An Abstract Approach to Consequence Relations

TL;DR

This paper generalizes the framework of structural consequence relations to include non-idempotent aggregations like multisets and fuzzy sets, using categorical methods to analyze their properties.

Contribution

It introduces a more general abstract framework for consequence relations that encompasses multiset and fuzzy set aggregations, extending prior set-based approaches.

Findings

Develops a categorical approach to consequence relations.

Provides matrix semantics and Hilbert system variations for multiset deductive relations.

Extends the theory of consequence relations to non-idempotent aggregations.

Abstract

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic.