Variable Binding Algebras for Nonclassical Logics

The abstract variable binding calculus (VB-calculus) introduced in [2] (see also [4]) provides a formal framework encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course in [2] and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculs by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that T ` s = t iff D |= s = t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifier exists uncountably many and a fragment of Riemann integration. In [3] H. Rasiowa identifies a large class of propositional logics, the standard implicational calculi (SIC’s), that can be algebraized in a meaningful way and which include most of the nonclassical logics considered in the literature: classical and intuitionistic logic and their various weakenings and fragments, the multiple-valued iogics of Post and Lukasiewicz, the modal logics that admit the rule of necessitation, BCK logic, etc. In [1] the authors characterize the polyadic algebras that arise when the first-order extension of a SIC S is algebraized. Each SIC S gives rise to the quasivariety of polyadic S-algebras. As application of the functional representation theorem for polyadic VB-algebras given in [2], the main result in [1] is a functional representation theorem for polyadic S-algebras that comprehends many of the known representation results for polyadic algebras over nonclassical logics.