N. Bezhanishvili - Academia.edu (original) (raw)

Topological models for belief by N. Bezhanishvili

We introduce a new topological semantics for evidence, evidence-based justifications, belief and ... more We introduce a new topological semantics for evidence, evidence-based justifications, belief and knowledge. This setting builds on the evidence model framework of van Benthem and Pacuit, as well as our own previous work on (a topological semantics for) Stalnaker's doxastic-epistemic axioms. We prove completeness, decidability and finite model property for the associated logics, and we apply this setting to analyze key issues in Epistemology: " no false lemma " Gettier examples, misleading defeaters, and undefeated justification versus undefeated belief.

We introduce a new topological semantics for belief logics in which the belief modality is interp... more We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete w.r.t. the class of all topological spaces. Moreover, we point out a problem regarding updates on extremally disconnected spaces that appears in the setting of [1] and show that our proposal for topological belief semantics on all topological spaces constitutes a solution for it. While generalizing the topological belief semantics proposed in [1] to all spaces, we model conditional beliefs and updates and give complete axiomatizations of the corresponding logics.

We present a new topological semantics for doxastic logic, in which the belief modality is interp... more 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 is the most general (exten-sional) semantics validating Stalnaker's epistemic-doxastic axioms [22] for " strong belief " , understood as subjective certainty. We prove two completeness results, and we also give a topological semantics for update (dynamic conditioning), i.e. the operation of revising with " hard information " (modeled by restricting the topology to a subspace). Using this, we show that our setting fits well with the defeasibility analysis of knowledge [18]: topological knowledge coincides with undefeated true belief. Finally, we compare our semantics to the older topological interpretation of belief in terms of Cantor derivative [23].

Stalnaker introduced a combined epistemic-doxastic logic that can formally express a strong conce... more Stalnaker introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept which captures the 'epistemic possibility of knowledge'. In this paper we first provide the most general extensional semantics for this concept of 'strong belief', which validates the principles of Stalnaker's epistemic-doxastic logic. We show that this general extensional semantics is a topological semantics, based on so-called extremally disconnected topological spaces. It extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. Formally, our belief modality is interpreted as the 'closure of the interior'. We further prove that in this semantics the logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. In the second part of the paper we study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our investigation of dynamic belief change, is based on hereditarily extremally disconnected spaces. The logic of belief KD45 is sound and complete with respect to the class of hereditarily extremally disconnected spaces (under our proposed semantics), while the logic of knowledge is required to be S4.3. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic.

Papers by N. Bezhanishvili

Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heytin... more Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspective does allow us, in the case of Heyting algebras, to derive Ghilardi’s (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices. 1

For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite... more For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite; (ii) A is finitely approximable, complete, and completely joinprime generated; (iii) A is isomorphic to the Heyting algebra Up(X) of upsets of an image-finite poset X. We also show that A is isomorphic to its profinite completion iff A is finitely approximable, complete, and the kernel of every finite homomorphic image of A is a principal filter of A.

Abstract. In this paper we discuss a uniform method for constructing free modal and distributive ... more Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1

We introduce a simple propositional calculus for compact Hausdor spaces. Our approach is based on... more We introduce a simple propositional calculus for compact Hausdor spaces. Our approach is based on de Vries duality. The main new connective of our calculus is that of strict implication. We de ne the strict implication calculus SIC as our base calculus. We show that the corresponding variety SIA of strict implication algebras is a discriminator and locally nite variety. We prove that SIC is strongly sound and complete with respect to the universal subclass RSub of SIA, where the modality associated with the strict implication only takes on the values of 0 and 1. We develop Π2-rules for strict implication algebras, and show that every Π2-rule de nes an inductive subclass of RSub. We prove that every derivation system axiomatized by Π2-rules is strongly sound and complete with respect to the inductive subclass of RSub it de nes. We introduce the de Vries calculus DVC and show that it is strongly sound and complete with respect to the class of compingent algebras, and then use MacNeill...

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (an... more It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (and hence modal box as interior), then S4 is the logic of any dense-initself metrizable space [14, 17]. The McKinsey-Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem [8, 10].

A deductive system is said to be structurally complete if its admissible rules are derivable. In ... more A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its finitary extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a selfcontained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.

We give a sufficient condition for deciding admissibility of non-standard inference rules inside ... more We give a sufficient condition for deciding admissibility of non-standard inference rules inside a modal calculus S with the universal modality. The condition requires the existence of a model completion for the discriminator variety of algebras which are models of S. We apply the condition to the case of symmetric strict implication calculus, i.e., to the modal calculus axiomatizing contact algebras. Such an application requires a characterization of duals of morphisms which are embeddings (in the model-theoretic sense). We supply also an explicit infinite set of axioms for the class of existentially closed contact algebras. The axioms are obtained via a classification of duals of finite minimal extensions of finite contact algebras.

We propose a Logic of Abstraction, meant to formalize the act of " abstracting away " t... more We propose a Logic of Abstraction, meant to formalize the act of " abstracting away " the irrelevant features of a model. We give complete axiomatiza-tions for a number of variants of this formalism, and explore their expressivity. As a special case, we consider the " logics of filtration " .

European Management Journal, 2005

The Journal of Symbolic Logic, 2020

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space... more The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Ston...

Archive for Mathematical Logic

The Journal of Symbolic Logic

Proceedings of the 26th Workshop on Logic, Language, Information and Computation (WoLLIC 2019), 2019

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semanti... more We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev's logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces) [2]. In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact...

Annals of Pure and Applied Logic

European Management Journal, 2005

The Journal of Symbolic Logic, 2020

Archive for Mathematical Logic

The Journal of Symbolic Logic

Proceedings of the 26th Workshop on Logic, Language, Information and Computation (WoLLIC 2019), 2019

Annals of Pure and Applied Logic

Studia Logica, 2004

We prove that every normal extension of the bi-modal system S5 2 is finitely axiomatizable and th... more We prove that every normal extension of the bi-modal system S5 2 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.

Studia Logica, 2012

We define analogues of modal Sahlqvist formulas for the modal mucalculus, and prove a corresponde... more We define analogues of modal Sahlqvist formulas for the modal mucalculus, and prove a correspondence theorem for them.

Logic and Logical Philosophy