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Operational Modal Logic
, 1995
"... Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact in ..."
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Cited by 79 (28 self)
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Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact interpretation as corresponding propositional operations on proofs; both S4 and Int turn out to be complete with respect to this proof realization. These results are based on operational reading of S4, where a modality is split into three operations. The logic of proofs with these operations is shown to be arithmetically complete with respect to the intended provability semantics and sufficient to realize every operation on proofs admitting propositional specification in arithmetic.
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.
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Cited by 10 (0 self)
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Peano's Smart Children  A provability logical study of systems with builtin consistency
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Logic of Proofs with the strong provability operator
 University of Amsterdam
, 1994
"... Logics with the strong provability operator ": : : is true and provable" together with the proof operators "p is a proof of : : :" are axiomatized. Kripkestyle completeness, decidability and arithmetical completeness of these logics are established. 1 Introduction Logics with th ..."
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Cited by 3 (3 self)
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Logics with the strong provability operator ": : : is true and provable" together with the proof operators "p is a proof of : : :" are axiomatized. Kripkestyle completeness, decidability and arithmetical completeness of these logics are established. 1 Introduction Logics with the provability operator ": : : is provable" and the proof operators "p is a proof of : : :" corresponding to certain natural classes of proof predicates in Peano arithmetic PA were introduced in [1]. In some respect (see e.g. [2],[3]) the strong provability operator ": : : is true and provable" provides a better model for provability than the operator ": : : is provable". The logic of the strong provability operator is known ([4]) to coincide with Grzegorczyk logic Grz. In this paper we present joint logics with both the strong provability operator and the proof operators. These logics are proved to be decidable; natural Research supported by the grant No. 2.1.21 of the Program "Universities of Russia"and ...
ON PROVABILITY LOGIC
, 2000
"... This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. I discuss the ideas that led to establishing this logic, I survey its history and the most important results, and I emphasize its applications in metamathematic ..."
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Cited by 2 (0 self)
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This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. I discuss the ideas that led to establishing this logic, I survey its history and the most important results, and I emphasize its applications in metamathematics. Stress is put on the use of Gentzen calculus for provability logic. I sketch my version of a decision procedure for provability logic and mention some connections to computational complexity.
Comparing strengths of beliefs explicitly
"... Inspired by a similar use in provability logic, formulas p ≻B q and p �B q are introduced in the existing logical framework for discussing beliefs to express that the strength of belief in p is greater than (or equal to) that in q. This explicit mention of the comparison in the logical language aid ..."
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Inspired by a similar use in provability logic, formulas p ≻B q and p �B q are introduced in the existing logical framework for discussing beliefs to express that the strength of belief in p is greater than (or equal to) that in q. This explicit mention of the comparison in the logical language aids in defining several other concepts in a uniform way, viz. older and rather clear concepts like the operators for universality (which possibilities ought to be considered), together with newer notions like plausibility (in the sense of ‘more plausible than not’) and disbelief. Moreover, it assists in studying the properties of the concept of greater strength of belief itself. A heavy part is played in our investigations by the relationship between the standard plausibility ordering of the worlds and the strength of belief ordering. If we try to define the strength of belief ordering in terms of the world plausibility ordering we get some undesirable consequences, so we have decided to keep the relation between the two orderings as light as possible to construct a system that allows for widely different interpretations. Finally, after a brief discussion on the multiagent setting, we move on to talk about the dynamics the change of ordering under the influence of hard and soft information.
THE ARITHMETICS OF A THEORY
"... Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of t ..."
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Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of the framework of the arithmetics of U. Finally, we provide a deeper study of the arithmetics
Bulletin of the Section of Logic Volume 33/1 (2004), pp. 11–21
"... To discuss Rosser sentences, Guaspari and Solovay [2] enriched the modal language by adding, for each 2A and 2B, the formulas 2A ≺ 2B and 2A 2B, with their arithmetic realizations the Σ1sentences “A ∗ is provable by a proof that is smaller than any proof of B∗”, and “A ∗ is provable by a proof th ..."
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To discuss Rosser sentences, Guaspari and Solovay [2] enriched the modal language by adding, for each 2A and 2B, the formulas 2A ≺ 2B and 2A 2B, with their arithmetic realizations the Σ1sentences “A ∗ is provable by a proof that is smaller than any proof of B∗”, and “A ∗ is provable by a proof that is smaller than or equal to any proof of B∗”. They axiomatized modal logics R−, R and Rω, each of which satisfies a kind of arithmetic completeness theorem concerning the above interpretation. Among these three logics, R − is the most preliminary one and provability of the other logics can be described by the provability of R−. Here we introduce a sequent system for R − with a kind of subformula property. 1. The logic R− In this section, we introduce the logic R−. We use logical constant ⊥ (contradiction), and logical connectives ∧ (conjunction), ∨ (disjunction), ⊃ (implication), 2 (provability), (witness comparison), and ≺ (witness comparison). Formulas are defined inductively as follows: