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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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9  
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
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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
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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
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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

2 Answers 2

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

First off, you've mentioned a traditional syllogism, NOT an Aristotelian one (an Aristotelian one would go "if All recursively enumerable sets are Diophantine., and if ..., then ...). See Jan Lukasiewicz, a scholar of the history of logic with access and knowledge of the Greek, in his Aristotle's Syllogistic: From the Standpoint of Modern Formal Logic.

Such syllogisms surely can get used. Consider the following: "all prime numbers greater than two are odd. Some natural numbers belonging to {a, b, c, d, e, f, g} are prime, where a, b, c, d, e, f, and g indicate distinct natural numbers greater than 2 and less than 12. Some numbers belonging to {a, b, c, d, e, f, g} are odd."

In short, it's not hard to claim that others "exist" in the sense that we can form true statements using traditional syllogisms, as basically traditional, Aristotelian, and modern predicate logic allow us to make all sorts of true statements even if no one has written them yet.

Whether this qualifies as "modern mathematics" or not all depends on one's point-of-view of the history of mathematics. One could very easily claim that Aristotle and actually all the ancient Greeks participate in "modern mathematics" in some way or another, since they place a priority on proofs, and so far as we know, the mathematics of homo habilis did NOT do this.

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