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Erschienen in:
The Useful MAM, a Reasonable Implementation of the Strong -Calculus Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

Justified Belief and the Topology of Evidence

verfasst von : Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün, Sonja Smets

Erschienen in: Logic, Language, Information, and Computation

Verlag: Springer Berlin Heidelberg

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Abstract

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.

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Vorheriges Kapitel Foundations of Mathematics: Reliability and Clarity: The Explanatory Role of Mathematical Induction
Nächstes Kapitel Semantic Acyclicity for Conjunctive Queries: Approximations and Constraints
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Fußnoten
1
Factive evidence is true in the actual world. In Epistemology it is common to reserve the term “evidence” for factive evidence. But we follow here the more liberal usage of this term in [20], which agrees with the common usage in day to day life, e.g. when talking about “uncertain evidence”, “fake evidence”, “misleading evidence” etc.
 
2
Another, purely technical advantage of our setting is that the resulting doxastic logic has finite model property, in contrast to the one in [21].
 
3
Indeed, the logic of Stalnaker’s knowledge is not S5, but the modal logic S4.2.
 
4
The notion of evidence model in [21] is more general, covering cases in which evidence depends on the actual world, but we stick with what they call ‘uniform’ models, since this corresponds to restricting to agents who are “evidence-introspective”.
 
5
This is a difference in notation with the setting in [20, 21], where E is used to denote the family of basic evidence sets (denoted here by \(E_0\)).
 
6
This is both to fit with the strength order on bodies of evidence (since \(F\subseteq F'\) implies \(\bigcap F\supseteq \bigcap F'\)), and to ensure that stronger evidence supports more propositions: since, if \(e\supseteq e'\), then every proposition supported by e is supported by \(e'\).
 
7
A preorder on X is a reflexive-transitive relation on X.
 
8
A subset \(A\subseteq X\) is said to be upward-closed wrt \(\le \) if \(\forall x, y\in X\, (x\in A \wedge x\le y\, \Rightarrow \, y\in A)\).
 
9
These families generate the same topology. We denote it by \(\tau _E\) only because the family E of combined evidence forms a basis of this topology.
 
10
In a multi-agent model, some worlds might be consistent with one agent’s information, while being ruled out by another agent’s information. So, in a multi-agent setting, \(\forall _i\) will only quantify over all the states in agent i’s current information cell (according to a partition \(\varPi _i\) of the state space reflecting agent i’s hard information).
 
11
They denote this by EP, but we use \(E_0 P\) for this notion, since we reserve the notation EP for having combined evidence for P.
 
12
Note that all the notions of belief we consider are global: they do not depend on the state of the world, i.e. we have either \(Bel P=X\) or \(Bel P=\emptyset \) (similar to the sets \(\forall P, E_0 P, E P\)). This expresses the assumption that belief is a purely internal notion, thus transparent and hence absolutely introspective. This is standard in logic and accepted by most philosophers.
 
13
Lehrer uses the metaphor of a Subjective Justification Game [13]: rational beliefs are based on justifications that survive a game between the Believer and an inner Critic, who tries to defeat them using the Believer’s own “acceptance system”.
 
14
As we’ll see, K and B satisfy all the Stalnaker axioms for knowledge and belief [1, 2, 16] and further generalizes our previous work on a topological interpretation of Stalnaker’s doxastic-epistemic axioms, which was based on extremally disconnected spaces.
 
15
Lehrer uses the metaphor of an ‘Ultra-Justification Game’ [13], according to which ‘knowledge’ is based on arguments that survive a game between the Believer and an omniscient truth-telling Critic, who tries to defeat the argument by using both the Believer’s current “justification system” and any new true evidence.
 
16
This shows that the semantics in this paper correctly generalizes the one in [1, 2, 16] for the system KB.
 
17
This axiom originates from [20], where it is stated as an equivalence rather than an implication. But the converse is provable in our system.
 
Literatur
1.
Baltag, A., Bezhanishvili, N., Özgün, A., Smets, S.: The Topology of belief, belief revision and defeasible knowledge. In: Grossi, D., Roy, O., Huang, H. (eds.) LORI. LNCS, vol. 8196, pp. 27–40. Springer, Heidelberg (2013)CrossRef
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Baltag, A., Smets, S.: Conditional doxastic models: a qualitative approach to dynamic belief revision. In: Proceedings of WOLLIC, vol. 165, pp. 5–21 (2006)
4.
Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. Texts Logic Games 3, 9–58 (2008)MathSciNetMATH
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Clark, M.: Knowledge and grounds: a comment on Mr. Gettier’s paper. Analysis 24, 46–48 (1963)CrossRef
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Foley, R.: When is True Belief Knowledge?. Princeton University Press, Princeton (2012)
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Gettier, E.: Is justified true belief knowledge? Analysis 23, 121–123 (1963)CrossRef
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Kelly, K.: The Logic of Reliable Inquiry. Oxford University Press, Oxford (1996)MATH
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Klein, P.: A proposed definition of propositional knowledge. J. Philos. 68, 471–482 (1971)CrossRef
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Klein, P.: Certainty, a Refutation of Scepticism. University of Minneapolis Press, Minneapolis (1981)
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Lehrer, K.: Theory of Knowledge. Routledge, London (1990)MATH
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Lehrer, K., Paxson, T.J.: Knowledge: undefeated justified true belief. J. Philos. 66, 225–237 (1969)CrossRef
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Özgün, A.: Topological Models for Belief and Belief Revision. Master’s thesis, University of Amsterdam, Amsterdam, The Netherlands (2013)
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van Benthem, J., Fernández-Duque, D., Pacuit, E.: Evidence and plausibility in neighborhood structures. Ann. Pure Appl. Logic 165, 106–133 (2014)MathSciNetCrossRefMATH
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Metadaten
Titel
Justified Belief and the Topology of Evidence
verfasst von
Alexandru Baltag
Nick Bezhanishvili
Aybüke Özgün
Sonja Smets
Copyright-Jahr
2016
Verlag
Springer Berlin Heidelberg
Buch
Logic, Language, Information, and Computation

Print ISBN: 978-3-662-52920-1

Electronic ISBN: 978-3-662-52921-8

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-662-52921-8_6

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