man and a woman.

Common mistakes

We tend not to talk about formal logic at all. It was discovered by Aristotle around 350 BC. We tend to use logic automatically without thinking about it, but this often leads to mistakes. Some of the most common mistakes involve objections against other arguments. In particular, people (a) don’t always object to premises and (b) don’t always object to conclusions. We can’t reject a conclusion to a valid argument unless we can reject both a premise and the conclusion.

How do we use logic automatically?

We often expect others to provide us with valid arguments. For example, the argument “Abortion is wrong because it kills a person” could appear invalid on close inspection:

1. Abortion kills a person.
2. Therefore, Abortion is wrong.

The argument form appears to be “A, therefore B.” This couldn’t possibly be valid because ‘A’ and ‘B’ can be any propositions, even if they are unrelated. For example, “The sky is blue, therefore lizards are mammals.” The premise is true, but the conclusion is false.

Even so, we don’t read the argument as invalid. Why? Because the fact that “killing people is wrong” is taken to be common knowledge. We don’t require people to “spell it out” for us.

How to provide objections

If we want to prove that a valid argument’s conclusion is false, then we must prove that (a) a premise is probably false and (b) the conclusion is probably false. If a valid argument provides us with a false premise, then it might have a false conclusion, so we need to show that it is probably false based on a new argument. However, a valid argument with true premises also have true conclusions.

For example, consider the following argument:

1. You have a duty to help people in every way you can.
2. You could help people if you cure cancer.
3. Therefore, you have a duty to cure cancer.

The problem is that the premise “you have a duty to help people in every way you can” is false because it’s too demanding. We don’t need to devote our lives to every single cause. If we help people too much, we could neglect our personal needs.

However, I haven’t yet proven that you don’t have a duty to cure cancer because there might be some other reason that you have such a duty. Perhaps curing cancer is the most effective way that you could personally help others. But that is probably not the case for you personally because you probably don’t have the expertise necessary to cure cancer. I could provide an additional argument to support the view that you probably don’t have a duty to cure cancer:

1. Your time would be more effectively used to help others in some way other than curing cancer.
2. If your time would be used more effectively doing something else, then you don’t have a duty to cure cancer.
3. Therefore, you don’t have a duty to cure cancer.

Two Common Logic Mistakes

Only objecting to a conclusion: Most people seem to want to totally ignore the argument provided by their opponent because they only want to object to the conclusion. However, a valid argument with true premises automatically provides us with true conclusions. Therefore, objecting to the conclusion is often insufficient to prove that the opposing argument is flawed.

For example, someone who argues that abortion is wrong because it kills a person might face the objection, “But forbidding abortion violates the rights of women!” The fact is that abortion could be wrong even if forbidding it does violate the rights of women. If we want to know if abortion should be forbidden, we might need a cost/benefit analysis to decide, and violating women’s rights would only be one factor in that analysis.

Only objecting to a premise: Some people realize that objecting to a premise is a good idea, but that doesn’t mean the conclusion is false.

For example, someone who argues that Socrates is mortal because he is a dog could face the objection, “But he isn’t a dog!” This objection isn’t against the conclusion, it’s only against a premise. The conclusion is still true. Socrates is mortal whether he is a dog or not.

When could this happen in real life? Consider the above anti-abortion argument:

1. Abortion kills a person.
2. Killing people is wrong.
3. Therefore, Abortion is wrong.

People could then object to this argument by trying to prove that abortion doesn’t kill people. Perhaps a fetus is not a person. Then they could conclude that abortion isn’t wrong after all. This objection is inadequate because the fact is that we might not know if the conclusion is true or false. Even if abortion doesn’t kill people, it might be wrong anyway. (Killing fetuses might also be wrong.) The assumption many people seem to have is, “If a premise is false, then the conclusion is false.” That assumption is false. If a premise is false, the conclusion might be true anyway.

Conclusion

Thinking deliberately in terms of logic can take a great deal of time. However, you can start to benefit from learning about logic now. There are common logical errors people make when they ignore logic rather than embrace it.

I have only briefly discussed formal logic here. The mistakes mentioned are the most common that I have noticed, but the benefits I have gained from learning about logic is greater than just to provide relevant objections to arguments.

There are free resources on the Internet to learn more about formal logic including Introduction to Logic by Stefan Waner and Steven R. Costenoble. I have also written a free introduction to propositional logic, found here.

(Updated 11/21/2013)