On an algebra of lattice-valued logic

Abstract

The purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.