# Proof-theoretic Semantics for Second-order Logic

arXiv: https://arxiv.org/abs/2508.07786 · HTML: https://mlsanity.com/paper/2508.07786/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2025.

## Built on by (most cited first, 1 of 1)

- [Towards an Inferentialist Account of Information Through Proof-theoretic Semantics](https://mlsanity.com/paper/2605.05368/) (2026)

## Abstract (arXiv)

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.