Merging fragments of classical logic

TL;DR

This paper explores whether combining incomplete Boolean logic fragments can produce a complete classical propositional logic system, concluding it is generally not possible unless one fragment is trivial.

Contribution

It demonstrates that merging Hilbert-style calculi for disjoint incomplete fragments cannot produce a complete logic unless one fragment contains only trivial connectives.

Findings

Merging incomplete fragments generally does not yield classical propositional logic.

The exception occurs if one fragment includes only top-like connectives.

The study provides formal proof of these limitations.

Abstract

We investigate the possibility of extending the non-functionally complete logic of a collection of Boolean connectives by the addition of further Boolean connectives that make the resulting set of connectives functionally complete. More precisely, we will be interested in checking whether an axiomatization for Classical Propositional Logic may be produced by merging Hilbert-style calculi for two disjoint incomplete fragments of it. We will prove that the answer to that problem is a negative one, unless one of the components includes only top-like connectives.