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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 nothing b ..."
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Cited by 114 (22 self)
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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 102 (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
Symmetric Logic of Proofs
 CUNY Ph.D. Program in Computer Science
, 2007
"... The Logic of Proofs LP captures the invariant propositional properties of proof predicates t is a proof of F with a set of operations on proofs sufficient for realizing the whole modal logic S4 and hence the intuitionistic logic IPC. Some intuitive properties of proofs, however, are not invariant an ..."
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Cited by 21 (9 self)
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The Logic of Proofs LP captures the invariant propositional properties of proof predicates t is a proof of F with a set of operations on proofs sufficient for realizing the whole modal logic S4 and hence the intuitionistic logic IPC. Some intuitive properties of proofs, however, are not invariant and hence not present in LP. For example, the choice function ‘+ ’ in LP, which is specified by the condition s:F ∨t:F → (s+t):F, is not necessarily symmetric. In this paper, we introduce an extension of the Logic of Proofs, SLP, which incorporates natural properties of the standard proof predicate in Peano Arithmetic: t is a code of a derivation containing F, including the symmetry of Choice. We show that SLP produces BrouwerHeytingKolmogorov proofs with a rich structure, which can be useful for applications in epistemic logic and other areas. 1
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 and compl ..."
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Cited by 17 (13 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 16 (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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Cited by 11 (8 self)
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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...
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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Cited by 7 (0 self)
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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
Definability and Commonsense Reasoning
 ARTIF. INTELL
, 1997
"... The definition of concepts is a central problem in commonsense reasoning. Many themes in nonmonotonic reasoning concern implicit and explicit definability. Implicit definability in nonmonotonic logic is always relative to the context  the current theory of the world. We show that fixed point equati ..."
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Cited by 6 (3 self)
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The definition of concepts is a central problem in commonsense reasoning. Many themes in nonmonotonic reasoning concern implicit and explicit definability. Implicit definability in nonmonotonic logic is always relative to the context  the current theory of the world. We show that fixed point equations provide a generalization of explicit definability, which correctly captures the relativized context. Theories expressed within this logical framework provide implicit definitions of concepts. Moreover, it is possible to derive these fixed points entirely within the logic.
A Finitary Treatment of the Closed Fragment of
, 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 6 (6 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.
Theoremhood Preserving Maps As A Characterisation Of Cut Elimination For Provability Logics.
, 1999
"... We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps from ..."
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Cited by 5 (5 self)
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We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps from the provability logics into properly displayable modal logics. 1 Australian Research Council International Research Fellow from Laboratoire LEIBNIZCNRS, Grenoble, France. 2 Supported by an Australian Research Council Queen Elizabeth II Fellowship. 1 Introduction Background. Display Logic (DL) is a prooftheoretical framework introduced by Belnap [Bel82] that generalises the structural language of Gentzen's sequents in a rather abstract way by using multiple complex structural connectives instead of Gentzen's comma. The term "display" comes from the nice property that any occurrence of a structure in a sequent can be displayed either as the entire antecedent or as the entire succedent...