The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

TL;DR

This paper introduces a dual logic to classical subset logic based on partitions of a set, exploring its theoretical foundations and establishing correctness and completeness of a tableau proof system.

Contribution

It develops the foundational theory of partition logic as dual to subset logic and proves key theorems for its formal tableau calculus.

Findings

Partition logic is dual to subset logic in categorical terms.

A tableau proof system for partition logic is sound and complete.

Abstract

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary "propositional" logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms--which is reflected in the duality between quotient objects and subobjects throughout algebra. If "propositional" logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.