A study of truth predicates in matrix semantics

Abstract

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic is associated with a matrix semantics. This article is a contribution to the systematic study of the so-called truth sets of the matrices in. In particular, we show that the fact that the truth sets of can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of are implicitly definable if and only if the Leibniz operator is injective on deductive filters of over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in that corresponds to the order-reflection of the Leibniz operator.