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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 logic of proofs, semantically
 Annals of Pure and Applied Logic
, 2005
"... web page: comet.lehman.cuny.edu/fitting ..."
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
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
On the complexity of the reflected logic of proofs
 Theoretical Computer Science
"... disjunctive property, complexity. Artemov’s system LP captures all propositional invariant properties of a proof predicate “x proves y ” ([1, 3]). Kuznets in [5] showed that the satisfiability problem for LP belongs to the class Π p 2 of the polynomial hierarchy. No nontrivial lower complexity bound ..."
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Cited by 14 (1 self)
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disjunctive property, complexity. Artemov’s system LP captures all propositional invariant properties of a proof predicate “x proves y ” ([1, 3]). Kuznets in [5] showed that the satisfiability problem for LP belongs to the class Π p 2 of the polynomial hierarchy. No nontrivial lower complexity bound for LP is known. We describe quite expressive syntactical fragment of LP which belongs to NP. It is rLP∧, ∨ – the set of all theorems of LP which are monotone boolean combinations of quasiatomic formulas (facts of sort “t proves F ”). A new decision algorithm for this fragment is proposed. It is based on a new simple independent formalization for rLP (the reflected fragment of LP) and involves the corresponding proof search procedure. Essentially rLP contains all the theorems of LP supplied with additional information about their proofs. We show that in many respects rLP is simpler than LP itself. This gives the complexity bound (NP) for rLP. In addition we prove a suitable variant of the disjunctive property which extends this bound to rLP∧,∨. 1 1
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
A logic of explicit knowledge
 In L. Behounek and M. Bilkova (Eds.), Logica Yearbook 2004
, 2005
"... A wellknown problem with Hintikkastyle logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpli ..."
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Cited by 6 (0 self)
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A wellknown problem with Hintikkastyle logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpliciter, to knowledge for a reason. Then, as consequences become ‘further away ’ from one’s basic knowledge, reasons for them become more complex, thus providing a kind of resource measurement. One kind of reason is a formal proof. Sergei Artemov has introduced a logic of explicit proofs, LP. I present a semantics for this, based on the idea that it is a logic of knowledge with explicit reasons. A number of fundamental facts about LP can be established using this semantics. But it is equally important to realize that it provides a natural logic of more general applicability than its original provenance, arithmetic provability. 1
The NPCompleteness of Reflected Fragments of Justification Logics
"... Abstract. Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities w ..."
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Cited by 4 (3 self)
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Abstract. Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the socalled reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NPcomplete, thereby proving a matching lower bound. 1 Introduction and Main Definitions Justification Logic is an emerging field that studies provability, knowledge, and belief via explicit proofs or justifications that are part of the language. A justification
Why do we need Justification Logic
 Norms and Reasons: Logic at the Crossroads, Synthese Library 353
, 2011
"... In this paper, we will sketch the basic system of Justification Logic, which is a general logical framework for reasoning about epistemic justification. Justification Logic renders a new, evidencebased foundation for epistemic logic. As a case study, we compare formalizations of the Kripke ‘Red Bar ..."
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Cited by 3 (0 self)
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In this paper, we will sketch the basic system of Justification Logic, which is a general logical framework for reasoning about epistemic justification. Justification Logic renders a new, evidencebased foundation for epistemic logic. As a case study, we compare formalizations of the Kripke ‘Red Barn ’ scenario in modal epistemic logic and Justification Logic and show here that the latter provides a deeper analysis. In particular, we argue that modal language fails to fully represent the epistemic closure principle whereas Justification Logic provides its adequate formalization. 1
Decidability for some Justification Logics with Negative Introspection
, 2011
"... Justification logics are modal logics that include justifications for the agent’s knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logi ..."
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Cited by 3 (3 self)
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Justification logics are modal logics that include justifications for the agent’s knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications. 1