Philosophy | Seminar in Logic
P751 | 3304 | Dunn

Topic: Algebraic Methods in Philosophical Logic

This course is aimed at introducing students to the use of algebraic
methods in the study of various logics of interest to philosophers
and computer scientists.  Among the logics to be considered are
classical logic (Boolean algebras), intutionistic logic (pseudo-
Boolean algebras), modal logics (closure algebras), relevance logic
(DeMorgan monoids).  Emphasis will be on a family of logics
called “substructural logics,” which includes intiutionistic logic,
relevance logic, BCK-logic, linear logic, and Lambek calculus.  Some
attention will also be paid to quantum logic in both its ortho-
modular and ortho versions.

A common theme will be that various meta-theoretic properties of the
logics are equivalent to standard algebraic results concerning the
corresponding algebras.  Of particular interest will be
representation results which translate into completeness theorems for
the logics.  A common abstraction will be presented that covers many
known logics (including the substructural logics).  The corresponding
algebras are called “gaggles” (for “generalized galois logics”).  For
these algebras an n-placed logical operation can be represented using
an n+l-placed accessibility relation (as for example possibility in
modal logic is represented using a binary accessibility relation).
This gives a Kripke-style semantics for the corresponding logics.

The text is the book Algebraic Methods in Philosophical Logic,
coauthored with Gary Hardegree at Umass-Amherst and just published by
Oxford Press.  In addition to this book we will look at various
recent papers, particularly a couple of mine (one joint with R.K.
Meyer) that use ternary Kripke semantics to give a representation of
combinatory logic and relation algebras, and that I have interpreted
as demonstrating a duality between information and computation.

The minimal prerequisite for the course is a good understanding of
the completeness theorem for classical logic and the ability to use
set intuitive set theory (the sort of knowledge given by P505).
Students are not expected to have had a course in abstract algebra.
The sorts of algebraic structures we shall study (primarily lattices)
are not usually covered in such a course anyway.  Some antecedent
knowledge of modal logic