A new introduction rule for disjunction

TL;DR

This paper introduces a new disjunction introduction rule in Natural Deduction that ensures all proofs end with an introduction, simplifies termination proofs, and has applications in quantum computing and cut reduction.

Contribution

It proposes a novel disjunction introduction rule, $ ext{vee-i3}$, that improves proof properties and extends applications in quantum logic without increasing complexity.

Findings

The $ ext{vee-i3}$ rule makes proofs end with an introduction.

Simplifies termination proof by avoiding ultra-reduction rules.

Enables expressing quantum measurement without new connectives.

Abstract

We extend Natural Deduction for intuitionistic logic with a third introduction rule for the disjunction, $\vee$-i3, with a conclusion $\Gamma\vdash A\vee B$, but both premises $\Gamma\vdash A$ and $\Gamma\vdash B$. This rule is admissible in Natural Deduction. This extension is interesting in several respects. First, it permits to solve a well-known problem in logics with interstitial rules that have a weak introduction property: closed cut-free proofs end with an introduction rule, except in the case of disjunctions. With this new introduction rule, we recover the strong introduction property: closed cut-free proofs always end with an introduction. Second, the termination proof of this proof system is simpler than that of the usual propositional Natural Deduction with interstitial rules, as it does not require the use of the so-called ultra-reduction rules. Third, this proof system, in its linear version, has applications to quantum computing: the $\vee$-i3 rule enables the expression of quantum measurement, without the cost of introducing a new connective. Finally, even in logics without interstitial rules, the rule $\vee$-i3 is useful to reduce commuting cuts, although, in this paper, we leave the termination of such reduction as an open problem.