Ethical Realism

May 13, 2013

Logically Valid Arguments

Filed under: epistemology,philosophy — JW Gray @ 10:28 pm
Tags: , ,

A formal logic class or textbook should teach us ways to know when an argument has a valid argument form, and that can take a significant amount of time to learn. I encourage everyone to learn formal logic one way or another because it is of central significance to rational argumentation, and it is not something we spontaneously understand instinctively or through personal experience. Perhaps the first philosopher to understand formal logic and the importance of validity was Aristotle, and philosophers would have liked to understand it sooner. It was a great achievement because it can be so difficult to figure out on our own. Even so, we can learn a lot about valid argument form very quickly. I will explain why we need to make sure our deductive arguments are valid, give examples of valid argument forms, and explain how we can improve our arguments.

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, 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.

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2 Comments »

  1. Thank you for the post and the examples. Eversince reading Plato for the first time I’ve known to look for errors in logic arguments, but so far I hadn’t found the logical form to check the validity of said arguments.

    There seems to be an error in the Disjunctive syllogism.
    Shouldn’t it be:

    Either A or B.
    Not-A.
    Therefore, B.
    Or is this me misunderstanding the logic?

    Apologies for the shabby english, its not my native language.

    Comment by Historie Amsterdam — May 30, 2013 @ 5:24 pm | Reply


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