N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions

TL;DR

This paper demonstrates that N-way joint mutual exclusion among propositions does not necessarily imply pairwise mutual exclusion, providing a counterexample and clarifying the logical relationship between these concepts.

Contribution

It introduces a new counterexample showing that N-way mutual exclusion does not imply pairwise mutual exclusion, challenging assumptions in propositional logic and set theory.

Findings

Counterexample exists for N-way mutual exclusion without pairwise exclusion

N-way mutual exclusion does not imply pairwise mutual exclusion in propositional calculus

Results apply to Boolean algebra models

Abstract

Given a set of N propositions, if any pair is mutual exclusive, then the set of all propositions are N-way jointly mutually exclusive. This paper provides a new general counterexample to the converse. We prove that for any set of N propositional variables, there exist N propositions such that their N-way conjunction is zero, yet all k-way component conjunctions are non-zero. The consequence is that N-way joint mutual exclusion does not imply any pairwise mutual exclusion. A similar result is true for sets since propositional calculus and set theory are models for two-element Boolean algebra.