Chapter 7

The categorical syllogism

The last of the major operations I am going to treat is the "categorical syllogism," (from the Greek word for "predicate") first formulated by Aristotle, and one of the great achievements of the human mind.

Since the predicate term implicitly refers to a set of objects, it can be thought of as a case of class inclusion--and in fact, this is how Aristotle thought of it. For this reason, in the traditional formulation of the syllogism, the proposition involving the larger classes is put first, and the one with the smaller second, as in "All human beings are mortal, but [i.e. "and"] sailors are human beings; therefore all sailors are mortal." But notice that in this arrangement the term that mediates between what will become the subject and predicate of the conclusion (called, traditionally, the "middle term") is at the extremes of the two propositions (the subject in the first, and the predicate in the second: on the "outside," so to speak, of the propositions).

But it makes sense if you think of it as saying that human beings are included as an indefinite part of the class of mortal beings; and sailors are included within that smaller class; and so obviously they are included within the larger class that the smaller is included within.

Still, I don't think this is really what is going on in predication, as I have said both in Chapter 5 of Section 3 of the third part 3.3.5, and earlier in this section on logic; and as a matter of fact, this way of looking at predication makes obvious inferences like "A horse is an animal, and the head of a horse is the head of a horse, therefore, the head of a horse is the head of an animal" not able to be done. It can be done easily by a little rule, if you take the approach to the categorical syllogism that I am going to take, following L. Susan Stebbing.

The idea is since the predicate as a word also in itself refers to a set of objects as well as expressing the meaning (which is its function as predicate), then predication involves a relation between the class of objects pointed to by the subject and the class of objects potentially pointed to by the predicate.

Now since the predicate does implicitly point to a class of objects, predicates could be applied to it as if it were a subject; and sometimes the relation between its subject and the relation of it to a given potential predicate of it is what they call "transitive."

A "transitive" relation is a relation that applies in a chain-wise fashion. Some relations are transitive and some aren't. For instance, the relation "is the ancestor of" is transitive, because if John is the ancestor of Frank, and Frank is the ancestor of James, then John is the ancestor of James. But the relation "is the father of" is not transitive, because if John is the father of Frank and Frank is the father of James, that does not make John the father of James, but his grandfather.

Basically, the rules for the categorical syllogism simply list the times when the relation of predication is transitive. If you violate one of the rules, you run into a case where the relation of predication is like using "is the father of" instead of "is the ancestor of." Not surprisingly, since you are looking at the predicate as if it were referring to a class (instead of in its actual meaning-function), this relation of predication will be very similar to the relation "is included within" and "is excluded from."

But if you think of the relation as one of predication and not class inclusion and exclusion, then the syllogism above reads (putting it into logical form now): "Every sailor is a human being and every human being is something mortal; and so every sailor is something mortal." Here it is obvious that "human being" is what mediates between sailors and mortal things; the extreme terms are where they belong: the subject first and the predicate last; and the middle term is in the middle.

Now of course you can think of class inclusion in this way also. In this case, the smallest class is inside the middle one and the middle one inside the largest. Aristotle started from the largest, that is all. Traditional logic therefore calls the premise containing the largest class the "major" premise, and the one containing the smallest class the "minor" premise'--and if you think in this way, it makes sense to put the larger first. But I think a shift in terminology will put things closer to the way we speak and the way we follow speech.

The subject premise is the premise that contains what will be the subject of the conclusion, whether this term is the subject of its premise or not.

The predicate premise is the premise containing what will be the predicate of the conclusion whether it is the predicate of its premise or not.

The subject term is the term that is to be the subject of the conclusion.

The predicate term is the term that is to be the predicate of the conclusion.

The middle term is the term that does not appear in the conclusion.

So the subject term may be the predicate of the subject premise, or it may be its subject; and similarly with the predicate term. If you just say "subject" or "predicate" without adding "term" then you are talking about the premise the term is in. If you add "term" to this, you are talking about the fact that it is the subject or predicate of the conclusion, not of the premise it is in.

This is what I warned you about in the beginning of this section when I was defining "subject" and "predicate." Even though there may be some confusion here, I think it is clearer than calling the subject term the "minor term" and the predicate term the "major term." With my terminology, you can see what the function of the term is.

Now then, as I say, the rules of the categorical syllogism are simply the conditions under which predication is transitive. Traditionally there are nine rules; but two of them can be derived from the others. Some also do not give the first two of the traditional rules as rules, because they are contained in the very form of the syllogism itself; but I think they belong because they define the form. Some also combine my rules six and seven with an "if and only if" statement; but since this reads as two implications, I think they are better separated.

Rules for the categorical syllogism

1. There must be three and only three propositions.

2. There must be three and only three terms.

3. The middle term must be definite at least once.

4. If a term is definite in the conclusion, it must be definite in its premise.

5. Both premises may not be negative.

6. If one premise is negative, the conclusion must be negative.

7. If both premises are affirmative, the conclusion must be affirmative.

Why these rules?

As far as the first rule is concerned, in a categorical sorites you don't have three propositions; but the point of the first two rules is that the number of terms used equals the number of propositions. But sticking with the syllogism itself, there is just one transitive relation, which will involve three propositions: two premises and a conclusion.

As to the second rule, it is most often violated by using the same word (or phrase or clause) as two different terms. No one would offer something like this as a syllogism: "Every sailor is a human being and every fish is mortal; and so every sailor is mortal." It is obvious that no mediation is going on here. But here, for instance, the fourth term is not obvious: "The conclusion of every valid argument is true, and everything that is true is factually the case; therefore the conclusion of every valid argument is factually the case." But as we saw "true" the first time means "follows according to the rules from the premises, and the second time means, of course, "is factually the case." But if the premises are factually false, then the conclusion is still "true" in the sense that it follows; but it may be factually false. So in some cases, the two middle words point to different sets of objects; and so the conclusion doesn't follow. This is, of course, what is traditionally called a "four term syllogism."

As to the third rule, the easiest and most common way to violate it is to have the middle term the predicate of two affirmative propositions. If it is, it is indefinite both times; and it doesn't have "at least one" to go with it to show this, because predicates don't carry the tag of what part of the class they point to, since they're not really pointing.

But that makes this the "guilt by association" type of fallacy. For instance, "Every murderer is someone who violates society's rules, and every drug addict is someone who violates society's rules; and so every murderer is a drug addict."

If you think of this in terms of class inclusion, you know that the whole class of murderers is somewhere inside that of violators of society's rules; and the class of drug addicts is also somewhere inside that same larger class; but what this doesn't tell you is whether they're inside each other, partially overlap, or are in completely separate parts of the larger set. Demagogues use this rule quite a bit, and usually in this form, because its invalidity is disguised.

As to the fourth rule, the reason a term that is indefinite in the premise has to be definite in the conclusion is what we saw earlier in discussing conversion; your conclusion would be going beyond your evidence. From "at least one" you can't argue to "every" member of the same class. Note, however, that the middle term can be indefinite in the subject premise and definite in the predicate premise, and so appear to be "going from indefinite to definite." But here you are not concluding to anything; both of these are premises, and so don't depend on each other. That is, "Every sailor is a human being (indefinite), and every human being (definite) is a mortal thing" yields a legitimate inference to "Every sailor is a mortal thing."

As to the fifth rule, it is easiest to see the reason for it in terms of class inclusion. If two classes are excluded from a third, they could be anywhere in the universe, and don't have to have anything to do with each other, even though it's possible that one could be wholly or partially included in the other. From the point of view of predication, it basically says that no mediation is possible when the subject doesn't belong to the "middle" and the "middle" doesn't belong to its predicate.

As to the sixth rule, its necessity can be seen from what happens if you obvert the negative premise, as in "Every sailor is a human being and every human being is not a horse." If you obvert the second premise you get, "Every human being is a non-horse" without changing the meaning; and it is obvious that from this you can't conclude to anything about horses.

Just as the sixth rule essentially says you can't argue to a connection from a disconnection, so this seventh rule says that you can't argue to a disconnection by connecting.

Those are the rules, and a little bit of why they are the rules. Applications can be quite intricate, of course. In fact, I think it is worth mentioning even in this sketch that there are several possible "figures" (arrangements of subject and predicate) that the categorical syllogism can take, the clearest of which is the first, and the most unclear the fourth. Here, the dots between the letters for Subject term Middle term and Predicate term simply indicate some kind of copula, either affirmative or negative.

I         II         III         IV

S . M    S . M    M . S    M . S

M . P    P . M    M . P    P . M

S . P      S . P      S . P      S . P

Logicians have developed special rules for each figure (such as that in the second, one premise and the conclusion must be negative); but they are simply applications of the general rules, and so I personally don't see any reason why anyone would be forced (as I was) to learn them.

I said that the last figure was the most unclear. An example would be, "Every horse is an animal, and every maverick is a horse; therefore at least one animal is a maverick." I wrote this into a textbook and in my first version, I drew the conclusion, "At least one horse is a maverick," which uses the middle term "horse" three times. It is just a very confusing way of arranging terms, and is to be avoided. If you see it, convert one of the premises and go on from there, and you will be able to follow what is happening.

Of course, there is also the traditional arrangement of these "figures" with the second line first and the first line second. It is, as I say, somewhat less clear than the arrangement I gave; and the clarity deteriorates with the less clear figures.

Now then, I mentioned that the addition of a rule (and a corollary of it) to traditional logic would make the categorical syllogism fit things like the head of a horse.

Now the reason this won't work in traditional logic is that the mediating term becomes embedded as part of the middle term; and the traditional way of treating the term deals with the middle term as a whole. For instance, if you talk about the head of a horse, the term is "head of a horse," and you can't argue from the "horseness" of the horse in this form. But it's obvious that in the real world of reasoning you can. Hence the following rule:

Rule of substitution: If a term appears as part of a more complex term, then any predicate of the part can, in its indefinite form, be substituted for the term which is the part.

This handles affirmative propositions. To flesh out the enthymeme "A horse is an animal; therefore the head of a horse is the head of an animal," we begin with a tautology for the subject premise: "Every head of a horse is a head of a horse; and every horse is an animal; therefore (by substitution), every head of a horse is a head of at least one animal."

Similarly, "John loves Mary and Mary is a woman, therefore John loves a woman" becomes "John is something that loves Mary, and (every) Mary is a woman; therefore John is something that loves at least one woman." Note that here you have not made Mary the only woman in John's life--which wouldn't in fact follow from the fact that he loves her, unfortunately.

I might point out that in contemporary logic as it stands, this kind of inference can be made; but you need twelve steps to do it.

Dealing with negative propositions is more complex. To make the substitution, you first start with the tautology, then obvert the negative proposition, giving you an affirmative one you can use for substitution, thus: To formalize, "A horse is not a fish, and so the head of a horse is not the head of a fish," we start, "Every head of a horse is a head of a horse, and every horse is a non-fish." Substituting gives us "Every head of a horse is a head of a non-fish."

But even though this says the same thing as "Every head of a horse is not the head of a fish, you need another rule to make it come out that way:

Rule of substitutional obversion: The obverse of a term can be substituted for a term contained within a more complex term if the whole proposition is changed from affirmative to negative or vice versa.

So the complete inference goes like this "Every head of a horse is a head of a horse, and every horse is a non-fish; therefore, every head of a horse is a head of at least one non-fish; therefore, (by substitutional obversion), every head of a horse is not a head of at least one fish."

Or, "John doesn't love Mary and Mary is a woman, therefore John doesn't love a woman" becomes, "John is a non-lover of Mary, and Mary is a woman; therefore John is a non-lover of at least one woman; therefore, John is not a lover of at least one woman," which as you can see is clearer than the conclusion drawn the informal way.

Next

I don't at the moment see how you can infer from the horse and fish that the head of a horse is not the head of any fish at all (which knowledge of horses and fishes tell me is true); but those more ingenious than I can probably modify these rules to get around the difficulty--or maybe, as in the case of John and Mary, you can't legitimately conclude that from the fact that a horse is not a fish to something general about their heads.

In any case, this is all I am going to say about logic and its relation to statements.