In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) non-axiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) non-axiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) non-axiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasi-polyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.

The paper presents a simple format for typed logics with states by adding a function for register update to standard typed lambda calculus. It is shown that universal validity of equality for this extended language is decidable (extending a well-known result of Friedman for typed lambda calculus) . This system is next extended to a full fledged typed dynamic logic, and it is illustrated how the resulting format allows for very simple and intuitive representations of dynamic semantics for natural language and denotational semantics for imperative programming. The proposal is compared with some alternative approaches to formulating typed versions of dynamic logics. Keywords: type theory, compositionality, denotational semantics, dynamic semantics 1 Introduction A slight extension to the format of typed lambda calculus is enough to model states (assignments of values to storage cells) in a very natural way. Let a set R of registers or storage cells be given. If we assume that the values...