# A study of truth predicates in matrix semantics

## TL;DR

This paper investigates the algebraic and order-theoretic properties of truth predicates in matrix semantics for propositional logics, revealing conditions under which truth sets are definable and how the Leibniz operator's properties transfer.

## Contribution

It provides new insights into the definability and transfer properties of truth sets in matrix semantics, especially regarding the Leibniz operator's behavior over deductive filters.

## Key findings

- Truth sets can be defined by equations with universally quantified parameters.
- Injectivity of the Leibniz operator over theories transfers to deductive filters in countable languages.
- An intermediate condition related to order-reflection of the Leibniz operator is analyzed.

## 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 L is associated with a matrix semantics Mod*(L). This paper is a contribution to the systematic study of the so-called "truth sets" of the matrices in Mod*(L). In particular, we show that the fact that the truth sets of Mod*(L) 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 L. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of Mod*(L) are implicitly definable if and only if the Leibniz operator is injective on deductive filters of L over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of L 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 Mod*(L) that corresponds to the order-reflection of the Leibniz operator.