Admissible Tools in the Kitchen of Intuitionistic Logic

Abstract

The usual reading of logical implication A → B as " if A then B " fails in intuitionistic logic: there are formulas A and B such that A → B is not provable, even though B is provable whenever A is provable. Intuitionistic rules apparently don't 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 behaviour 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.

Algorithmic Theories of Problems. A Constructive and a Non-Constructive Approach

“…Concerning related work, Condoluci and Manighetti (2018) proposed a typed rule for the (non-generalized) Harrop rule that follows the same general pattern as our typed Split rule (i.e., it is based on the typed disjunction elimination rule). Interestingly, they arrived at this pattern differently: while our approach is bottom-up (we have started by studying the inferential behavior of the Split rule and then generalized it), their approach was top-down (they have started with Visser rules (Roziére (1993), Iemhoff (2005)) as a basis for all admissible rules and considered the Harrop rule as a special case).…”

“…Despite its significance, the Split rule itself remains mostly unexplored, especially in terms of its proof-theoretic meaning and computational content (a recent exception to this is Condoluci and Manighetti (2018) examining the admissibility of the related Harrop rule from the computational view). In this paper, we fill this gap and propose a computational interpretation of the Split rule.…”