Aristotelian syllogisms in modern mathematics?

Question

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.

Answer 1

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

Answer 2

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.