  One of the big problems in philosophy is the problem of meaning. The turning
point in thinking on meaning was Tarski's definition of truth. The rapid
development of logical semantics and model theory was a consequence of this
achievement. It is less known among classical logicians and philosophers that
it is category theory that provides adequate mathematical tools to study the
relationship between the syntax of formalized theories and their semantics. The
aim of this article is to change this situation and make a preliminary
philosophical analysis of the results obtained so far. They concern formalized
algebraic theories with axioms in the form of equational laws, theories based
on propositional logic and coherent Boolean logic, as well as decidable logic
not necessarily Boolean. The syntactic-semantics relation for these theories
takes the form of dualisms between the respective syntactic and semantic
categories. These dualisms are given by the appropriate adjoint functors. In
all the considered cases, the syntax--semantics dualism corresponds to the
algebra-geometry dualism. We analyze the philosophical significance of these
results. They allow us to look at the problem of meaning in a new light and
formulate a criterion distinguishing formal theories from empirical theories.
The disputes that once were fought over Tarski's definition of truth are
transferred to a new context. The polemics as to whether Tarski actually
succeeded in reducing the problem of meaning to purely syntactic terms has been
superseded: between the syntactic and semantic categories there is not a
relation of reducibility but rather that of interaction, and this relation is
given by adjoint functors. We also touch upon other philosophical aspects of
the categorical approach to the problem of meaning.
