A comparison of three kinds of monotonic proof-theoretic semantics and the base-incompleteness of intuitionistic logic
Antonio Piccolomini d'Aragona
TL;DR
The paper addresses how three monotone proof-theoretic semantics—Prawitz's reducibility semantics, standard base semantics, and Sandqvist's base semantics—relate and what this implies for completeness in intuitionistic logic IL. It develops rigorous formalisms for atomic bases, base semantics (including Sandqvist's variant), and reducibility semantics with argument-structures and reductions, and then analyzes how these frameworks compare. The main results show that reducibility semantics and standard base semantics are equivalent at the consequence level, enabling transfer of incompleteness results for IL; however, Sandqvist's variant is not pointwise equivalent to Prawitz's framework, yielding base-incomparability for higher levels. A deeper examination of base-completeness reveals it to be inconsistent for IL under these settings, connected to export principles and GDP, which narrows the prospects for a universal base-completeness account. Overall, the work clarifies when semantic results transfer across frameworks, highlights the distinct role of disjunction handling, and delineates limits on achieving completeness in constructive logics via base-theoretic means.
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.