Four Terms

Charlene Elsby

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.

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, Part 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. This chapter deals with the fallacy of four terms (FT). Also see the other chapters on the exclusive premises fallacy (Chapter 4), the illicit major and minor terms fallacies (Chapter 6), and the fallacy of the undistributed middle term (Chapter 7).

The fallacy of FT violates the very first rule of constructing a valid syllogism: any syllogism must contain three and only 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.

Aristotle’s reasoning for the idea that each syllogism must have three, and only three, terms is that three is the minimum number of terms required to make a deduction. 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. Of course, it is possible to have an argument with more than three terms, but a syllogism is but one deduction. If we are constructing a syllogism with more than three terms, then we either have multiple syllogisms or no syllogism at all.

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.

Without the middle term to connect the premises, no deduction is possible. That is why if we have two premises with four completely unrelated terms, then we cannot form any conclusion. That is, one of the terms needs to repeat in the first and second premises in order for any deduction to be made. If it doesn’t, then we have no way of making a reasonable connection. If instead we have four terms, then no deduction is possible. 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 FT, 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 the animal that is the offspring of a horse and 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.

On the converse side of things, a syllogism can’t have any fewer than three terms either.

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. When there are four terms, either the terms are unrelated and no deduction is possible, or there are multiple syllogisms.

There’s another thing about negative premises, that is, 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.

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 (1840) 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.

But we don’t know anything about cat habits from the information given. From those two premises, we can’t conclude anything at all.

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.

This doesn’t even make sense. In fact, we could conclude something from the premises given, namely that there are some clowns who don’t do drugs. That’s because if all cats are clowns, then some clowns are cats. And if no cats do drugs, then those ones that are clowns don’t do drugs. And therefore there are some clowns who don’t do drugs. But there’s absolutely no reason to think that you could prove the opposite.

References

Alexander of Aphrodisias. 1991. On Aristotle Prior Analytics 1.1-7, translated by Jonathan Barnes. London: Gerald Duckworth.

Aristotle. 1984. Prior Analytics. In The Complete Works of Aristotle: The Revised Oxford Translation, edited by Jonathan Barnes. Princeton: Princeton University Press.

Whately, Richard. 1840. Elements of Logic. London: B. Fellowes.

William of Sherwood. 1966. Introduction to Logic, translated by Norman Kretzmann. Minneapolis: University of Minnesota.