# Is there something in syllogism that is not covered by set theory?

I was looking at the wikipedia article on Syllogism where it describes the different types of syllogisms.

I understand the usefulness of syllogism when first introducing logic, and I also understand that it is much older than set theory. But I find all those descriptions much harder to understand than if I simply think in terms of sets. I spent a few minutes trying to understand one of the types (Modus Celarent) only to realize it was obvious if you think about it using sets.

So my question is the same as in the title: is there something in syllogism that is not covered by set theory? Or, more weakly, is it still useful to learn about it once you have the basic intuitions about sets?

• I am way out of my depth here, since I have no training in symbolic logic. But I am wondering if the classical syllogism retains a relation to ordinary spoken and written language that is useful. I am always a bit suspicious of "complete" reductions. In Hegel SL, for example, we see the forms of logic contorted towards the "ironic" or "paradoxical" half-meanings of language, which are simply not captured in mathematics. Some of the "failures" of the old logic may be expressive of the "useful ambiguities" of language itself. Obviously, this is not useful "in logic" but perhaps in some sense? Oct 11 '15 at 18:22
• There's more to logic than that contained in Set Theory; syllogisms are only one part of Aristotles Organon - though the most well-known. Oct 15 '15 at 11:06

As regards the first question, the answer is a negative one, since terms can be interpreted as sets, and the syllogistic relations as relations between sets. I have mentioned one way of achieving this elsewhere, so I'll just repeat Definition 1 from that post here for your convenience:

Definition 1. (Set Theoretic Semantics for Aristotle's Categorical Syllogistic)

1. AaB  =df         B ⊆ A              ('all Bs are As'),

2. AeB  =df         B ∩ A = ∅       ('no Bs are As'),

3. AiB   =df       ¬(B ∩ A = ∅)     ('some B is A'),

4. AoB  =df       ¬(B ∩ A = B)     ('some B is not A').

Usefulness though is a relative notion; just because we can find set-theoretic models of the syllogistic, it doesn't mean that taking the syllogistic relations as primitives is a useless idea. Compare this situation with modal logic: there is what's called a standard translation of modal formulas into first-order logic. Despite that, we find it useful to take at least one of the usual modal operators as a primitive and define things in terms of it.

• I agree with this answer, but there is one subtlety which should be noted. Many medieval commentators took an affirmative universal proposition like "All men are mortal" to entail the affirmative singular "Some man is mortal" which entails "some man exists." This is one important difference between the older syllogistic logic and modern first order languages, such as the language of set theory. Although this distinction is somewhat recherché, it is important. This has less to do with memorizing the figures and moods and more to do with philosophy of language though.
– user5172
Sep 24 '14 at 19:37
• @shane I'm not sure I understand. It sounds like the idea in Russell's Theory of Descriptions, is it related? Sep 25 '14 at 13:33

Two mistakes in traditional usage of syllogisms were discovered by Peano, Freg, Whitehead & Russell, and were subsequently removed from set theory.

Mistake One:

Before Peano and Freg, traditional syllogisms mistook the relation of membership (∈) for a particular case of inclusion (⊂). This was pointed out by Peano. As a matter of fact, in virtual of traditional Barbara(AAA), "Socrates is mortal" can never be asserted. The following is an excerpt from Russell & Whitehead's Principia Mathematica:

Mistake two:

Whitehead & Russell pointed out that:

It should be observed that syllogisms are traditionally expressed with "therefore," as if they asserted both premises and conclusion, this is, of course, a slipshod way of speaking. Since what is really asserted is only the connection of premisses with conclusion. (Source: Whitehead & Russell, Principia Mathematica. Merchant Books, 1910)

For example, the following implication is true even though both its premises and conclusion are false:

If all horses have wings and goats are horses, then goats have wings.

• Interesting, I hadn't thought of that case, of something wrong in syllogism being removed. Sep 25 '14 at 13:37
• @Koeng, regarding your last question, here's what Russell says: "If you wish to become a logician, there is one piece of advice which I cannot urge too strongly, and that is: Do NOT learn the traditional formal logic. In Aristotle's day it was a creditable effort, but so was the Ptolemaic astronomy. To teach either in the present day is a ridiculous piece of antiquarianism." (Russell, Bertrand. The Art of Philosophizing and other essays. New York: Philosophical Library, 1968) Sep 25 '14 at 19:01
• But modern interpretation of Aristotle Logic has greatly improved since Russell's time. Regarding "Mistake One" : to be honest "Socrates is a man" is not a categorical propositions (i.e. with both terms general) and so not part of the theory of categorical syll. Thus, we can say that the set-theoretic interpretation of A's theory still works. Regarding "Mistake Two" : it is highly plausible that the "correct" interpretation of A's syll is not as a propositional law (if A and B, then C), but as an argument : 1/2 Sep 26 '14 at 14:27
• ... i.e."from premises A and B, conclusion C follows". If so, "therefore" is correct. Sep 26 '14 at 14:28
• @Mauro, To be fair, A's categorical Syll can never reach individuals, but is nevertheless correct; only the traditional usage was mistaken. Same can be said about "mistake two.' It was the interpretation that was ambiguous, as was pointed out by Lewis Carroll. Sep 26 '14 at 19:04