Proof-theoretic Semantics for Second-order Logic
Abstract
We develop a proof-theoretic semantics (P-tS) for second-order logic (S-oL), providing an inferentialist alternative to both full and Henkin model-theoretic interpretations. Our approach is grounded in base-extension semantics (B-eS), a framework in which meaning is determined by inferential roles relative to atomic systems -- collections of rules that encode an agent's pre-logical inferential commitments. We show how both classical and intuitionistic versions of S-oL emerge from this set-up by varying the class of atomic systems. These systems yield modular soundness and completeness results for corresponding Hilbert-style calculi, which we prove equivalent to Henkin's account of S-oL. In doing so, we reframe second-order quantification as systematic substitution rather than set-theoretic commitment, thereby offering a philosophically lightweight yet expressive semantics for higher-order logic. This work contributes to the broader programme of grounding logical meaning in use rather than reference and offers a new lens on the foundations of logic and mathematics.
Keywords
Cite
@article{arxiv.2508.07786,
title = {Proof-theoretic Semantics for Second-order Logic},
author = {Alexander V. Gheorghiu and David J. Pym},
journal= {arXiv preprint arXiv:2508.07786},
year = {2025}
}