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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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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
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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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.
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 26 (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
Prefixed Tableaus and Nested Sequents
, 2010
"... Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tabl ..."
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Cited by 14 (2 self)
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Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.
Kripke Models of Transfinite Provability Logic
"... For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no nontrivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variablefree ..."
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For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no nontrivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variablefree fragment with natural number modalities, denoted GLP 0 ω. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each ordinals Θ, Λ we build a Kripke model I Θ Λ and show that GLP 0 Λ is sound for this structure. In our notation, Ignatiev’s original model becomes I ε0 ω.
Ordinal Completeness of Bimodal Provability Logic GLB
"... Bimodal provability logic GLB, introduced by G. Japaridze, currently plays an important role in the applications of provability logic to prooftheoretic analysis. Its topological semantics interprets diamond modalities as derived set operators on a scattered bitopological space. We study the quest ..."
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Cited by 5 (2 self)
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Bimodal provability logic GLB, introduced by G. Japaridze, currently plays an important role in the applications of provability logic to prooftheoretic analysis. Its topological semantics interprets diamond modalities as derived set operators on a scattered bitopological space. We study the question of completeness of this logic w.r.t. the most natural space of this kind, that is, w.r.t. an ordinal α equipped with the interval topology and with the socalled club topology. We show that, assuming the axiom of constructibility, GLB is complete for any α ≥ ℵω. On the other hand, from the results of A. Blass it follows that, assuming the consistency of “there is a Mahlo cardinal, ” it is consistent with ZFC that GLB is incomplete w.r.t. any such space. Thus, the question of completeness of GLB w.r.t. natural ordinal spaces turns out to be independent of ZFC.
Explicit Proofs in Formal Provability Logic
 Logical Foundations of Computer Science. International Symposium, LFCS 2007
"... In this paper we answer the question what implicit proof assertions in the provability logic GL can be realized by explicit proof terms. In particular we show that the fragment of GL which can be realized by generalized proof terms of GLA is exactly S4∩GL and equals the fragment that can be realized ..."
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In this paper we answer the question what implicit proof assertions in the provability logic GL can be realized by explicit proof terms. In particular we show that the fragment of GL which can be realized by generalized proof terms of GLA is exactly S4∩GL and equals the fragment that can be realized by proofterms of LP. Additionally we show that the problem of determining which implicit provability assertions in a given modal formula can be made explicit is decidable. In the final sections of this paper we establish the disjunction property for GLA and give an axiomatization for GL ∩ S4. 1
Public Communication in Justification Logic
, 2007
"... Justification Logic is the study of a family of logics used to reason about justified true belief. Dynamic Epistemic Logic is the study of a family of logics obtained by adding various kinds of communication to the language of multimodal logic, yielding languages for reasoning about communication a ..."
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Cited by 4 (1 self)
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Justification Logic is the study of a family of logics used to reason about justified true belief. Dynamic Epistemic Logic is the study of a family of logics obtained by adding various kinds of communication to the language of multimodal logic, yielding languages for reasoning about communication and true belief. This paper is a firststep in merging these two areas, in that it brings the most basic kind of communication studied in Dynamic Epistemic Logic—the public announcement—over to Justification Logic. This gives us a language for reasoning about public announcements and justified true belief. After giving an overview of Justification Logic, the paper introduces a notion of bisimulation for Justification Logic. Bisimulation allows us to study the affect on language expressivity when we add various kinds of communication to the language. Among a number of expressivity results, we show that adding public announcements to the language of Justification Logic strictly increases language expressivity. This stands in contrast to the PlazaGerbrandy Theorem, which states that adding public announcements to multimodal logic does not increase language expressivity. This leads us to extend the language of Justification Logic in order to provide a PlazaGerbrandy analog of multimodal logic that we can use to reason about justified true belief. 1
Principles of Constructive Provability Logic
, 2010
"... We present a novel formulation of the modal logic CPL, a constructive logic of provability that is closely connected to the GödelLöb logic of provability. Our logical formulation allows modal operators to talk about both provability and nonprovability of propositions at reachable worlds. We are in ..."
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We present a novel formulation of the modal logic CPL, a constructive logic of provability that is closely connected to the GödelLöb logic of provability. Our logical formulation allows modal operators to talk about both provability and nonprovability of propositions at reachable worlds. We are interested in the applications of CPL to logic programming; however, this report focuses on the presentation of a minimal fragment (in the sense of minimal logic) of CPL and on the formalization of minimal CPL and its metatheory in the Agda programming language. We present both a natural deduction system and a sequent calculus for minimal CPL and show that the presentations are equivalent.