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Equational definitions of logical filters

Michele Pra Baldi, Adam Přenosil

Abstract

A finitary propositional logic can be given an algebraic reading in two different ways: by translating formulas into equations and logical rules into quasi-equations, or by translating logical rules directly into equations. The former type of algebraic interpretation has been extensively studied and underlies the theory of algebraization. Here we shall develop a systematic theory of the latter type of algebraic interpretation. More precisely, we consider a semantic form of this property which we call the equational definability of compact filters (EDCF). Paralleling the well-studied hierarchy of variants of the deduction--detachment theorem (DDT), this property also comes in local, parametrized, and parametrized local variants. Our main results give a semantic characterization of each of these variants of the EDCF in a spirit similar to the existing characterizations of the DDT. While the EDCF hierarchy and the DDT hierarchy coincide for algebraizable logics, part of the interest of the EDCF stems from the fact it is often enjoyed even by logics which are not well-behaved in terms of other existing classifications in algebraic logic.

Equational definitions of logical filters

Abstract

A finitary propositional logic can be given an algebraic reading in two different ways: by translating formulas into equations and logical rules into quasi-equations, or by translating logical rules directly into equations. The former type of algebraic interpretation has been extensively studied and underlies the theory of algebraization. Here we shall develop a systematic theory of the latter type of algebraic interpretation. More precisely, we consider a semantic form of this property which we call the equational definability of compact filters (EDCF). Paralleling the well-studied hierarchy of variants of the deduction--detachment theorem (DDT), this property also comes in local, parametrized, and parametrized local variants. Our main results give a semantic characterization of each of these variants of the EDCF in a spirit similar to the existing characterizations of the DDT. While the EDCF hierarchy and the DDT hierarchy coincide for algebraizable logics, part of the interest of the EDCF stems from the fact it is often enjoyed even by logics which are not well-behaved in terms of other existing classifications in algebraic logic.
Paper Structure (8 sections, 24 theorems, 106 equations)

This paper contains 8 sections, 24 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parametrized and parametrized local EDCF
  4. Global and local EDCF
  5. Finitary EDCF
  6. The Leibniz hierarchy
  7. The EDCF and the DDT
  8. Equational definition of consequence

Key Result

Theorem 3.7

Every logic has a parametrized local EDCF on the class of all algebras (in the signature of the logic).

Theorems & Definitions (83)

  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • proof
  • proof
  • Theorem 3.7
  • proof
  • proof
  • ...and 73 more