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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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Cited by 132 (24 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 representat ..."
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Cited by 25 (9 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 ..."
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Cited by 7 (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 polynom ..."
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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 g ..."
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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.
Computation, consciousness and the quantum
 Teorie e Modelli
, 2001
"... Abstract: It is sometimes said that Everett’s formulation of Quantum Mechanics dispenses us with the need of a theory of consciousness in the foundation of physics. This is false as Everett himself clearly recognized in his paper. Indeed he has build its quantum mechanics formulation by using expli ..."
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Abstract: It is sometimes said that Everett’s formulation of Quantum Mechanics dispenses us with the need of a theory of consciousness in the foundation of physics. This is false as Everett himself clearly recognized in his paper. Indeed he has build its quantum mechanics formulation by using explicitly the mechanist or computationalist hypothesis in psychology. Everett and his followers have then derived the subjective appearance, in the mind of machineobservers, of indeterminacy and nonlocality from the Schrödinger Equation. I argue in this paper that if we take the computationalist hypothesis seriously enough then the Schrödinger equation itself should be derivable from the computationalist theory of consciousness, making ultimately physics a branch of machine’s psychology. I sketch the basic argument and illustrate it with two embryonic derivations. In some sense I criticize Everett for his lack of radicality. 1 Quantum Realism Let me put it in this way: all sufficiently realist1 interpretations of quantum
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
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 pr ..."
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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.