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Completeness of Second-Order Intuitionistic Propositional Logic with Respect to Phase Semantics for Proof-Terms

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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.

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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.

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Authors and Affiliations

  1. Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan

    Yuta Takahashi

  2. Nihon University, 5-2-1, Kinuta, Setagaya-ku, Tokyo, 157-8570, Japan

    Ryo Takemura

Authors
  1. Yuta Takahashi
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  2. Ryo Takemura
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Correspondence to Yuta Takahashi.

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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

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  • DOI: https://doi.org/10.1007/s10992-018-9484-z

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