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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics pp 535–555Cite as

Epistemic Logic with Evidence and Relevant Alternatives

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 12))

Abstract

Starting with Jaakko Hintikka’s seminal work, epistemic logic has now grown up to a huge academic industry. When we look back to its history, however, there is a practical paradox. While Hintikka’s original purpose was to facilitate epistemological discussion, most subsequent work of epistemic logic has been done outside of philosophy, and has been ignored by most mainstream epistemologists. At the current point, one might naturally wonder, does epistemic logic still have something to do with epistemology? Along with Hintikka and a growing number of formal epistemologists, we believe the answer is “yes”. Contemporary epistemology has largely focused on the goal of providing an analysis of knowledge and an answer to skepticism. To avoid skepticism, many epistemologists prefer to define knowledge in terms of evidence and relevant alternatives. A typical saying is that “in order to know that p, one only needs evidence to eliminate all relevant alternatives, rather than all alternatives”. We see such a project as perfectly congenial to the spirit of Hintikka’s epistemic logic. In this paper, we propose an epistemic logic (EREL) to deal with evidence and relevant alternatives explicitly, and extend EREL to its dynamic version (PARL) to investigate how different notions of knowledge behavior with the decrease and increase of relevant alternatives. To illustrate its philosophical application, we provide a new formalization of the so called factivity problem of epistemic contextualism. We conclude with some remarks on further developments of our framework.

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Notes

  1. 1.

    Note that, Hintikka originally interprets epistemic formulas in model sets ([24], pp. 40–41). Roughly speaking, model sets are sets of formulas which partially describe possible states of affairs. Therefore, Hintikka’s semantics of epistemic logic are different from nowadays routine. Thank Professor Gabriel Sandu to remind us this.

  2. 2.

    According to Timothy Williamson, if one wants to be fully realistic, “knowledge implies truth” might be the only valid principle of knowledge (personal communication in 2009).

  3. 3.

    As far as we know, such a link-cutting dynamics was first introduced by van Benthem and Liu in the literature of preference upgrade [44].

  4. 4.

    As proved by Wang Yanjing, without axiom P5, one need the rule of equivalence replacement (RE: from \(\vdash \psi \leftrightarrow \chi \), derive \(\vdash \varphi \leftrightarrow \varphi (\psi /\chi )\)) to get a complete axiomatization of PAL [46].

  5. 5.

    Why not denying (c)? According to the first author, such an option, if not more, is equally plausible. It’s nature to think that, if one knows that p but does not know that q, then it is epistemically possible for her that p and not q, even if it is not logically possible.

  6. 6.

    We only prove \(CS(O\wedge \lnot H)=CS(O)\). \(CS(O\vee \lnot H)=CS(\lnot H)\) can be proved similarly. For any \(P\in CS(O)\), according to the definition of CS, O entails \(\lnot P\). Since O is clearly entailed by \(O\wedge \lnot H\), so \(O\wedge \lnot H\) also entails \(\lnot P\). By definition of CS again, \(P\in CS(O\wedge \lnot H)\). On the other hand, for any \(P\in CS(O\wedge \lnot H)\), according to the definition of CS, \(O\wedge \lnot H\) entails \(\lnot P\). Since O entails \(\lnot H\), O also entails \(O\wedge \lnot H\), and thus entails \(\lnot P\). Again by definition of CS, \(P\in CS(O)\).

  7. 7.

    One could easily generalize P and \(K_{s}\) to account for relevant alternatives and knowledge attributions under multiple contexts, e.g., let \(P^{c}\varphi \) represents “\(\varphi \) is true at all relevant alternatives in context c”, and \(K^{c}_{s}\varphi \) represents “s know that \(\varphi \) in context c”. In that case, our \(P\varphi \) and \(K_{s}\varphi \) can be seen as abbreviation of \(P^{c}\varphi \) and \(K^{c}_{s}\varphi \) for some fixed context c.

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Acknowledgements

The work is supported by the Chinese National Social Science Fund (12AZX008; 12AZD072), the Ministry of Education Humanities and Social Sciences Youth Project (15YJC72040001), and Sichuan University (skzx2015-sb05; skqy201645). Thank Professor Gabriel Sandu for helpful and detailed comments.

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Authors and Affiliations

  1. School of Public Administration, Sichuan University, Chengdu, 610065, China

    Zhaoqing Xu

  2. Center of Logic and Information, Sichuan Normal University, Chengdu, 610068, China

    Zhaoqing Xu

  3. Department of Philosophy, Peking University, Beijing, 100871, China

    Bo Chen

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  1. Zhaoqing Xu
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Correspondence to Zhaoqing Xu .

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Editors and Affiliations

  1. LORIA, CNRS, Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Hans van Ditmarsch

  2. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Gabriel Sandu

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Xu, Z., Chen, B. (2018). Epistemic Logic with Evidence and Relevant Alternatives. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_22

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