According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix A is (...) Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

We effectively construct the finitely generated free equivalential algebras corresponding to the equivalential fragment of intuitionistic propositional logic.

In some varieties of algebras one can reduce the question of finding most general unifiers to the problem of the existence of unifiers that fulfil the additional condition called projectivity. In this article, we study this problem for Fregean varieties that arise from the algebraization of fragments of intuitionistic or intermediate logics. We investigate properties of Fregean varieties, guaranteeing either for a given unifiable term or for all unifiable terms, that projective unifiers exist. We indicate the identities which fully characterize (...) congruence permutable Fregean varieties having projective unifiers and describe an effective procedure for finding such unifiers. In particular, we show that for a congruence permutable Fregean variety there exists the largest subvariety that has projective unifiers. (shrink)