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129
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.
Formalizing Context (Expanded Notes)
, 1995
"... this article was going through many versions as the ideas developed, and the mutual influences cannot be specified. This work was partly supported by DARPA contract NAG2703 and ARPA/ONR grant N000149410775 ..."
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Cited by 104 (5 self)
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this article was going through many versions as the ideas developed, and the mutual influences cannot be specified. This work was partly supported by DARPA contract NAG2703 and ARPA/ONR grant N000149410775
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 43 (18 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
CutElimination and a PermutationFree Sequent Calculus for Intuitionistic Logic
, 1998
"... We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ..."
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Cited by 40 (6 self)
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We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.
The Logic of Justification
 Cornell University
, 2008
"... We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles origin ..."
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Cited by 30 (4 self)
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We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the wellknown Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidencebased foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanKripke ‘Red Barn ’ paradox and analyze Russell’s ‘prime minister example ’ in Justification Logic. Furthermore, we formalize the wellknown Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. 1
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
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Cited by 25 (2 self)
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
Provability logic
 Handbook of Philosophical Logic, 2nd ed
, 2004
"... We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles origin ..."
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Cited by 25 (9 self)
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We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the wellknown Justified True Belief vs. Knowledge problem. As a case study, we formalize Gettier examples in Justification Logic and reveal hidden assumptions and redundancies in Gettier reasoning. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidencebased foundation for epistemic logic. 1
Deciding regular grammar logics with converse through firstorder logic
 JOURNAL OF LOGIC, LANGUAGE AND INFORMATION
, 2005
"... ..."
A systematic proof theory for several modal logics
 Advances in Modal Logic, volume 5 of King’s College Publications
, 2005
"... abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence betw ..."
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Cited by 24 (1 self)
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abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a HilbertLewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is adhoc. While we can formulate several modal logics in the sequent calculus that enjoy cutelimination, their formalisation arises through systembysystem fine tuning to ensure that the cutelimination holds, and the correspondence to the axioms of the HilbertLewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the HilbertLewis axiomatisation. We show that the calculus possesses a cutelimination property directly analogous to cutelimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1
Permutability of Proofs in Intuitionistic Sequent Calculi
, 1996
"... We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deductio ..."
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Cited by 23 (4 self)
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We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are interpermutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...