A new introduction rule for disjunction

2025-02-26

AI Summary

This paper addresses the long-standing absence of the strong introduction property—i.e., the requirement that every closed cut-free proof must end with an introduction rule—in natural deduction for disjunction. We introduce a third disjunction introduction rule (∨-i₃), applicable only when both disjuncts are provable. This rule is the first in natural deduction to guarantee the strong introduction property for disjunction, ensuring all closed cut-free proofs terminate with an introduction step. It eliminates reliance on hyper-reduction rules in termination proofs, enables faithful modeling of quantum measurement within linear logic—without introducing new connectives—and reduces the complexity of exchange-cut elimination in substructural logics. Collectively, these contributions enhance structural regularity, provability-theoretic robustness, and computational expressivity in proof theory.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

Abstract

We extend Natural Deduction with a third introduction rule for the disjunction, $vee$-i3, with a conclusion $Gammavdash Avee B$, but both premises $Gammavdash A$ and $Gammavdash B$. 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.

Problem

• Introduces a new disjunction rule in Natural Deduction.
• Simplifies termination proofs in propositional systems.
• Applies to quantum computing for expressing measurements.

Innovation

• New disjunction introduction rule
• Simplifies termination proof
• Enables quantum measurement expression