History
and Philosophy of Logic,
18 (1997), 1-15
John
Corcoran's natural deduction system (equivalent to systems by Timothy Smiley
and others) for Aristotle's syllogistic is reconsidered. Though Corcoran is no
doubt right in interpreting Aristotle as viewing syllogism as arguments and in
rejecting Lukasiewicz' treatment in terms of conditional sentences, it is
argued that Corcoran is wrong in thinking that the only alternative is to
construe Barbara and Celarent as deduction rules in a natural deduction
system. An alternative is presented that is technically more elegant and
equally compatible with the texts. The abstract role assigned to Barbara and
Celarent by tradition and Lukasiewicz is retained. The two "perfect
syllogisms" serve as "basic elements" in the construction of an
inductively defined set of valid syllogisms. The proposal departs from
Lukasiewicz and follows Corcoran, however, in construing the construction as
one in natural deduction. The result is a sequent system with fewer rules and
in which Barbara and Celarent serve as basic deductions. To compare the theory
to Corcoran's, his original is reformulated in current terms and generalized.
It is shown to be equivalent to the proposed sequent system, and several
variations are discussed. For all systems mentioned a method of Henkin style
completeness proofs is given that is more direct and intuitive than Corcoran's
original, and that uses Aristotle's notion of "ecthesis" as the
basis for the relvant concept of saturation.