The purpose of this monograph is to develop a very general approach to
the algebra- ization of sententiallogics, to show its results on a
number of particular logics, and to relate it to other existing
approaches, namely to those based on logical matrices and the equational
consequence developed by Blok, Czelakowski, Pigozzi and others. The main
distinctive feature of our approachlies in the mathematical objects used
as models of a sententiallogic: We use abstract logics, while the
dassical approaches use logical matrices. Using models with more
structure allows us to reflect in them the metalogical properties of the
sentential logic. Since an abstract logic can be viewed as a "bundle" or
family of matrices, one might think that the new models are essentially
equivalent to the old ones; but we believe, after an overall
appreciation of the work done in this area, that it is precisely the
treatment of an abstract logic as a single object that gives rise to a
useful -and beautiful- mathematical theory, able to explain the
connections, not only at the logical Ievel but at the metalogical Ievel,
between a sentential logic and the particular dass of models we
associate with it, namely the dass of its full models. Traditionally
logical matrices have been regarded as the most suitable notion of model
in the algebraic studies of sentential logics; and indeed this notion
gives sev- eral completeness theorems and has generated an interesting
mathematical theory.
