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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.
On Epistemic Logic with Justification
 NATIONAL UNIVERSITY OF SINGAPORE
, 2005
"... The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This ..."
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Cited by 20 (7 self)
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The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This
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...
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which ..."
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Cited by 2 (2 self)
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Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which constitute the system of proof terms in LP. Proof polynomials are built from variables and constants by three operations on proofs: "\Delta" (application), "!" (proof checker), and "+" (choice). Here constants stand for canonical proofs of "simple facts", namely instances of propositional axioms and axioms of LP in a given proof system. We show that every operation on proofs that (i) can be specified in a propositional modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial.
Explicit Modal Logic
 in Proceedings AiMLII, Philosophical Institute
, 1998
"... In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give t ..."
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Cited by 1 (0 self)
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In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's suggestion (1938) of replacing formulas "F is provable" by the propositions for explicit proofs "t is a proof of F" (t : F ). LP admits the reflection of explicit proofs t : F ! F thus circumventing restrictions imposed on the provability operator by Godel's second incompleteness theorem. LP formalizes the Kolmogorov calculus of problems and proves the Kolmogorov conjecture that intuitionistic logic coincides with the classical calculus of problems.
Basic systems of epistemic logic with justification
, 2005
"... An issue of an epistemic logic with justification has been discussed since the early 1990s. Such a logic, along with the usual knowledge operator ✷F (F is known), should contain assertions t:F (t is a justification for F), which gives a more nuanced and realistic model of knowledge. In this paper, w ..."
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An issue of an epistemic logic with justification has been discussed since the early 1990s. Such a logic, along with the usual knowledge operator ✷F (F is known), should contain assertions t:F (t is a justification for F), which gives a more nuanced and realistic model of knowledge. In this paper, we build two systems of epistemic logic with justification: the minimal one—S4LP—which is an extension of the basic epistemic logic S4 by an appropriate calculus of justification corresponding to the logic of proofs LP, and S4LPN—which is S4LP augmented by the explicit negative introspection principle ¬(t:F) → ✷¬(t:F). Epistemic semantics for both systems are suggested. Completeness and specific properties of S4LP and S4LPN, reflecting the explicit character of those systems, are established. 1
The Modal Logic of Pure Provability
, 1990
"... We introduce a propositional modal logic PP of "pure" provability in arbitrary theories (propositional or firstorder) where the # operator means "provable in all extensions". This modal logic has been considered in another guise by Kripke. An axiomatization and a decision procedure are given an ..."
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We introduce a propositional modal logic PP of "pure" provability in arbitrary theories (propositional or firstorder) where the # operator means "provable in all extensions". This modal logic has been considered in another guise by Kripke. An axiomatization and a decision procedure are given and the ## subtheory is characterized.
On the BlokEsakia Theorem
"... Abstract We discuss the celebrated BlokEsakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of genera ..."
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Abstract We discuss the celebrated BlokEsakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the BlokEsakia theorem to extensions of intuitionistic logic with modal operators and coimplication. In memory of Leo Esakia 1