Representing Conjunctive Deductions by Disjunctive Deductions

TL;DR

This paper demonstrates a categorical equivalence showing that conjunctive deductions with many propositional letters can be represented by disjunctive deductions with a single letter, revealing a deep structural connection.

Contribution

It establishes a categorical isomorphism between categories generated by conjunctive and disjunctive logic, providing a proof-theoretical translation between them.

Findings

Conjunctive deductions with multiple propositional letters can be represented by disjunctive deductions with one letter.

Categories with finite coproducts and products are shown to have equivalent subcategories.

Dual results are obtained by reversing the categorical constructions.

Abstract

A skeleton of the category with finite coproducts D freely generated by a single object has a subcategory isomorphic to a skeleton of the category with finite products C freely generated by a countable set of objects. As a consequence, we obtain that D has a subcategory equivalent with C. From a proof-theoretical point of view, this means that up to some identifications of formulae the deductions of pure conjunctive logic with a countable set of propositional letters can be represented by deductions in pure disjunctive logic with just one propositional letter. By taking opposite categories, one can replace coproduct by product, i.e. disjunction by conjunction, and the other way round, to obtain the dual results.