Some Remarks on Proof-Theoretic Semantics

This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential “constant” \(\bullet \) to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on “general elimination rules” and harmony, with a retraction of one of the main ideas of that work, i.e. the use of “flattened” general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

[1]  Luca Tranchini,et al.  Ekman's Paradox , 2017, Notre Dame J. Formal Log..

[2]  Ralph Matthes,et al.  Interpolation for Natural Deduction with Generalized Eliminations , 2001, Proof Theory in Computer Science.

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

[4]  Per Martin-Löf,et al.  Intuitionistic type theory , 1984, Studies in proof theory.

[5]  Jan Ekman,et al.  Propositions in Prepositional Logic Provable Only by Indirect Proofs , 1998, Math. Log. Q..

[6]  Michael Dummett,et al.  The logical basis of metaphysics , 1991 .

[7]  Stephen Read,et al.  Harmony and Autonomy in Classical Logic , 2000, J. Philos. Log..

[8]  Neil Tennant,et al.  Ultimate Normal Forms for Parallelized Natural Deductions , 2002, Log. J. IGPL.

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

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

[11]  J. Girard,et al.  Proofs and types , 1989 .

[12]  Zhaohui Luo,et al.  Computation and reasoning , 1994 .

[13]  Stephen Read,et al.  General-Elimination Harmony and Higher-Level Rules , 2015 .

[14]  Jan von Plato,et al.  Natural deduction with general elimination rules , 2001, Arch. Math. Log..

[15]  Peter Schroeder-Heister,et al.  Generalized Definitional Reflection and the Inversion Principle , 2007, Logica Universalis.

[16]  Jan von Plato,et al.  Explicit Composition and Its Application in Proofs of Normalization , 2016, Advances in Proof-Theoretic Semantics.

[17]  Roy Dyckhoff,et al.  Generalised elimination rules and harmony , 2009 .

[18]  Pierre Castéran,et al.  Interactive Theorem Proving and Program Development , 2004, Texts in Theoretical Computer Science An EATCS Series.

[19]  Enrico Moriconi,et al.  On Inversion Principles , 2008 .

[20]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[21]  Peter Schroeder-Heister,et al.  Harmony in Proof-Theoretic Semantics: A Reductive Analysis , 2015 .

[22]  Frank Pfenning,et al.  A judgmental reconstruction of modal logic , 2001, Mathematical Structures in Computer Science.

[23]  Ralph Matthes,et al.  Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T , 2003, Arch. Math. Log..

[24]  Grigory K. Olkhovikov,et al.  ON FLATTENING ELIMINATION RULES , 2014, The Review of Symbolic Logic.

[25]  Jeffery I. Zucker,et al.  The adequacy problem for inferential logic , 1978, J. Philos. Log..

[26]  Franz Kutschera,et al.  Die Vollständigkeit des Operatorensystems {¬, ∨, ⊃} für die Intuitionistische Aussagenlogik im Rahmen der Gentzensematik , 1968 .

[27]  R. Dyckho,et al.  Proof search in constructive logics , 1998 .

[28]  Peter Schroeder-Heister,et al.  A natural extension of natural deduction , 1984, Journal of Symbolic Logic.

[29]  P. Mendler.  Inductive Definition in Type Theory , 1988 .

[30]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[31]  Sara Negri,et al.  Structural proof theory , 2001 .

[32]  Jan von Plato,et al.  NORMAL DERIVABILITY IN CLASSICAL NATURAL DEDUCTION , 2012, The Review of Symbolic Logic.

[33]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[34]  D. Prawitz.  Proofs and the Meaning and Completeness of the Logical Constants , 1979 .

[35]  Luís Pinto,et al.  Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic , 1998, Stud Logica.

[36]  Nissim Francez,et al.  A Note on Harmony , 2012, J. Philos. Log..

[37]  Luís Pinto,et al.  Proof search in constructive logics , 1999 .

[38]  Helmut Schwichtenberg,et al.  Proofs and Computations , 2012, Perspectives in logic.

[39]  Peter Schroeder-Heister,et al.  Generalized Elimination Inferences, Higher-Level Rules, and the Implications-as-Rules Interpretation of the Sequent Calculus , 2014 .