Logics for Belief as Maximally Plausible Possibility
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Logics for Belief as Maximally Plausible Possibility
Abstract. We consider a basic logic with two primitive uni-modal operators: one for certainty and the other for plausibility. The former is assumed to be a normal operator (corresponding—semantically—to a binary Kripke relation), while the latter is merely a classical operator (corresponding—semantically—to a neighborhood function). We then define belief, interpreted as “maximally plausible possibility”, in terms of these two notions: the agent believes φ if (1) she cannot rule out φ (that is, it is not the case that she is certain that ¬φ), (2) she judges φ to be plausible and (3) she does not judge ¬φ to be plausible. We consider four interaction properties between certainty and plausibility and study how these properties translate into properties of belief (e.g. positive and negative introspection and their converses). We then prove that all the logics considered are minimal logics for the highlighted theorems. We also consider a number of possible interpretations of plausibility, identify the corresponding logics and show that some notions considered in the literature are special cases of our framework. Keywords: Knowledge, Belief, Certainty, Plausibility, Positive introspection, Negative introspection.
Introduction and Overview There is a large body of literature in philosophy devoted to investigating the notions of knowledge and belief. One strand in the literature takes belief as primitive and addresses the question “what is needed in order for belief to constitute knowledge?”.1 Another strand takes knowledge as primitive and defines belief in terms of knowledge.2 Yet another strand in the literature takes both knowledge and belief as primitive notions and investigates the interaction between the two: for example, whether it is reasonable to postulate that if the individual believes φ then she should believe that she knows φ.3 1 2 3
See, for example, [19, 27, 39, 46]. For example, belief has been defined as the epistemic possibility of knowledge: [31, 38]. See, for example, [21, 22, 31, 32, 38].
Presented by Richmond Thomason; Received May 8, 2019
Studia Logica https://doi.org/10.1007/s11225-019-09887-w
c Springer Nature B.V. 2019
G. Bonanno
Another body of literature (mainly in modal logic, computer science and game theory) takes belief as primitive but distinguishes two different, yet coexisting, types of belief, one often referred to as knowledge and the other as belief.4 Knowledge is interpreted as a stronger doxastic attitude, for example reflecting strong evidence, while belief represents a weaker doxastic attitude, reflecting an assessment of likelihood or plausibility. Some of this literature suggests that the difference between the two notions reflects a distinction between hard information and soft information. For example, van Benthem writes [40, p.2]: “[. . . ] hard information, [. . . ] changes what I know. If I see that the Ace of Spades is played on the table, I come to know that no one holds it any more. [. . . ] Soft information, [. . .
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