ON THE NOTION OF CANONICAL DERIVATIONS FROM OPEN ASSUMPTIONS AND ITS ROLE IN PROOF-THEORETIC SEMANTICS

Abstract The paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

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

[2]  Jan von Plato,et al.  Translations from natural deduction to sequent calculus , 2003, Math. Log. Q..

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

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

[5]  Peter Schroeder-Heister,et al.  The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics , 2012, Synthese.

[6]  Dag Prawitz,et al.  Meaning Approached Via Proofs , 2006, Synthese.

[7]  Heinrich Wansing,et al.  Dag Prawitz on Proofs and Meaning , 2015 .

[8]  Stephen Read,et al.  General-Elimination Harmony and the Meaning of the Logical Constants , 2010, J. Philos. Log..

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

[10]  Nils Kürbis.  How Fundamental is the Fundamental Assumption , 2012 .

[11]  Kosta Dosen,et al.  Uniqueness, definability and interpolation , 1988, Journal of Symbolic Logic.

[12]  Nissim Francez,et al.  PROOF-THEORETIC SEMANTIC VALUES FOR LOGICAL OPERATORS , 2011, The Review of Symbolic Logic.

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

[14]  Arthur N. Prior,et al.  The Runabout Inference-Ticket , 1960 .

[15]  Lars Hallnäs,et al.  A Proof-Theoretic Approach to Logic Programming. I. Clauses as Rules , 1990, J. Log. Comput..

[16]  Hartry Field.  Introduction to Author Meets Critics Session on Saving Truth from Paradox , 2011, Rev. Symb. Log..

[17]  Nissim Francez,et al.  Views of proof-theoretic semantics: reified proof-theoretic meanings , 2016, J. Log. Comput..

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

[19]  Ozan Kahramanogullari,et al.  Nondeterminism and Language Design in Deep Inference: A Proof Theoretic Approach to Logic Programming , 2012 .

[20]  Lars Hallnäs,et al.  A Proof-Theoretic Approach to Logic Programming. II. Programs as Definitions , 1991, J. Log. Comput..

[21]  P. Schroeder-Heister.  Proof-Theoretic versus Model-Theoretic Consequence , 2009 .

[22]  Peter Schroeder-Heister,et al.  Validity Concepts in Proof-theoretic Semantics , 2006, Synthese.

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

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

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