Abstract
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 modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on hereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. 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.
Highlights
Edmund Gettier’s famous counterexamples against the justified true belief (JTB) account of knowledge [32] invited an interesting and extensive discussion among formal epistemologists and philosophers concerned with understanding the correct relation between knowledge and belief, and, in particular, with identifying the exact properties and conditions that distinguishes a piece of belief from a piece of knowledge and vice versa
We model knowledge and belief on hereditarily extremally disconnected spaces and propose a topological semantics for conditional beliefs and updates based on these spaces
Steinsvold [54] does not elaborate on to what extend his proposed semantics for knowledge could give new insight into the Gettier problem and leaves this point open for discussion. We propose another topological semantics for belief, in particular, for Stalnaker’s notion of belief as subjective certainty [53], in terms of the closure of the interior operator on extremally disconnected spaces
Summary
Edmund Gettier’s famous counterexamples against the justified true belief (JTB) account of knowledge [32] invited an interesting and extensive discussion among formal epistemologists and philosophers concerned with understanding the correct relation between knowledge and belief, and, in particular, with identifying the exact properties and conditions that distinguishes a piece of belief from a piece of knowledge and vice versa. We formalize a notion of conditional belief Bφψ by relativizing the semantic clause for a simple belief modality to the extension of the learnt formula φ and first give a complete axiomatization of the logic of knowledge and conditional beliefs based on extremally disconnected spaces. This topological interpretation of conditional belief allows us to model static belief revision of a more general type than axiomatized by the AGM theory: the topological model val-. We model knowledge and belief on hereditarily extremally disconnected spaces and propose a topological semantics for conditional beliefs and updates based on these spaces.. We refer the reader who is not familiar with the aforementioned topics to [19, Chapters 1 and 4]
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