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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 130 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
FirstOrder Logic of Proofs
, 2011
"... The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capa ..."
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Cited by 27 (11 self)
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The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capable of realizing firstorder modal logic S4 and, therefore, the firstorder intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for firstorder S4 and HPC compliant with BrouwerHeytingKolmogorov requirements. FOLP opens the door to a general theory of firstorder justification.
Functionality in the Basic Logic of Proofs
, 1993
"... This report describes the elimination of the injectivity restriction for functional arithmetical interpretations as used in the systems PF and PFM in the Basic Logic of Proofs. An appropriate axiom system PU in a language with operators "x is a proof of y" is defined and proved to be sound ..."
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Cited by 22 (16 self)
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This report describes the elimination of the injectivity restriction for functional arithmetical interpretations as used in the systems PF and PFM in the Basic Logic of Proofs. An appropriate axiom system PU in a language with operators "x is a proof of y" is defined and proved to be sound and complete with respect to all arithmetical interpretations based on functional proof predicates. Unification plays a major role in the formulation of the new axioms.
The interpretability logic of Peano arithmetic
 The Journal of Symbolic Logic
, 1990
"... prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtai ..."
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Cited by 19 (0 self)
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prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
A Pragmatic Interpretation Of Substructural Logics
"... Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from ..."
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Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from mixed contexts of assertions and obligations and we prove the cutelimination theorem for it. For the formalization of reallife normative systems it is essential to consider inferences from mixed contexts of assertions and obligations, and also of assertions justi able relatively to a given state of information and obligations valid in a given normative system. In order to provide a formalization of the notion of causal implication and its interaction with obligations, the sequents of ILP have two areas in the antecedent, expressing the relevant and the ordinary intuitionistic consequence relation, respectively. To provide a pragmatic interpretation of reasoning with the lin...
A Finitary Treatment of the Closed Fragment of Japaridze's Provability Logic
, 2005
"... We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some nonfinitary principles such as transfinite induction up to #0 or reflection principles. In fact, the cl ..."
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Cited by 10 (8 self)
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We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some nonfinitary principles such as transfinite induction up to #0 or reflection principles. In fact, the closed fragment of GLP gives rise to a natural system of ordinal notation for #0 that was used in [1] for a prooftheoretic analysis of Peano arithmetic and for constructing simple combinatorial independent statements. In this
Kripke Models of Transfinite Provability Logic
"... For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no nontrivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variablefree ..."
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For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no nontrivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variablefree fragment with natural number modalities, denoted GLP 0 ω. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each ordinals Θ, Λ we build a Kripke model I Θ Λ and show that GLP 0 Λ is sound for this structure. In our notation, Ignatiev’s original model becomes I ε0 ω.
Referential logic of proofs
 Theoretical Computer Science
"... We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness ..."
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We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness and completeness with respect to the intended provability semantics. We show that FLPref provides a complete test of admissibility of inference rules in a sound extension of arithmetic. Key words: proof theory, explicit modal logic, single conclusion logic of proofs, proof term, reference, unification, admissible inference rule. 1