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Knowledge, Time, and Paradox: Introducing Sequential Epistemic Logic

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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics

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

Abstract

Epistemic logic in the tradition of Hintikka provides, as one of its many applications, a toolkit for the precise analysis of certain epistemological problems. In recent years, dynamic epistemic logic has expanded this toolkit. Dynamic epistemic logic has been used in analyses of well-known epistemic “paradoxes”, such as the Paradox of the Surprise Examination and Fitch’s Paradox of Knowability, and related epistemic phenomena, such as what Hintikka called the “anti-performatory effect” of Moorean announcements. In this paper, we explore a variation on basic dynamic epistemic logic—what we call sequential epistemic logic—and argue that it allows more faithful and fine-grained analyses of those epistemological topics.

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Notes

  1. 1.

    For a different variation on dynamic epistemic logic in a similar spirit, see [8, 9].

  2. 2.

    We will return to Hintikka’s point that “You may come to know that what I say was true” in Sect. 15.5.

  3. 3.

    An analogous statement is true for a third semantics for PAL based on arrow-elimination (as in [19, 32]), where the new relation \(R_{a \uparrow Q}\) is defined from the original relation \(R_a\) by: \(vR_{a \uparrow Q}u\) iff both \(vR_au\) and \(u\in Q\). Note, however, that non-trivial arrow-elimination will turn the reflexive \(R_a\) into a non-reflexive \(R_{a \uparrow Q}\), so the updated model will not be an epistemic model. For this reason, it is used to model the updating of belief rather than knowledge.

  4. 4.

    Also see [39, 47] on the role of time in knowability principles.

  5. 5.

    By contrast, the analysis in Sect. 3 of [18] involves syntactic derivations, as does the analysis in [24].

  6. 6.

    Also cf. [13], which uses an existential branching next-time operator, parametrized by epistemic actions, in connecting dynamic epistemic and epistemic temporal logic.

  7. 7.

    One way to see this is to note that the truth of our formulas without update operators is preserved under taking disjoint unions of models, defined in an obvious way, whereas \(\langle p\rangle \top \) concerns the model globally, not just what is reachable from the point of evaluation, so it is not preserved under taking disjoint unions of models. (If we had a universal modality, there would be more to say.).

  8. 8.

    The ‘\(\dots \)’ after \(S_{n-1}\) indicates that all coordinates of the new \(\omega \)-sequence are \(S_{n-1}\) thereafter, representing the supposition that “nothing else happens” after the update with \(\varphi _{n-1}\).

  9. 9.

    Even if \(\varphi \) does not contain futuristic operators, the fact that \(\langle !\rangle \) brings in a hypothetical future that may differ from the actual future means that we must be careful with the claim that \(Y\varphi \) expresses at \(t+1\) in the hypothetical sequence the “same proposition” that \(\varphi \) expresses at t in the actual sequence. This is correct if we mean that the set of worlds Q at which \(Y\varphi \) is true at \(t+1\) in the hypothetical sequence is the same as the set of worlds at which \(\varphi \) is true at t in the actual sequence. But the “worlds” in Q may have different futures—i.e., with respect to what epistemic relations these worlds will stand in—in the hypothetical sequence versus the actual sequence, so in a finer-grained sense, the set Q does not represent the same proposition relative to the hypothetical sequence and relative to the actual sequence. Still, the worlds in Q will have the same past up to t in both the hypothetical sequence and the actual sequence, so for a formula \(\varphi \) not containing futuristic operators, there is a reasonable sense in which the proposition expressed by \(Y\varphi \) at \(t+1\) in the hypothetical sequence is “the same” as the proposition expressed by \(\varphi \) at t in the actual sequence. This point deserves further discussion, but we do not have room for it here.

  10. 10.

    Cf. Hintikka on \(p\wedge \lnot Kp\) in Example 15.1: “You may come to know that what I say was true”.

  11. 11.

    This uses the fact that we are treating p as an eternal sentence, so \(KYp\rightarrow Kp\) is valid. If we were not treating p as eternal, then we would need to eternalize p with temporal operators: where \(S\varphi := P\varphi \vee \varphi \vee F\varphi \) (“sometime, \(\varphi \)”), the formula \(S p\wedge \lnot F K Sp\) is unascertainable.

  12. 12.

    See [24] for an analysis of the designated student paradox using static multi-agent epistemic logic. In that analysis, the assumptions about the initial epistemic states of the agents are not given by the model \(\mathscr {M}\), but rather by a weaker set of syntactically specified assumptions.

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Acknowledgements

This paper is the written version of a talk with the same title that I gave at the LogiCIC mini-workshop on The Dynamics of Information States, organized by Ben Rodenhäuser in Amsterdam on June 19, 2014. I am grateful to Ben for the invitation to the workshop and to Alexandru Baltag and Sonja Smets for their encouraging feedback on the talk. I also benefited from the experience of co-teaching a course with Eric Pacuit on Ten Puzzles and Paradoxes of Knowledge and Belief at ESSLLI 2013 in Dusseldorf. It was during a sleepless jet-lagged night in Dusseldorf, before my lecture on dynamic epistemic logic, that ideas in this paper occurred to me. Finally, I appreciate having had the pleasure to meet Jaakko Hintikka at Stanford in March 2012, when he very kindly spent a lunch discussing epistemic logic with me. For helpful comments on this paper, I thank Johan van Benthem and Hans van Ditmarsch.

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Authors and Affiliations

  1. Department of Philosophy and Group in Logic and the Methodology of Science, University of California, Berkeley, USA

    Wesley H. Holliday

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Correspondence to Wesley H. Holliday .

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  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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Holliday, W.H. (2018). Knowledge, Time, and Paradox: Introducing Sequential Epistemic Logic. 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_15

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