A judgmental reconstruction of modal logic

We reconsider the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

[1]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[2]  W. V. Quine,et al.  Natural deduction , 2021, An Introduction to Proof Theory.

[3]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[4]  D. Prawitz.  Ideas and Results in Proof Theory , 1971 .

[5]  P. Martin-Lof,et al.  Constructive mathematics and computer programming , 1984, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[6]  Per Martin-Löf,et al.  Constructive mathematics and computer programming , 1984 .

[7]  Eugenio Moggi,et al.  Computational lambda-calculus and monads , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[8]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[9]  Dale Miller,et al.  Logic programming in a fragment of intuitionistic linear logic , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[10]  P. Martin-Löf.  Analytic and Synthetic Judgements in Type Theory , 1994 .

[11]  Dale Miller,et al.  Logic Programming in a Fragment of Intuitionistic Linear Logic , 1994, Inf. Comput..

[12]  Michael Mendler,et al.  An Intuitionistic Modal Logic with Applications to the Formal Verification of Hardware , 1994, CSL.

[13]  Frank Pfenning.  Structural Cut Elimination in Linear Logic. , 1994 .

[14]  F. Pfenning,et al.  On a Modal -calculus for S4 ? , 1995 .

[15]  F. Pfenning,et al.  On a Modal λ-Calculus for S41 1This work is supported by NSF Grant CCR-9303383 and the Advanced Research Projects Agency under ARPA Order No. 8313. , 1995 .

[16]  Frank Pfenning,et al.  A modal analysis of staged computation , 1996, POPL '96.

[17]  S. Martini,et al.  A Computational Interpretation of Modal Proofs , 1996 .

[18]  P. Wadler,et al.  A Reeection on Call-by-value , 1996 .

[19]  P. Martin-Löf.  On the meanings of the logical constants and the justi cations of the logical laws , 1996 .

[20]  Philip Wadler,et al.  A reflection on call-by-value , 1997, TOPL.

[21]  A reflection on call-by-value , 1997, TOPL.

[22]  Michael Mendler,et al.  First-order Lax Logic as a framework for Constraint Logic Programming , 1997 .

[23]  Michael Mendler,et al.  Propositional Lax Logic , 1997, Inf. Comput..

[24]  Satoshi Kobayashi,et al.  Monad as Modality , 1997, Theor. Comput. Sci..

[25]  Frank Pfenning,et al.  Primitive recursion for higher-order abstract syntax , 1997, Theoretical Computer Science.

[26]  Luca Viganò.  A framework for non-classical logics , 1997 .

[27]  Peter Lee,et al.  Modal types as staging specifications for run-time code generation , 1998, CSUR.

[28]  Nick Benton,et al.  Computational types from a logical perspective , 1998, Journal of Functional Programming.

[29]  Frank Pfenning,et al.  Primitive recursion for higher-order abstract syntax , 1997, Theor. Comput. Sci..

[30]  F. Pfenning,et al.  On a Modal λ-Calculus for S 4 ? , 2022 .