Structures for Epistemic Logic

Bezhanishvili, Nick; Hoek, Wiebe van der

doi:10.1007/978-3-319-06025-5_12

Cited by 13 publications

(14 citation statements)

References 80 publications

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“…We also recall that a topological space (X, τ ) is called an Alexandroff space if the intersection of open sets of X is open. It is well known that Alexandroff spaces correspond to reflexive and transitive Kripke frames, see e.g., [3], [20] or [8]. Moreover, the evaluation of modal formulas in an Alexandroff space coincides with their evaluation in the corresponding Kripke frame.…”

Section: Extensions and Improvementsmentioning

confidence: 99%

“…SEE [3], [20] and [8] for an overview of the results on the co-derived set semantics. Here we only mention that the logic wK4 = K + ((p ∧ Bp) → BBp) is complete wrt all topological spaces [12] and that the doxastic logic KD45 is complete wrt so-called DSO-spaces, where a topological space X, τ is called a DSO-space if it satisfies the T D -separation axiom, 9 for every A ⊆ X the set d (A) is open, and (X, τ ) is dense-in-itself, i.e., d(X) = X [24].…”

Section: Comparison With Related Workmentioning

confidence: 99%

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The Topology of Belief, Belief Revision and Defeasible Knowledge

Baltag¹,

Bezhanishvili

Özgün³

et al. 2013

Logic, Rationality, and Interaction

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Abstract. We present a new topological semantics for doxastic logic, in which the belief modality is interpreted as the closure of the interior operator. We show that this semantics validates Stalnaker's epistemicdoxastic axioms [23], and indeed it is the most general (extensional) semantics validating them. We prove, among other things, that in this semantics the doxastic logic KD45 is sound and complete with respect to the class of all extremally disconnected topological spaces. We also give a topological semantics for conditional belief and show its connection to the operation of updating with "hard information" (modeled by restricting the topology to a subspace). We show that our topological notions fit well with the defeasibility analysis of knowledge: topological knowledge coincides with undefeated true belief. We compare our semantics to the older topological interpretation of belief in terms of Cantor derivative (Steinsvold 2006), arguing in favor of our new semantics.

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Section: Extensions and Improvementsmentioning

confidence: 99%

Section: Comparison With Related Workmentioning

confidence: 99%

The Topology of Belief, Belief Revision and Defeasible Knowledge

Baltag¹,

Bezhanishvili

Özgün³

et al. 2013

Logic, Rationality, and Interaction

Self Cite

View full text Add to dashboard Cite

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“…Then, by the condition 4 of Definition 3.1, {z | X ∈ N (z)} ∈ N (w). Hence, X ∈ N (u), by (1). So v ∈ X, by (2).…”

Section: Invariance Results For Neighborhood Modelsmentioning

confidence: 85%

“…This semantics for some families of modal logic and its relation to the neighborhood semantics have been studied extensively, see e.g. [1]. The book [3] contains much materials on the connection between algebraic and topological semantics for IPL.…”

Section: Neighborhood Semantics Vs Topological Semanticsmentioning

confidence: 99%

Neighborhood Semantics for Basic and Intuitionistic Logic

Moniri

Maleki

2015

LLP

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Abstract. In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.

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“…Note also that none of the modalities alter either the epistemic or the doxastic range; this is essentially what guarantees the validity of the strong introspection axioms. 10 In order to distinguish these semantics from those previous, we refer to them as epistemicdoxastic (e-d) semantics for topological subset spaces. 9 If we want to insist on consistent beliefs, we should add the axiom (D B ): Bϕ →Bϕ (or, equivalently,B ) and require that V = ∅.…”

Section: Proposition 32 Let T : L K2b → L Kb Be the Map That Repmentioning

confidence: 99%

Logic and Topology for Knowledge, Knowability, and Belief

Bjorndahl

Özgün

2019

The Review of Symbolic Logic

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In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [30]. Building on Stalnaker's core insights, and using frameworks developed in [11] and [4], we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an "evidence-in-hand" conception of knowledge and a weaker "evidence-out-there" notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.

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Structures for Epistemic Logic

Cited by 13 publications

References 80 publications

The Topology of Belief, Belief Revision and Defeasible Knowledge

The Topology of Belief, Belief Revision and Defeasible Knowledge

Neighborhood Semantics for Basic and Intuitionistic Logic

Logic and Topology for Knowledge, Knowability, and Belief

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