Some more theorems on structural entailment relations and non-deterministic semantics

TL;DR

This paper extends classical results on matrix models of entailment relations to non-deterministic matrices, highlighting key differences and generalizations in the non-deterministic setting, with implications for algebraic logic.

Contribution

It generalizes the characterization of matrix models to Nmatrices, introduces new properties, and discusses the adaptation of algebraic logic to non-deterministic semantics.

Findings

Nmatrix quotients can be obtained using any compatible equivalence relation.

A logic is finitely based iff both its Nmatrix models and their complement are closed under ultraproducts.

Operations of taking images and preimages in Nmatrices cannot be interchanged, affecting model generation.

Abstract

We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations - nowadays more commonly called multiple-conclusion logics - to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a congruence); the problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras.