Fibring by functions as a method for combining matrix logics

TL;DR

This paper adapts the fibring method for combining matrix logics using functions, proving it results in a matrix logic and exploring its properties as a conservative extension.

Contribution

It introduces a novel approach to fibring matrix logics via functions, establishing key properties and conditions for conservative extensions.

Findings

Fibring of two matrix logics results in a matrix logic.

Fibring is a weak conservative extension of the original logics.

Conditions for strong extension are provided.

Abstract

We present in this paper an adaptation of the process of combination of logics known as fibring introduced by D. Gabbay. We are focused on the combination of two logics defined by matrix semantics, and based on pairs of functions that relate the logics to be combined. A number of technical results are proved. Among them, we demonstrate that the fibring of two matrix logics is also a matrix one. In addition, we prove that fibring is a (weak) conservative extension of the original logics, and we give conditions for such extension to be strong. We also study the case of fibring identifying two connectives as being the same. Several examples referred to fibring of some well-known matrix logics are shown along this paper.