Negation and Implication in Partition Logic

TL;DR

This paper explores the concepts of negation and implication within the less-studied mathematical logic of partitions, which is dual to Boolean subset logic and offers a different perspective on logical operations.

Contribution

It introduces and analyzes the notions of negation and implication specifically in the context of partition logic, expanding the understanding of non-classical logical systems.

Findings

Negation in partition logic differs from classical negation.

Implication in partition logic exhibits unique properties compared to Boolean logic.

Partition logic provides a dual perspective to subset-based Boolean logic.

Abstract

The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.