Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conditional algebras

Sergio Celani, Rafał Gruszczyński, Paula Menchón

TL;DR

This work develops a comprehensive algebraic theory for a binary conditional operator $\rightarrowtriangle$ that generalizes Chellas's conditional logic by aligning it with a unary necessity-like modality. It introduces conditional algebras $\mathsf{CA}$, builds ultrafilter and t-frame representations, and proves a representation theorem placing $\mathsf{CA}$ within full complex algebras $\operatorname{Em}(\mathfrak{A})$, while establishing canonical extensions and a π-/σ-extension framework. A central achievement is the term-equivalence between $\mathsf{CA}$ and multi-modal antitone algebras $\mathsf{MMA}$, enabling a rich duality theory—topological, categorical, and relational—between CA and categories of conditional spaces; this yields robust dualities, subalgebra/congruence characterizations, and a detailed analysis of subvarieties such as PSB, PsC, SIA, and $S^2IA$, including canonicity results. The results offer a unified, dualizable account of conditional reasoning in algebraic terms with precise correspondences between equational laws and spatial/first-second-order properties, paving the way for quasi-conditional extensions and connections to related contact/subordination frameworks. The framework has potential implications for modal correspondence theory, logic-algebra dualities, and semantic modeling of conditionals in formal epistemology and computer science.

Abstract

Drawing on the classic paper by Chellas "Basic conditional logic" (1975), we propose a general algebraic framework for studying a binary operation of conditional that models universal features of the "if..., then..." connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of conditional algebras, and we develop its duality and canonical extensions theory.

Conditional algebras

TL;DR

This work develops a comprehensive algebraic theory for a binary conditional operator that generalizes Chellas's conditional logic by aligning it with a unary necessity-like modality. It introduces conditional algebras , builds ultrafilter and t-frame representations, and proves a representation theorem placing within full complex algebras , while establishing canonical extensions and a π-/σ-extension framework. A central achievement is the term-equivalence between and multi-modal antitone algebras , enabling a rich duality theory—topological, categorical, and relational—between CA and categories of conditional spaces; this yields robust dualities, subalgebra/congruence characterizations, and a detailed analysis of subvarieties such as PSB, PsC, SIA, and , including canonicity results. The results offer a unified, dualizable account of conditional reasoning in algebraic terms with precise correspondences between equational laws and spatial/first-second-order properties, paving the way for quasi-conditional extensions and connections to related contact/subordination frameworks. The framework has potential implications for modal correspondence theory, logic-algebra dualities, and semantic modeling of conditionals in formal epistemology and computer science.

Abstract

Drawing on the classic paper by Chellas "Basic conditional logic" (1975), we propose a general algebraic framework for studying a binary operation of conditional that models universal features of the "if..., then..." connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of conditional algebras, and we develop its duality and canonical extensions theory.
Paper Structure (16 sections, 46 theorems, 104 equations, 1 figure)

This paper contains 16 sections, 46 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Conditional algebras and their canonical extensions
  4. Representation of conditional algebras
  5. Canonical extensions of the conditional operator
  6. Conditional algebras and multi-modal antitone algebras
  7. A representation theorem for t-frames
  8. Topological duality for conditional algebras
  9. Categorical duality for conditional algebras
  10. Characterization of conditional subalgebras
  11. Congruences of conditional algebras
  12. Characterizations of some subvarieties of CA
  13. Dual spaces of pseudo-subordination algebras
  14. Correspondences
  15. Canonicity
  16. ...and 1 more sections

Key Result

Lemma 2.1

In every conditional algebra $\mathfrak{A}\coloneqq\langle A,\rightarrowtriangle\rangle$:

Figures (1)

  • Figure 1: The poset of subvarieties of the variety of conditional algebras analyzed in Section \ref{['sec:subvarieties']}.

Theorems & Definitions (102)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 92 more