Topic: Substructural Logics

Gentzen formalized deducibility with a distinction between structural and operational rules. The
latter are associated each with a particular logical connective and show both how it is introduced
into, and eliminated from, rational discourse. The former have to do with manipulating the
premisses, no matter what connectives they contain. Examples are changing the order of the
premises, adding a premise, or removing duplicate premises. Various researchers came to realize
that many well-known non-classical logics can be categorized as omitting one or more of the
structural rules. These include intuitionistic logic, relevance logic, linear logic, BCK logic,
and the Lambek calculus, and some others not as well known. Dosen and Schroeder-Heister
crystalized this point of view with their 1993 Oxford Press book Substructural Logics.

In this seminar we will examine such logics both proof-theoretically and model-theoretically. We
will look at the proof of Gentzen=s cut-theorem for these logic, and will also look at semantical
interpretations of these logics. Principal among these last will be a Kripke-style semantics, but
using a ternary rather than a binary relation as Kripke used in his semantics for modal logic and
for intuitionistic logic. We shall see how this grows out of the Routley-Meyer semantics for
relevance logic, and how it is a special case of generalized galois logics (gaggles) as developed by
Dunn. I will also present some work on Astructurally-free logic@ due to R. K. Meyer and myself,
wherein structural rules are replaced by their corresponding combinators, and also some results in
this area due to K. Bimbo.

Any student who knows the equivalent of P505 should be able to take this course, though some
exposure to modal logic and other non-classical logics would be helpful. Some knowledge of
abstract algebra, especially lattice theory, would also be useful. But Aeverything@ will be explained.

We shall preview an as yet unpublished book by Greg Restall, and examine classic papers and
some stuff written by me. I plan to run this course as a true seminar. After a few weeks I will ask
students to make reports on things we are reading. Students will be graded on the basis of these
reports, but principally on the quality of a written paper presenting original results and/or
surveying the literature relating to some topic or problem.