Numerically definite syllogisms

[waveletter]

Hello aristotle-logic group:

A paper by Ian Pratt-Hartmann, 'On the computational complexity of the
numerically definite syllogistic and related logics,' appeared in the
recent issue of "The Bulletin of Symbolic Logic," vol. 14, no. 1,
March 2008, p. 1. I thought it was interesting in that its valid
arguments are remarkably simple, like Aristotle's syllogisms, but that
it is strangely difficult to discern their validity. In any case, it
shows that syllogistic, invented by Aristotle 24 centuries ago, still
has some kind of life.

Here's the first example from Pratt-Hartmann's article:

At least 13 artists are beekeepers.
At most 3 beekeepers are carpenters.
At most 4 dentists are not carpenters.
-------------------------------------------------------- (therefore)
At least 6 artists are not dentists.

Every statement is really simple, like "All men are mortal," but the
head just swims with the finite set theoretic inclusions and
exclusions.

Despite the simplicity of the statements, Aristotle did not
investigate such deductions. We have to extend his syllogistic with
numerical quantifiers "At most N", "At least M", and so forth.
Actually, as Pratt-Hartmann points out, this logic was not examined
until de Morgan did so in the mid-19th century: "Formal logic: or, the
calculus of inference, necessary and probable," London: Taylor &
Watson, 1847.

Pratt-Hartmann goes on to consider whether there is some proof system
that could make this logic complete...in the usual sense that any
valid argument, like the first one above (valid means that if the
premises are true, then the conclusion must be true), has a proof in
the logic. The author begins by trying to adapt Aristotle's syllogisms
to the logic with new quantifiers. In Pratt-Hartmann's development,
there are even rules of inference that are reminiscent of Aristotle's
*Barbara* and *Darii* rules.

Well, it's a rather deep article for those without a good background
in university-level mathematical logic...it's not an easy article for
me, and I supposedly meet that criterion!...but it's nice to see the
work of The Philosopher twisted and turned and wrung out to
precipitate new ideas.

Thanks!
--Ron