A comparison of three kinds of monotonic proof-theoretic semantics and the base-incompleteness of intuitionistic logic

TL;DR

This paper compares two proof-theoretic semantics approaches—reducibility and base semantics—and explores their relationship and implications for the completeness of intuitionistic logic.

Contribution

It introduces conditions under which one can shift between the two semantics and discusses their impact on the base-incompleteness of intuitionistic logic.

Findings

Conditions for shifting between semantics are established.

Implications for the completeness of recursive logical systems are analyzed.

The relationship between proof-objects and semantics types is clarified.

Abstract

I deal with two approaches to proof-theoretic semantics: one based on argument structures and justifications, which I call reducibility semantics, and one based on consequence among (sets of) formulas over atomic bases, called base semantics. The latter splits in turn into a standard reading, and a variant of it put forward by Sandqvist. I prove some results which, when suitable conditions are met, permit one to shift from one approach to the other, and I draw some of the consequences of these results relative to the issue of completeness of (recursive) logical systems with respect to proof-theoretic notions of validity. This will lead me to focus on a notion of base-completeness, which I will discuss with reference to known completeness results for intuitionistic logic. The general interest of the proposed approach stems from the fact that reducibility semantics can be understood as a labelling of base semantics with proof-objects typed on (sets of) formulas for which a base semantics consequence relation holds, and which witness this very fact. Vice versa, base semantics can be understood as a type-abstraction of a reducibility semantics consequence relation obtained by removing the witness of the fact that this relation holds, and by just focusing on the input and output type of the relevant proof-object.