Topological Subset Space Models for Belief

The Review of Symbolic Logic, 2019

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.

Electronic Proceedings in Theoretical Computer Science

Bjorndahl, A.; Özgün, A.

2015

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.

Journal of Philosophical Logic

Stalnaker (Philosophical Studies, 128(1), 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. We prove that the belief 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. We also study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modaliti...

2013

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].

We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the closure of the interior operator. We show that our semantics validates the axioms of Stalnaker's combined system of knowledge and belief, in fact, that it constitutes the most general extensional (and compositional) semantics validating these axioms. We further prove that in this semantics the logic KD45 is sound and complete with respect to the class of extremally disconnected spaces. We have a critical look at the topological interpretation of belief in terms of the derived set operator [45] and compare it with our proposal. We also provide two topological semantics for conditional beliefs of which especially the latter is quite successful in capturing the rationality postulates of AGM theory. We further investigate a topological analogue of dynamic belief change, namely, update. In addition, we provide a completeness result of the system wKD45, a weakened version of K...

2015

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.

2018

This thesis studies several aspects of the topological semantics for evidencebased belief and knowledge introduced by Baltag, Bezhanishvili, Özgün, and Smets (2016). Building on this work, we introduce a notion of generic models, topological spaces whose logic is precisely the sound and complete logic of topological evidence models. We provide generic models for the different fragments of the language. Moreover, we give a multi-agent framework which generalises that of single-agent topological evidence models. We provide the complete logic of this framework together with some generic models for a fragment of the language. Finally, we define a notion of group knowledge which differs conceptually from previous approaches.

Erkenntnis

The first aim of this paper is to prove a topological completeness theorem for a weak version of Stalnaker's logic KB of knowledge and belief. The weak version of KB is characterized by the assumption that the axioms and rules of KB have to be satisfied with the exception of the axiom (NI) of negative introspection. The proof of a topological completeness theorem for weak KB is based on the fact that nuclei (as defined in the framework of point-free topology) give rise to a profusion of topological belief operators that are compatible with the familiar topological knowledge operator. Thereby a canonical topological model for weak KB can be constructed. For this canonical model a truth lemma for the K and B holds such that a completeness theorem for KB can be proved in the familiar way. The second aim of this paper is to show that the topological interpretation of knowledge K comes along with a complete Heyting algebra of belief operators B that all fit the knowledge operator K in the sense that the pairs (K, B) satisfy all axioms of weak KB. This amounts to a pluralistic relation between knowledge and belief: Knowledge does not fully determine belief, rather it designs a conceptual space for belief operators where different (competing) belief operators coexist.

Synthese, 2022

We propose a new topological semantics for evidence, evidence-based justifications, belief, and knowledge. Resting on the assumption that an agent's rational belief is based on the available evidence, we try to unveil the concrete relationship between an agent's evidence, belief, and knowledge via a rich formal framework afforded by topologically interpreted modal logics. We prove soundness, completeness, decidability, and the finite model property for the associated logics, and apply this setting to analyze key epistemological issues such as "no false lemma" Gettier examples, misleading defeaters, undefeated justification versus undefeated belief, as well as the defeasibility theories of knowledge.