2nd Semester 2024/25: Measuring Meaning: Proof-Theoretic Semantics and Substructurality

Instructors

Sophie Nagler

ECTS

6

Description

This project explores proof-theoretic semantics (P-tS). P-tS characterises the meaning of the logical connectives based on their use in reasoning. This use is modelled via proof rules, in contrast to orthodox model-theoretic methodologies.

We will survey the existing P-tS landscape, covering both proof-theoretic validity (P-tV) and base extension semantics (B-eS). We will particularly focus on the technique of inference-behaviour semantics (I-bS). To capture inferential use, I-bS measures the behaviour of a logical constant in minimal substructures of its proof system, employing techniques from (sub)structural proof theory. While initial results are promising, there is still ongoing work in this area.

This project provides students with practical research experience, offering the opportunity to tackle open problems in I-bS and contribute new findings to the field. Interested students who exhibit exceptional understanding and make significant contributions may have the chance to co-author a research paper after the project's completion.

By the end of the project, students will be able to:

• Outline different techniques in P-tS, including Pt-V and Be-S;
• Apply I-bS to measure connective use and give semantic clauses;
• Conduct independent research on a given research question;
• Present their knowledge and findings clearly and effectively.

Organisation

• Week 1: A Survey of Proof-Theoretic Semantics (2 group meetings)
• Week 2: Deep Dive - Inference Behaviour Semantics (2 group meetings)
• Weeks 3+4: Mini-Projects on Inference Behaviour Semantics (weekly check-in group meetings, and individual meetings as required)
• Monday of Week 5: Presentation of Result Reports
• Post-Project: Potential opportunity to contribute to a co-authored paper (not part of the project)

Prerequisites

• Students should be familiar with sequent calculus, including simple Cut-elimination proofs.
• Background readings will be circulated in advance to allow students to catch up or refresh their memory.
• Whilst beneficial, experience with substructural logics is not required.
• Mini-projects can be tailored to student skillsets, allowing for more technical or more philosophical projects.
• Students will need access to a computer with LaTeX for writing reports and presentations.

Assessment

1. Presentation of (part of) a reading in Week 1 or 2 (presentation length depending on signups, 10-30 minutes; slides and/or handouts should be prepared) [20%].
2. Result report, due at the end of Week 4 [60%].
3. Presentation of the result report (10 minutes) [20%].

References

(Optional) Pre-Reading:

[1] Mancosu, Paolo, Sergio Galvan and Richard Zach (2021, Chapters 5-6). An Introduction to Proof Theory. Oxford: 168-268.

[2] Negri, Sara, and Jan von Plato (2001, Chapters 1-3). Structural Proof Theory. Cambridge: 1-60.

[3] Troelstra, Anne Sjerp, and Helmut Schwichtenberg (2000, Chapters 3-4.1). Basic Proof Theory. 2nd ed. Cambridge: 60-105.

Week 1, Meeting 1:

[4] Wansing, Heinrich (2000). “The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations”, Studia Logica, 64: 3–20. doi:10.1023/A:1005217827758

[5] Schroeder-Heister, Peter (2024). “Proof-Theoretic Semantics”, The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL = <https://plato.stanford.edu/archives/sum2024/entries/proof-theoretic-semantics/>.

[6] Sandqvist, Tor (2015). “Base-extension semantics for intuitionistic sentential logic.” Logic Journal of the IGPL, 23 (5):719-731.

[7] Schroeder-Heister, Peter (2016). “Open Problems in Proof-Theoretic Semantics.” In: Piecha and Schroeder-Heister, eds. (2016) Advances in Proof-Theoretic Semantics, pp. 253–283. doi:10.1007/978-3-319-22686-6_16

Week 1, Meeting 2:

[8] Restall, Greg (2014). “Pluralism and Proofs,” Erkenntnis 79:2 (2014) 279–291.

[9] Kouri, Teresa (2016). “Restall’s Proof-Theoretic Pluralism and Relevance Logic.” Erkenntnis 81.6: 1243–1252.

[10] Dicher, Bogdan (2016). “A Proof-Theoretic Defence of Meaning-Invariant Logical Pluralism.” Mind 125.499: 727–757.

[11] Ferrari, Filippo, and Eugenio Orlandelli (2021). “Proof-Theoretic Pluralism.” Synthese 198.S20: S4879–S4903.

Week 2:

[12] Nagler, Sophie (2025). “Measuring Meaning.” Under review.

[13] Nagler, Sophie (2025). “Inference Behaviour Semantics for All* Connectives in Two-Dimensional Sequent Calculi.” Work in progress.