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Introduction to Formal Philosophy pp 351–369Cite as

Epistemic Logic and Epistemology

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Abstract

This chapter provides a brief introduction to propositional epistemic logic and its applications to epistemology. No previous exposure to epistemic logic is assumed. Epistemic-logical topics discussed include the language and semantics of basic epistemic logic, multi-agent epistemic logic, combined epistemic-doxastic logic, and a glimpse of dynamic epistemic logic. Epistemological topics discussed include Moore-paradoxical phenomena, the surprise exam paradox, logical omniscience and epistemic closure, formalized theories of knowledge, debates about higher-order knowledge, and issues of knowability raised by Fitch’s paradox. The references and recommended readings provide gateways for further exploration.

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Notes

  1. 1.

    To reduce clutter, I will not put quote marks around symbols and sentences of the formal language, trusting that no confusion will arise.

  2. 2.

    In our formal models, “scenarios” will be unstructured points at which atomic sentences can be true or false. We are not committed to thinking of them as Lewisian possible worlds.

  3. 3.

    Hintikka presented his original formal framework somewhat differently. Such details aside, we use the now standard relational structure semantics for normal modal logics.

  4. 4.

    It is important to draw a distinction between epistemic accessibility and other notions of indistinguishability. Suppose that we replace R K by a binary relation E on W, where our intuitive interpretation is that wEv holds “iff the subject’s perceptual experience and memory” in scenario v “exactly match his perceptual experience and memory” in scenario w [28, 553]. Suppose we were to then define the truth of Kφ in w as in (MC), but with R K replaced by E. In other words, the agent knows φ in w iff φ is true in all scenarios that are experientially indistinguishable from w for the agent. (Of course, we could just as well reinterpret R K in this way, without the new E notation.) There are two conceptual differences between the picture with E and the one with R K. First, given the version of (MC) with E, the epistemic model with E does not simply represent the content of one’s knowledge; rather, it commits us to a particular view of the conditions under which an agent has knowledge, specified in terms of perceptual experience and memory. Second, given our interpretation of E, it is plausible that E has certain properties, such as symmetry (wEv iff vEw), which are questionable as properties of R K (see Sect. 17.7). Since the properties of the relation determine the valid principles for the knowledge operator K (as explained in Sects. 17.3 and 17.7), we must be clear about which interpretation of the relation we adopt: epistemic accessibility, experiential indistinguishability, or something else. Here we adopt the accessibility interpretation.

    Finally, note that while one may read wR K v as “for all the agent knows in w, scenario v might be the scenario he is in,” one should not read wR K v as “in w, the agent considers scenario v possible,” where the latter suggest a subjective psychological notion. The spymaster may not subjectively consider it possible that his spy, whom he has regarded for years as his most trusted agent, has defected. It obviously does not follow that he knows that his spy has not defected, as it would according to the subjective reading of R K together with (MC).

  5. 5.

    For any theory of knowledge that can be stated in terms of R K and (MC), the rule RK of Sect. 17.3 must be sound. Therefore, theories for which RK is not sound, such as those discussed in Sect. 17.6, cannot be stated in this way. Given a formalization of such a theory, one can always define a relation R K on scenarios such that wR K v holds iff everything the agent knows in w according to the formalization is true in v. It is immediate from this definition that if φ is not true in some v such that wR K v, then the agent does not know φ in w. However, it is not immediate that if φ is true in all v such that wR K v, then the agent knows φ in w. It is the right-to-left direction of (MC) that is not neutral with respect to all theories of knowledge.

  6. 6.

    Throughout we use the nomenclature of modal logic for schemas and rules.

  7. 7.

    For additional ways of understanding idealization in epistemic logic, see [45].

  8. 8.

    A similar formalization applies to the designated student paradox [33, 317], a genuinely multi-agent version of the surprise exam paradox.

  9. 9.

    We skip steps for the sake of space. E.g., we obtain (4) by applying RK 2 to the tautology ((e 1 ∨ e 2) ∧¬e 1) → e 2. We then obtain (5) directly from (4) using the special case of RK 1 where n = 0 in the premise (φ 1 ∧⋯ ∧ φ n) → ψ, known as Necessitation: if ψ is a theorem, so is K 1 ψ. It is important to remember that RK i can only be applied to theorems of the logic, not to sentences that we have derived using undischarged assumptions like (A), (B), and (C). To be careful, we should keep track of the undischarged assumptions at each point in the derivation, but this is left to the reader as an exercise. Clearly we have not derived (8) as a theorem of the logic, since the assumptions (A), (B), and (C) are still undischarged. What we have derived as a theorem of the logic is the sentence abbreviated by ((A) ∧ (B) ∧ (C)) → (8).

  10. 10.

    We can derive (8) from (A), (B), and (C) in a doxastic logic (see Sect. 17.5) without the T axiom, substituting B i for K i. Thus, insofar as B 1(e 1 ∧¬B 1 e 1) is also problematic for an ideal agent, the surprise exam paradox poses a problem about belief as well as knowledge.

  11. 11.

    In fact, the sentence ¬B(p ∧¬Bp) precisely corresponds to a condition on R B, namely that for every w, there is a v such that wR B v and for every u, vR B u implies wR B u.

  12. 12.

    Note that the K axiom is derivable from the RK rule with the tautology (φ ∧ (φ → ψ)) → ψ.

  13. 13.

    Assume ¬φ is true in w, so φ is not true in w. Consider some v with wR K v. By symmetry, vR K w. Then since φ is not true in w, Kφ is not true in v by (MC), so ¬Kφ is true in v. Since v was arbitrary, ¬Kφ is true in all v such that wR K v, so K¬Kφ is true in w by (MC).

  14. 14.

    Note that if we reject the requirement that R K be symmetric in every epistemic model, we can still allow models in which R K is symmetric (such as the model in Fig. 17.1), when this is appropriate to model an agent’s knowledge. The same applies for other properties.

  15. 15.

    Assuming Kφ → Bφ, D, and 5, the principle BKφ → Kφ is derivable (see [16, §2.4]). Given the same assumptions, if an agent is a “stickler” [30, 246] who believes something only if she believes that she knows it (Bφ → BKφ), then one can even derive Bφ ↔ Kφ (see [26] and [17]). Given Kφ → Bφ, D, B, and Bφ → BKφ, one can still derive Bφ → φ (see [17, 485]).

  16. 16.

    To see that 4 is valid over the class of transitive models, assume that Kφ is true in w in such a model, so by (MC), φ is true in all v such that wR K v. Consider some u with wR K u. Toward proving that Kφ is true in u, consider some v with uR K v. By transitivity, wR K u and uR K v implies wR K v. Hence by our initial assumption, φ is true in v. Since v was arbitrary, φ is true in all v such that uR K v, so Kφ is true in u by (MC). Finally, since u was arbitrary, Kφ is true in all u such that wR K u, so KKφ is true in w by (MC).

  17. 17.

    Assuming D, 4, and M for B, we have: (i) B(p ∧¬Bp), assumption for reductio; (ii) Bp ∧ B¬Bp, from (i) by M for B and PL; (iii) BBp ∧ B¬Bp, from (ii) by 4 for B and PL; (iv) ¬B¬Bp ∧ B¬Bp, from (iii) by D and PL; (v) ¬B(p ∧¬Bp), from (i)-(iv) by PL.

  18. 18.

    Stalnaker shows that the epistemic logic of the defeasibility analysis as formalized is S4.3, which is intermediate in strength between Lenzen’s lower and upper bounds of S4.2 and S4.4.

  19. 19.

    Note that the universal quantifiers in (9), (10), (15), and (21) are not part of our formal language. They are merely shorthand to indicate a schema of sentences.

  20. 20.

    For discussion of such “unsuccessful” announcements in the context of the surprise exam paradox, see [15].

  21. 21.

    See [22, Ch. 6], [16, §5], and [1] for discussion of quantified epistemic logic. Hintikka [23] has proposed a “second generation” epistemic logic, based on independence-friendly first-order logic, aimed at solving the difficulties and fulfilling the epistemological promises of quantified epistemic logic.

  22. 22.

    For helpful discussion and comments, I wish to thank Johan van Benthem, Tomohiro Hoshi, Thomas Icard, Alex Kocurek, Eric Pacuit, John Perry, Igor Sedlár, Justin Vlasits, and the students in my Fall 2012 seminar on Epistemic Logic and Epistemology at UC Berkeley.

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  1. Department of Philosophy, University of California, Berkeley, CA, USA

    Wesley H. Holliday

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  1. Division of Philosophy, Royal Institute of Technology (KTH), Stockholm, Sweden

    Sven Ove Hansson

  2. Center for Information and Bubble Studies, University of Copenhagen, Copenhagen, Denmark

    Vincent F. Hendricks

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Holliday, W.H. (2018). Epistemic Logic and Epistemology. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_17

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