On the Decidability of Intuitionistic Tense Logic without Disjunction

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

On the multi-dimensional modal logic of substitutions

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

Yet some more non-finite axiomatizability results for algebras of relations and ways to avoid them

Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of Lω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ωω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

Vaught's Theorem on Axiomatizability by a Scheme

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.