Abstract
Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. In this paper, we first formulate phase semantics for proof-terms of second-order intuitionistic propositional logic by modifying Tait-Girard’s saturated sets method. Next, we prove the completeness theorem with respect to this semantics, which implies a strong normalization theorem.
Similar content being viewed by others
References
Gallier, J. (1990). On Girard’s Candidats de reductibilité. In Odifreddi, P. (Ed.) Logic and computer science (pp. 123–203). London: Academic Press.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.
Laird, J. (2005). Game semantics and linear CPS translation. Theoretical Computer Science, 333, 199–224.
Okada, M. (2002). A uniform semantic proof for cut-elimination and completeness of various first and higher order logics. Theoretical Computer Science, 281, 471–98.
Okada, M., & Takemura, R. (2007). Remarks on semantic completeness for proof-terms with Lairds dual affine/intuitionistic λ-calculus. In Comon-Lundh, H., Kirchner, C., Kirchner, H. (Eds.) Rewriting, computation and proof: essays dedicated to Jean-Pierre Jouannaud on the occasion of his 60th birthday. Volume 4600 of lecture notes in computer science (pp. 167–81). Berlin: Springer.
Prawitz, D. (1971). Ideas and results in proof theory. In Fenstad, J.E. (Ed.) Proceedings of the second scandinavian logic symposium, studies in logic and the foundations of mathematics, (Vol. 63, pp 235–307. Amsterdam: North-Holland.
Prawitz, D. (1973). Towards a foundation of a general proof theory. In Suppes, P., Henkin, L., Joja, A., Moisil, GC (Eds.) Logic, methodology and philosophy of science IV (pp. 225–250). Amsterdam: North-Holland.
Riba, C. (2008). Toward a general rewriting-based framework for reducibility. <hal-00779623>.
Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148, 525–71.
Schroeder-Heister, P. (2016). Proof-theoretic semantics. In Zalta, E.N. (Ed.) The Stanford Encyclopedia of Philosophy, Winter 2016 Edition. https://plato.stanford.edu/archives/win2016/entries/proof-theoretic-semantics/.
Sørensen, M.H., & Urzyczyn, P. (2006). Lectures on the Curry-Howard isomorphism. Amsterdam: Elsevier.
Acknowledgements
We would like to thank the anonymous referee for very valuable comments and remarks. We owe the remark about assignments to the referee. The content of this paper was presented at the following three workshops: the joint Conference of The 3rd Asian Workshop on Philosophical Logic (AWPL-2016) & The 3rd Taiwan Philosophical Logic Colloquium (TPLC-2016) at National Taiwan University in October 2016, Workshop on philosophy of logic: Logic, computation and normativity at Université Paris 1 Panthéon-Sorbonne in November 2016 and French-Japanese Workshop “Philosophy of logic and Mathematics – Towards Philosophy of Proofs” at Keio University in January 2017. We profited from the discussions at these workshops and would like to thank in particular Ryota Akiyoshi, Alberto Naibo, Mitsuhiro Okada and Peter Schroeder-Heister for their very valuable comments and remarks. The first author is supported by KAKENHI (Grant-in-Aid for JSPS Fellows) 16J04925.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Takahashi, Y., Takemura, R. Completeness of Second-Order Intuitionistic Propositional Logic with Respect to Phase Semantics for Proof-Terms. J Philos Logic 48, 553–570 (2019). https://doi.org/10.1007/s10992-018-9484-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-018-9484-z