Fallacy of Four Terms

5 Four Terms

The war against ISIS is no news. And no news is good news. So, the war against ISIS is good news.

Used by Rob Arp in every logic primer he gives to students

Humans are natural classifiers, sorting all kinds of things into categories so as to understand, predict, and control reality better. A category is a class, group, or set containing things (or members or elements) that share some feature or characteristic in common. We can construct a category of things that are dogs, a category of things that are human, a category of things that are red and left shoes, a category of things that taste sweet and fly, and on and on. In fact, it’s possible to classify anything that exists or you can think of into one category or another.

Aristotle was one of the first thinkers to lay out rules for describing the relationships between and among categories of things, as well as for reasoning with categories – thus, he’s considered the father of categorical logic. As he notes in Book I of the Prior Analytics, Aristotle’s principal tool for reasoning and for explanation is the syllogism: “discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.” And in Book II, he lays out probably the most well‐known syllogism: “If then it is true that A belongs to all that to which B belongs, and that B belongs to all that to which C belongs, it is necessary that A should belong to all that to which C belongs, and this cannot be false.” We recognize this syllogism today as:

1. All A are B.
2. All B are C.
3. All A are C.

Aristotle also describes other syllogisms that are well formed as well as ones that are not well formed and hence, fallacious. The fallacy of four terms (FT), also known by its Latin name quaternio terminorum, breaks this structure. An ambiguous middle term is one way it can arise.

The standard categorical syllogism contains exactly three terms. These terms have, since Aristotle, been called the major, the minor, and the middle. The major and minor are also called the “extremes” of a syllogism, since they lie on either extreme of the middle term. This rule is already explicit in Aristotle’s Prior Analytics, where he deduces at 41b36–8 that it “is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions.” And again at 42a30–35:

So it is clear that every demonstration and every deduction will proceed through three terms only. This being evident, it is clear that a conclusion follows from two propositions and not from more than two for the three terms make two propositions unless a new proposition is assumed, as was said at the beginning, to perfect the deductions.

The three-term structure lets a categorical syllogism relate two terms through a shared middle term. In order to form a valid syllogism, we must at least be able to relate two different terms (the major and minor) to one and the same term (the middle) such that we can make a valid inference that relates the major term and the minor. An argument with more terms may still be valid. It may, for example, contain a chain of categorical syllogisms.

In this example, there are two syllogisms, each one of which is valid on its own:

1. All cats are cute.
2. All cute things are blue.
3. All blue things are smelly.
4. Therefore, all cats are smelly.

In order to put this argument into proper syllogistic form, we would have to elucidate an implicit conclusion – that all cats are blue. The argument reduces to two syllogisms:

1. All cats are cute.
2. All cute things are blue.
3. Therefore, all cats are blue.

And then

1. All cats are blue.
2. All blue things are smelly.
3. Therefore, all cats are smelly.

In each of the above syllogisms, we have one term that is in both the premises (the middle term) and disappears in the conclusion, and two other terms, one of which is in the first premise, and the other of which is in the second premise, which both appear in the conclusion. The conclusion of each of these syllogisms is a necessary result of two premises, and the syllogism as a whole contains only three terms.

Two premises with no shared term cannot supply the middle-term link that a categorical syllogism needs. For example:

1. All cats are blue.
2. Your mom smells funny.
3. Therefore, …?

But there is a sneakier way to commit the fallacy of four terms, which is to use in place of the middle some ambiguous term that actually has two meanings, when we need it to have one. For example:

1. All asses eat hay.
2. Aristotle is an ass.
3. Therefore, Aristotle eats hay.

Looked at from the perspective of bare language, this looks like a valid syllogism, but the meaning of the word “ass” shifts between the first and second premise. Where in the first, “ass” refers to a donkey, the second use of “ass” is metaphorical, referring to a human having some of the same qualities as the animal from which we derive this usage. What we have in reality is another supposed syllogism where the two premises aren’t related at all, since the middle term is actually two terms. This is a specific instance of the equivocation fallacy: the same word is used with two different meanings, effectively introducing a fourth term. While the argument appears to have only three terms, the shift in meaning makes it invalid. The apparent middle term does not connect the two premises.

The same traditional account requires three terms, not fewer. This is a rule for categorical syllogisms, rather than a claim that every kind of deduction needs three terms. William of Sherwood (1966), in his Introduction to Logic of the thirteenth century, summarizes it as follows:

There will be no more and no fewer than three terms, since propositions cannot share two terms, for then they would be one and the same proposition. This is because there are exactly two terms in one proposition – viz., the subject and the predicate – for the proposition is analyzed (resolvitur) into the term. So there will be three terms in every syllogism. (60)

To summarize, every good syllogism has three and exactly three terms. There are two premises, one of which relates the major term to the middle, and the other of which relates the minor term to the middle. In a valid syllogism, the major and minor terms are related in the conclusion, and their relation is absolutely necessary and completely justified by the premises alone. An apparent syllogism with a fourth term needs a different analysis: has a meaning shifted, is the connecting premise missing, or is this a longer chain of reasoning?

A separate syllogistic error concerns negative premises—premises that deny a relationship between the subject and predicate. As soon as you have a negative premise, your conclusion must be negative as well. If you try to conclude something affirmative when one (or more) of your premises is negative, you commit the fallacy of illicit negative. This fallacy is the violation of Whately’s (1840) sixth rule of syllogisms:

6th. If one premiss be negative, the conclusion must be negative; for in that premiss the middle term is pronounced to disagree with one of the extremes, and in the other premiss (which of course is affirmative by the preceding rule) to agree with the other extreme; therefore, the extremes disagreeing with each other, the conclusion is negative. In the same manner it may be shown, that to prove a negative conclusion one of the Premises must be a negative. (87)

In general, Aristotle claims that to prove any sort of conclusion, there must be a premise of the same kind previously in the argument.

And it is clear also that in every syllogism either both or one of the premises must be like the conclusion. I mean not only in being affirmative or negative, but also in being necessary, pure, or problematic (Prior Analytics, 41b27–30).

This fallacy is the converse of that rule: If you want to prove a negative conclusion, one of your premises must be negative, and if you have a negative premise, your conclusion must also be negative. The modern logicians like to speak of this fallacy in the “agreement/disagreement” terminology, as Whately does above. An affirmation is an agreement, and a negation is a disagreement. Now, if one of the terms agrees with the middle, and the other disagrees, then they necessarily disagree with each other.

Just take a look at this terrible forgery of a syllogism:

1. All cats are clowns.
2. Some clowns don’t do drugs.
3. Therefore, some cats do drugs.

The premises do not tell us whether any cats do drugs. The clowns who do not do drugs might include cats, or they might not.

And then there’s this other terrible sham of a syllogism:

1. All cats are clowns.
2. No cats do drugs.
3. Some clowns do drugs.

If cats exist, these premises imply that some clowns do not do drugs: the cats are clowns, and none of them do drugs. Without that existence assumption, the universal premises do not establish that there are any clowns. In either case, they do not establish the stated affirmative conclusion. These cats-and-clowns examples illustrate errors involving negative premises; they do not contain four distinct terms.