Duality Theory for Selfextensional Logics
by
Alessandra Palmigiano
University of Barcelona

The theory of Abstract Algebraic Logic (AAL) is about finding general properties - criteria of algebrizability, above all - of classes of logics, which, for sake of generality, are understood as tuples S = <Fm_S, \vdash_S>, where Fm is an algebra of formulas of a given similarity type \tau, and \vdash_S is a consequence relation on Fm_S.

Thus AAL shifts the conceptual focus from truth (theoremhood) to derivability.

A class of logics of particular relevance in this context is the class of selfextensional logics, i.e. those logics S such that their associated interderivability relation is a congruence on Fm_S.

All the most important logics - such as CPC, IPC, the local consequence relations associated with many normal modal logics - are selfextensional. However, there are also interesting logics which are not, such as those that arise from the global consequence relations associated with almost all classes of Kripke frames.

Wójcicki characterized selfextensional logics as those logics that arise from the local consequence relations associated with some classes of referential algebras, i.e. tuples H = <W, A>, where W is a nonempty set, and A is a (\tau-) algebra of subsets of W.

We wish to present the following results:

1) A duality that involves certain natural subcategories of referential algebras as dual structures. This duality behaves well w.r.t. the consequence relations associated with the objects involved, and uniformly accounts for every selfextensional logic. Thus it imports Wójcicki's result in the context of AAL.

2) A general procedure on how to ``embed'' Priestley-style dualities (i.e. dualities involving classes of algebras which have a distributive lattice part) into the one of point 1). Thus, the duality result of point 1) can be regarded as a general duality theory for all those selfextensional logics whose associated algebraic semantics is given by classes of algebras that have a distributive lattice part.

Date received: February 27, 2003