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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics pp 499–533Cite as

Beyond Knowing That: A New Generation of Epistemic Logics

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

Abstract

Epistemic logic has become a major field of philosophical logic ever since the groundbreaking work by Hintikka [58]. Despite its various successful applications in theoretical computer science, AI, and game theory, the technical development of the field has been mainly focusing on the propositional part, i.e., the propositional modal logics of “knowing that”. However, knowledge is expressed in everyday life by using various other locutions such as “knowing whether”, “knowing what”, “knowing how” and so on (knowing-wh hereafter). Such knowledge expressions are better captured in quantified epistemic logic, as was already discussed by Hintikka [58] and his sequel works at length. This paper aims to draw the attention back again to such a fascinating but largely neglected topic. We first survey what Hintikka and others did in the literature of quantified epistemic logic, and then advocate a new quantifier-free approach to study the epistemic logics of knowing-wh, which we believe can balance expressivity and complexity, and capture the essential reasoning patterns about knowing-wh. We survey our recent line of work on the epistemic logics of ‘knowing whether”, “knowing what” and “knowing how” to demonstrate the use of this new approach.

A mindmap of this paper is here: http://www.phil.pku.edu.cn/personal/wangyj/mindmaps/.

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Notes

  1. 1.

    Hintikka was never happy with the term “possible worlds”, since in his models there may be no “worlds” but only situations or states, which are partial descriptions of the worlds. However, in this paper we will still use the term “possible worlds” for convenience.

  2. 2.

    How is in general also considered as a wh-question word, besides what, when, where, who, whom, which, whose, and why.

  3. 3.

    The “knows X” search term can exclude the phrases such as “you know what” and count only the statements, while “know X” may appear in questions as well.

  4. 4.

    Later on he singled out “knowing why” in his framework of interrogative models [65].

  5. 5.

    It also makes sense to understand knowing-wh constructions by first understanding the semantics of questions, see [53] and references therein. Knowing-wh is then knowing a/the answer of the corresponding wh-question.

  6. 6.

    [58] argued that the quantification into the modal context is necessary and not misleading, in contrast to Quine who was against such quantification due to the lack of substitution of identity in modal context.

  7. 7.

    How the domain varies may affect the corresponding quantified modal axioms, see [17] for a overview on this issue in first-order modal logic.

  8. 8.

    The above \(\textit{K}\forall x (\exists y\slash \textit{K})(M(x)\rightarrow T(x,y))\) is an example that cannot be expressed in standard first-order epistemic logic.

  9. 9.

    For the background of first-order modal logic, the readers are referred to the handbook chapter [17] and the book [46]. For the discussions on the philosophical issues of quantified first-order epistemic logic, see [72] and references therein.

  10. 10.

    Modeling it globally can be done in propositional modal logic with new axioms like \(K_jp\rightarrow K_ip\), cf. e.g., [80].

  11. 11.

    The absence of equality symbols also make the substitution of equal constants apparently irrelevant.

  12. 12.

    See [3] in this volume for a quantified epistemic logic treatment of this context-sensitivity of knowing who, using conceptual covers proposed by Aloni [2].

  13. 13.

    On the other hand, a slightly different axiom holds intuitively: knowing whether \(\varphi \leftrightarrow \psi \) and knowing whether \(\varphi \) does entail knowing whether \(\psi \).

  14. 14.

    Fitch proved that you cannot know all the truths, e.g., \(p\wedge \lnot K_i p\) is not knowable by i, which is demonstrated by the inconsistent Moore sentence: \(K_i(p\wedge \lnot K_i p)\) in the basic epistemic logic [45].

  15. 15.

    For example, [28, 29, 73, 76, 86, 105], see [43] for a survey.

  16. 16.

    See [5] for more general versions of the knowing whether operator.

  17. 17.

    Similar semantics has been applied to neighborhood structures [41].

  18. 18.

    Note that if there is at most one successor of s then every \(\varDelta \varphi \) formula holds.

  19. 19.

    The transitivity is hard, you need to enrich the two bisimulations a bit in connection with the middle model when proving it, see [40].

  20. 20.

    See [43] for comparisons with other equivalent systems in the literature.

  21. 21.

    Here we can also see the parallel of deduction and interrogation that [64] discussed.

  22. 22.

    For some equivalent proof systems in the literature, see the survey and comparisons in [43].

  23. 23.

    As we mentioned earlier, knowing the value can be seen as knowing the answer to a concealed question, see [4] and references therein for some recent discussions.

  24. 24.

    Two people S and P are told respectively the sum and product of two natural numbers which are known to be below 100. The following conversation happens: P says: “I do not know the numbers.” S says: “I knew you didn’t.” P says: “I now know the numbers.” S says: “I now also know it.”.

  25. 25.

    On the other hand, replacing \(\textit{Kv}\) with the knowing whether operator results in a consistent formula.

  26. 26.

    [87] gave the following two extra introspection axioms on top of \(\mathbb {S}5\) to capture this announcement-free fragment without a proof: \(\textit{Kv}_ic\rightarrow \textit{K}_i\textit{Kv}_i c\) and \(\lnot \textit{Kv}_i c\rightarrow \textit{K}_i\lnot \textit{Kv}_i c\). Our later language will supersede this simple language.

  27. 27.

    For example, I know that I have hands given that I am not a brain in a vat.

  28. 28.

    Uniform substitution does not work for these new schemas.

  29. 29.

    The decidability of \(\mathbf{ELKv }^r\) over epistemic models was shown by [102].

  30. 30.

    In some applications in computer science, the exact value is also not that important, but people care about whether two values are equivalent, e.g., see logic works on data words [15, 16]. The author thanks Martin Otto for pointing this out.

  31. 31.

    Instead of the ternary relation, it seems also natural to introduce an anti-equivalence relation \(R^c\) such that \(sR^ct\) intuitively means that s and t do not agree on the value of c. However, this approach faces troubles due to the limited expressive power of the modal language, see [52] for a detailed discussion.

  32. 32.

    Clearly the corresponding models also satisfy more properties, such as \(sR_{i}^{c}uv\) only if \(v\not =u\). However, (1)-(3) are enough to keep the logic intact, see [52] for details.

  33. 33.

    Note that here the maximal consistent sets are enough to build the canonical model due to the change of models, compared to canonical model for \(\mathbf{ELKv }^r\).

  34. 34.

    Due to \(\texttt {SYM}\), we only need \(\texttt {DISTKv}^b\) and \(\texttt {NECKv}^b\) w.r.t. the first argument.

  35. 35.

    A similar operator with propositional arguments was proposed by [51] in the setting of knowing whether, which can express that given the truth values of \(\varphi _1,\dots ,\varphi _n\) the agent i knows whether \(\varphi \).

  36. 36.

    See [95] for a more detailed survey.

  37. 37.

    See the collection of papers on the topic at philpaper edited by by John Bengson: http://philpapers.org/browse/knowledge-how.

  38. 38.

    Such examples motivated intellectualists to propose an account other than treating knowledge-how simply as ability. A notable approach proposed in [90] breaks down “knowing how to F” into: “There is a way such that I know it is a way to do F, and I entertain it in a practical mode of presentation.” Note that it essentially has the familiar shape \(\exists x\textit{K}\varphi (x)\), which also inspired the semi-formal treatment in [77].

  39. 39.

    See [56] for some existing solutions, e.g. by using epistemic STIT logic proposed by [18].

  40. 40.

    We can derive \({\textit{U}}p\rightarrow {\textit{U}}{\textit{U}}p\) and \(\lnot {\textit{U}}p\rightarrow U\lnot {\textit{U}}p\) [95].

  41. 41.

    For each maximal consistent set we build a canonical model [95].

  42. 42.

    Recall that \(\lnot \textit{Kv}_i c\) is true if there are two states which disagree on c.

  43. 43.

    In Hintikka’s terms, maybe it can be called the 1.5th generation of epistemic logics, since it is not as general as Hintikka’s idea of the second generation epistemic logics.

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Acknowledgements

The author acknowledges the support from the National Program for Special Support of Eminent Professionals and NSSF key projects 12&ZD119. The author is grateful to Hans van Ditmarsch for his very detailed comments on an early version of this paper. The author also thanks the anonymous reviewer who gave many constructive suggestions including the observation in footnote 14.

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    Yanjing Wang

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    Hans van Ditmarsch

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

    Gabriel Sandu

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Wang, Y. (2018). Beyond Knowing That: A New Generation of Epistemic Logics. 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_21

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