Admissible Tools in the Kitchen of Intuitionistic Logic

TL;DR

This paper introduces a calculus that captures admissible inferences in intuitionistic logic, extending its capabilities while preserving intuitionistic properties, and demonstrates this with the Kreisel-Putnam logic KP.

Contribution

It presents a novel calculus representing admissible inferences in intuitionistic logic, maintaining normal forms as intuitionistic terms, and proves key properties for the extended logic KP.

Findings

The calculus can represent admissible inferences within intuitionistic logic.

Strong normalization and disjunction property are proved for the extended logic KP.

The approach advances understanding of admissible rules in intuitionistic logic.

Abstract

The usual reading of logical implication "A implies B" as "if A then B" fails in intuitionistic logic: there are formulas A and B such that "A implies B" is not provable, even though B is provable whenever A is provable. Intuitionistic rules apparently do not capture interesting meta-properties of the logic and, from a computational perspective, the programs corresponding to intuitionistic proofs are not powerful enough. Such non-provable implications are nevertheless admissible, and we study their behavior by means of a proof term assignment and related rules of reduction. We introduce V, a calculus that is able to represent admissible inferences, while remaining in the intuitionistic world by having normal forms that are just intuitionistic terms. We then extend intuitionistic logic with principles corresponding to admissible rules. As an example, we consider the Kreisel-Putnam logic KP, for which we prove the strong normalization and the disjunction property through our term assignment. This is our first step in understanding the essence of admissible rules for intuitionistic logic.