Chapter 12: Formal Fallacies & Symbolic Logic

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Before beginning this chapter, please evaluate the following examples.

Argument 1: All cats are animals, therefore all animals are cats. 

Argument 2: All bears are strange creatures, so all strange people are bears.

Argument 3: All whales are mammals, so all mammals are whales. 

Argument 4: If I am happy then I am singing. I am singing. Therefore, I am happy. 

Argument 5: If I am taller than Sue, then I am short. I am short. Therefore, I am taller than Sue.

Argument 6: If God exists then there is good in the world. There is good in the world. Therefore, God exists.  

Argument 7: If you are correct then pigs are flying. You are correct, therefore pigs are flying. 

Argument 8: If goodness rules the world then God exists. Good does not rule the world. Therefore, God does not exist.

 

In Chapter 3, we explore informal fallacies. In this chapter, we will explore formal fallacies.

Informal fallacies (Chapter 3) are those in which you must examine the content; you cannot tell if it is fallacious by form alone. For example, the composition fallacy is informal because it is sometimes valid to infer the quality of the whole from the parts and sometimes invalid. You must examine the content in each case.

Formal fallacies, on the other hand, are arguments with a bad form or inference. You do not have to think about the meaning of the words, you can see the arguments are fallacious by their form alone. For example, the following argument is fallacious by its form alone:

 

All A are B, therefore all B are A. 

 

This form of argument is always and absolutely fallacious or invalid. It does not matter what A and B represent (as long as A and B represent different things). Because we can tell by its form alone that it is always fallacious, it is called a formal fallacy.

To clarify, let's plug in some meanings for A and B in the argument: "All A are B, therefore all B are A:"

 

Argument 1: All cats are animals, therefore all animals are cats. 

Argument 2: All bears are strange creatures, so all strange creatures are bears.

Argument 3: All triangles are three sided figures, so all three sided figures are triangles. 

 

Notice all three of these arguments have bad form or, to use the vocabulary from Chapter 2, a bad inference. So, even if we assume the premises are true in each of these arguments, the conclusion does not follow. We do not even need to think about the content to evaluate these arguments because we can immediately see they are fallacious/invalid by their form alone. That is, we do not need to think about the relationship between cats and animals, bears and strange creatures, or triangles and three sided figures because we can immediately see these arguments take the same bad form (i.e. All A are B, therefore all B are A). By the way, this formal fallacy is called “illicit conversion.” You can learn more about it by studying Categorical or Aristotelian Logic, which is the first form of symbolic/formal logic.

Perhaps you can now see one reason why studying symbolic/formal logic is valuable. It trains you to see arguments in form, so you can more quickly and accurately evaluate their validity. In this case, you can say it is always and absolutely invalid to infer "All B are A" from "All A are B" as long as A and B represent different things.

Ok, I started with a clear and easy example. Let's consider a more difficult argument form that often arises on IQ and entrance exams. Such exams may present one of the three arguments below and ask, "Is this a good argument?"

 

Argument 4: If I am happy then I am singing. I am singing. Therefore, I am happy. 

Argument 5: If I am taller than Sue, then I am tall. I am tall. Therefore, I am taller than Sue.

Argument 6: If God exists then there is good in the world. There is good in the world. Therefore, God exists.  

 

Interestingly, over sixty five percent of people mark these arguments as good, but that is incorrect. Of course, a person who studies formal logic will know this immediately because these arguments take the same fallacious form called affirming the consequent:

 

If A then B. 

B.

Therefore, A.

 

So, students of symbolic logic do not have to work out the relationship between singing and happiness in the first argument or think too deeply about Sue's height in the second. Nor will they be distracted by the highly emotional and connotative words in the third argument (e.g. God and goodness). They simply recognize that it takes this invalid form and are done.

Ok, let's look more closely at affirming the consequent.

Consider again the three fallacious arguments.

 

Argument 4: If I am happy then I am singing. I am singing. Therefore, I am happy.

Argument 5: If I am taller than Sue, then I am short. I am short. Therefore, I am taller than Sue.

Argument 6: If God exists then there is good in the world. There is good in the world. Therefore, God exists.

 

Can you explain how you would change each to make it a good/valid argument form? Let's focus on argument 1.

One possible answer is to change the second sentence to "I am happy" instead of "I am singing." This changes the argument form to the following:

 

If A then B.

A.

Therefore, B.

 

This argument form is called modus ponens and it is always valid. Instead of the fallacious argument, we now have a good argument:

 

If I am happy then I am singing. I am happy. Therefore, I am singing.

 

A note on vocabulary: these arguments contain conditional statements(if-then statements). The antecedent is the part that comes first and the consequent is the part that comes second. For example, "If A then B" is a conditional statement because it is an if-then statement. A is called the antecedent because it comes first (just after "if" and before "then") and B is the consequent because it comes last (just after "then"). This should help you better understand why this fallacy is called "affirming the consequent."

Ok, so we have now discovered two argument forms. One is fallacious and is called affirming the consequent. The other is valid and is called modus ponens. Let's put them side by side:

 

Modus Ponens (always a good/valid form)   If A then B. A. Therefore, B. Example: If it is a cat then it is an animal. Fido is a cat. Therefore, Fido is an animal.

Affirming the Consequent (always fallacious)   If A then B. B. Therefore, A. Example: If it is a cat then it is an animal. Fido is an animal. Therefore, Fido is a cat.

 

Now, substitute anything you want for A and B. The modus ponens argument on the left will always be valid because the conclusion must follow if you assume the premises are true. But, on the right, the affirming the consequent form of argument is a poor imitation of modus ponens. When we assume the premises are true, the conclusion simply does not follow from those premises. Read over the cat example in the box and substitute in your own examples for A and B. You will find this argument form is always fallacious/invalid no matter what A and B represent, as long as they represent different things.

Interestingly, we can now see why formal logic is absolutist. Modus ponens is always and absolutely valid while affirming the consequent is always and absolutely invalid. No matter what we plug in for A and B, modus ponens has good form/inference, and affirming the consequent does not.

Now, remember what we learned in chapter 2. There are two ways arguments can go bad: bad premises or a bad inference/bad form. Formal/Symbolic logic is only concerned with inferences/forms (i.e. validity and invalidity). To drive this point home, consider the following argument:

 

Argument 7: If you are correct then pigs are flying. You are correct, therefore pigs are flying.

 

This argument is actually valid, it is not a formal fallacy. I immediately know this is a good form/inference because it takes the form of modus ponens: "If A then B, A, therefore B." A represents "you are correct" and B represents "pigs are flying." Again, this argument and any other argument that takes the form of modus ponens is valid because the conclusion must be true if we assume the premises are true (see last chapter for a review of what validity means).

Now, you might protest and argue "the first premise is false." It is false that "If you are correct then pigs are flying."

This is true, but formal fallacies are not about whether the premises or conclusion are true. They are about whether the conclusion really follows when we assume the premises are true. In the vocabulary of chapter 2, formal logic is about inferences or form, not whether the premises are true or not. In the vocabulary of the last chapter, formal fallacies are about validity and invalidity, not soundness or unsoundness.

So, I would reject this argument because it has a false premise and is therefore unsound. The argument is valid, but unsound.

The important point is formal logic gives us the ability to immediately determine whether an argument is valid or fallacious/invalid. If it is valid, we can then leave the realm of formal logic and think about whether the premises are true.

Ok, let's consider two more argument forms.

Modus Tollens is a good/valid argument form, but denying the antecedent is formally fallacious.

 

Modus Tollens   If P then Q. Not Q. Therefore, not P.   Example: If it is a cat, then it is an animal. It is not an animal. Therefore, it is not a cat.

Denying the Antecedent   If P then Q. Not P. Therefore, not Q.   Example: If it is a cat then it is an animal. It is not a cat. Therefore, it is not an animal.

 

 

By the way, I am now using P and Q instead of A and B, but it doesn't matter what letters you use. I could have used A and B.

Denying the antecedent is always fallacious, it does not matter what P and Q represent. Nor does it matter if the premises are true or false, the form/inference is bad.

 

Consider argument 8: If goodness rules the world then God exists. Good does not rule the world. Therefore, God does not exist.

 

Many people will be confused and distracted by the terms in this argument. But students of formal logic- if they had a good teacher- bypass that because they can immediately see this argument is a formal fallacy calleddenying the antecedent. Such a person might say, “Even if the premises are true, the conclusion does not follow.” On the other hand, people who have not studied formal logic will ask complicated questions about the nature of god, goodness, and what it means to rule the world. So, what I am trying to sell you is the idea that the study of formal logic is indeed valuable, but most people don’t believe it until they actually learn and practice it in the proper way.

So, there you have it. You learned three formal fallacies (illicit conversion, affirming the consequent, and denying the antecedent) and two valid argument forms (modus ponens and modus tollens). When you study the formally fallacious and formally valid argument forms, you are developing a new tool. . .  or metaphorically loading new software unto your wet brain. You are learning to see the world in a new way. With this ability, you can more quickly and accurately evaluate most arguments.

So, as you listen to the arguments coming from loved ones, the media, your brilliant professors, and so on, take a moment to put the argument in form. Test the form and learn the valid and fallacious forms. You will eventually have the advantage of seeing in form as well as content… you will have an advantage over creatures who only see the world in terms of content and cannot therefore transport themselves into the Platonic and formal universe full of palm trees and ocean essences.

In the following exercise, you can practice identifying the formal fallacies from this chapter, as well as some informal fallacies from chapter 3. To learn even more formal logic, I recommend watching my Youtube video in the next chapter  and reading Patrick Hurley’s Concise Introduction to Logic or Harry Gensler’s Introduction to Logic. Finally, don’t forget to put the arguments you hear every day in form, and then evaluate the form before you evaluate the content. Enjoy!

 

Exercise 1: Evaluate the following arguments. Are they valid or fallacious? If fallacious, is it an informal (Ch. 3) or formal fallacy?

 

1. If I'm happy then I'm singing & clogging. I'm singing & clogging. Therefore, I'm happy.
2. I I ace the test, I will ace the course. I aced the course, so I aced the test.
3. If Adam is a man, he is mortal. He is a man. Therefore, he is mortal.
4. If you are correct then pigs are flying. You are correct, so pigs are flying.
5. If God exists, then there is good in the world. There is good in the world. Therefore, God exists.
6. If Bob is washing dishes then Bob is angry. Bob isn't washing dishes, so he isn't angry.
7. If Bob is washing dishes then he is angry. Bob isn't angry, so he isn't washing dishes.
8. My consciousness is physical because my neurons are physical.
9. Every brick in the wall is red, so the whole wall is red.
10. I'm either an absolute success (100) or an absolute failure (0).  I'm not an absolute success, so I must be an absolute failure.
11. Well, I don't believe cigarettes are harmful. After all, my grandpa smoked and drank for over 80 years and lived to be 100! My uncle, on the other hand, exercised every day and didn't smoke. He died at the age of 40. I wish people would be more logical & use their brains. Fools!
12. According to the Theory of Evolution, the fittest creatures will survive. Therefore we shouldn't make special efforts to feed the poor. If they can't survive on their own, that just means they aren't as fit as us. It's nature's way of weeding out defective human models.
13. If you don't work hard, you will be poor. Since Mary is poor, it's obvious she doesn't work hard.
14. Everybody knows vaccines are dangerous.
15. All clowns are dangerous people, so all dangerous people are clowns.

 

Answers

1. Formal fallacy, affirming the consequent
2. Formal fallacy, affirming the consequent
3. Valid argument, modus ponens
4. Valid argument, modus ponens. Again, it's valid because the form is good. I still reject the argument as unsound because I believe the first premise is false.
5. Formal fallacy, affirming the consequent.
6. Formal fallacy, denying the antecedent
7. Valid argument, modus tollens.
8. Informal fallacy, composition.
9. Good argument, does not seem to be a composition fallacy since all the parts of the wall are red and redness seems to transfer from the parts to the whole.
10. Informal fallacy, black and white fallacy (also called false dilemma, false dichotomy, polarized thinking, or the either/or fallacy).
11. Informal fallacy, cherry picking.
12. Appeal to nature fallacy. Do you think this fallacy is formal or informal? Can we ever infer goodness from naturalness alone?
13. Formal fallacy, affirming the consequent
14. Informal fallacy, ad populum.
15. Formal fallacy.

 

Application and Value

Studying formal logic (i.e. the forms of arguments) will help you more quickly and accurately evaluate all arguments.It will also give you the ability to teleport to distant planets and dimensions, such as Plato’s Formal Realm.

 

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