# Aristotelian syllogisms in modern mathematics?

Somewhere (?) in the writings of Gian-Carlo Rota, I recall a statement that old-fashioned Aristotelean syllogisms are not used in modern mathematics. I know of one gaudy counterexample, and wondered whether there are others.

The major premise is Matiyasevich's theorem, proved in 1970:

All recursively enumerable sets are Diophantine.

The minor premise is a discovery in the 1930s, I think by several people including maybe, Kleene, Turing, and Church:

Some recursively enumerable sets are non-recursive.

(Matiyasevich built on work of Julia Robinson, Hillary Putnam, and Martin Davis, done over a couple of decades.)

The conclusion crosses the 10th item off of Hilbert's famous list of problems:

Some Diophantine sets are non-recursive.

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They are used every day. They are just not mentioned. –  André Nicolas May 18 '12 at 16:50
@AndréNicolas does that logic he used follow. –  simplicity May 18 '12 at 16:55
No, @MarianoSuárez-Alvarez It is $A\subset B$ and $A\cap C\neq \emptyset$ implies $B\cap C\neq \emptyset$ –  Thomas Andrews May 18 '12 at 17:14
I would argue that almost every "informal proof" (as opposed to formal proofs in a formal language, like the kind that can be machine-verified) is in fact, a syllogism, or strings of syllogisms parsed together. The verbiage added to make these "readable" often obfuscates this. –  David Wheeler May 18 '12 at 17:57
Every normal space is regular; some normal spaces are nonmetrizable; hence some regular spaces are nonmetrizable. There are lots of these. –  Carl Mummert May 21 '12 at 0:21

Today, they take on the form of theorems in predicate logic.

From Wiki:

'"In Aristotle, each of the premises is in the form 'All A are B,' Some A are B', 'No A are B' or 'Some A are not B,' where 'A' is one term and 'B' is another."

http://en.wikipedia.org/wiki/Syllogism

Translated into the notation of predicate logic, they are (respectively):

$\forall x: A(x) \rightarrow B(x)$

$\exists x: A(x) \wedge B(x)$

$\forall x: A(x) \rightarrow \neg B(x)$

$\exists x: A(x) \wedge \neg B(x)$

Here is a link to proofs of three classical syllogisms using predicate logic in my DC Proof system:

http://www.dcproof.com/ClassicalSyllogisms.htm

EDIT: To resolve the syllogistic fallacies, you will need to use the set-theoretic equivalents to construct counterexamples. See, for example, my resolution of the existential fallacy at:

http://www.dcproof.com/ExistentialFallacy.htm

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I used the word "gaudy", which is not very precise but I think it can be understood. However, I think I'm wondering which things worth knowing find their simplest expression in the language of syllogisms rather than in modern logical notation? –  Michael Hardy May 18 '12 at 22:10
The modern notation is very powerful. I have found that, with a bit of practice, the modern version of the syllogism is much easier (simpler) to work with. And, of course, the modern notation can be applied to a much wider class of problems. –  Dan Christensen May 19 '12 at 3:36
Your latest comment asserts the obvious and doesn't address the question. –  Michael Hardy May 20 '12 at 16:54
I think I have shown how Aristotelian syllogisms and the resolution the fallacies are handled in modern logic and set theory. –  Dan Christensen May 21 '12 at 4:37