Inexact knowledge and dynamic introspection


Cases of inexact observations have been used extensively in the recent literature on higher-order evidence and higher-order knowledge. I argue that the received understanding of inexact observations is mistaken. Although it is convenient to assume that such cases can be modeled statically, they should be analyzed as dynamic cases that involve change of knowledge. Consequently, the underlying logic should be dynamic epistemic logic, not its static counterpart. When reasoning about inexact knowledge, it is easy to confuse the initial situation, the observation process, and the result of the observation; I analyze the three separately. This dynamic approach has far reaching implications: Williamson’s influential argument against the KK principle loses its force, and new insights can be gained regarding synchronic and diachronic introspection principles.

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  1. 1.

    See Horowitz (2014), Elga (2013), Dorst (2020), Lasonen-Aarnio (2014, 2015), Roush (2017).

  2. 2.

    For recent defense of the KK thesis, see Das and Salow (2018), Dorst (2019), Goodman and Salow (2018), Greco (2014a, 2014b, 2015a, 2016)), Stalnaker (2015).

  3. 3.

    Williamson’s argument is arguably the most influential criticism of the KK principle in contemporary epistemology, but it is not the only one. My analysis does not address other arguments against the KK principle, such as Liu (2020) and the arguments reviewed in Greco (2015b).

  4. 4.

    This is true for both logical and probabilistic formulations of introspection. One can add evidential probabilities in the style of Williamson (2014) to the epistemic models I present here. It can then be shown that on such models the probabilistic diachronic reflection principle fails when inexact observations occur. This is analogous to Williamson’s (2014) demonstration that the probabilistic (synchronic) reflection principle fails on Williamson’s static models. I leave this out due to space limitations.

  5. 5.

    For these non-dynamic critiques, see Mott (1998), Brueckner and Fiocco (2002), Neta and Rohrbaugh (2004), Conee (2005), Dutant (2007), Greco (2014a) Halpern (2008), Bonnay and Egre (2008, 2009, 2011), Sharon and Spectre (2008) and Stalnaker (2015). Egre and Bonnay use dynamic epistemic logic, but not in order to model the act of observation. The only exception is Baltag and van-Benthem (2018), who take a dynamic approach different than mine.

  6. 6.

    Although not relevant to the argument presented here, it can be worthwhile to expand the dynamic analysis I develop in this paper for multi-agent epistemic logic, in order to reevaluate a recent attack on common knowledge based on inexact knowledge (Lederman 2017).

  7. 7.

    We assume that the implication in P2 is material.

  8. 8.

    The KK principle is sometimes presented in weaker formulations, e.g. with the language of being in a position to know instead of knowledge, or with some additional doxastic constraints. For discussions, see, e.g., Greco (2014b: pp. 173–174) and Stalnaker (2015: p. 28). These subtleties will not affect the arguments in this paper.

  9. 9.

    The closure of knowledge will not be the focus of this paper, and will be assumed throughout as an idealization, even if its failure can break Williamson’s derivation. See (Williamson 2000: p. 117) for a defense of closure in this context. A different approach would be to assume closure for the concept of being in a position to know, but see Yli-Vakkuri (2020) for complications.

  10. 10.

    If P2 is based on the safety condition of knowledge, then one way of blocking Williamson’s argument is by rejecting safety. See Neta and Rohrbaugh (2004) for this direction. But even if the safety condition is false in general, it seems like a very reasonable constraint in cases of inexact observations (Williamson 2008).

  11. 11.

    I am not claiming that the model in Fig. 1 is the intended model for the unmarked clock example, rather that it is a good enough simplified model. The model introduces many assumptions that go beyond P1, P2 and P3. Further complications and considerations might be added, (see Williamson 2014)—but this is a good picture to start with.

  12. 12.

    A similar point is made by Sharon and Spectre (2008).

  13. 13.

    Bonnay and Egre (2011) were first to consider the tools of DEL in order to analyze inexact knowledge. Their approach of using a non-standard semantics for the static base epistemic logic is very different from the one I develop here, as they do not use updates to model inexact observations. The very recent work of Baltag and van-Benthem (2018) uses what I call exact updates to analyze Williamson’s inexact knowledge. The latter approach is quite different than mine as it does not try to offer an alternative or an explanation to the margin-for-error principle.

  14. 14.

    Although it is possible, in this paper we will not develop the analysis of the DEL update operator as expressing a non-material conditional. See Icard and Holliday (2017) for an analysis of the relationship between the update operator of DEL and indicative conditionals. Even though the main function of the [e] operator is to describe the effect of observing e on the agent’s epistemic state, the formal syntax of DEL allows for expressions like [e]p (where p is an atomic, non-epistemic formula). In those cases, there is clearly no dependency between observing e and p, and the conditional reading of the operator is more appropriate.

  15. 15.

    The debate whether all evidence is veridical is not at issue here. Even if there is false evidence, such evidence does not generate knowledge (at most it can generate false beliefs). Our focus is on knowledge update, so we can safely restrict our attention to veridical evidence and veridical update operators. We care to model knowledge update and not belief revision given any kind of (possibly false) piece of information. That being said, it is technically possible to extend the formal apparatus to accommodate such cases.

  16. 16.

    Alternatively, one could use the framework of epistemic temporal logic (instead of dynamic epistemic logic) to represent the knowledge stages before and after the update with two distinct knowledge operators: \(K_0\) and \(K_1\). See van-Benthem et al. (2009) for details about the relationship between the two frameworks.

  17. 17.

    For a more systematic discussion about dynamic introspection, externalism, and skepticism see Cohen (2020a).

  18. 18.

    It is thus also an independent question whether the Tree example is compatible with the KK principle. I leave this question aside. My point is that the example is clearly not compatible with Dynamic Introspection.

  19. 19.

    Since the initial the model is an S5 model, positive and negative introspection hold for any \(\varphi \), not just for the \(p_i\)’s. Since what we care about is Ann’s knowledge of the clock (i.e. about \(p_i\)), this idealization seems harmless.

  20. 20.

    See Halpern (2008) for different approach that uses two relations in order to analyze related cases of perpetual vagueness. Dutant (2007) builds on Halpern’s approach and provides an infallibilist critique of Williamson’s margin-for-error argument.

  21. 21.

    We could complicate the structure of the P relation to allow for varying margin-for-error the same way Williamson is varying his R relation in (Williamson 2014).

  22. 22.

    In Cohen (2020b), I develop a more general logic for inexact and opaque updates, which can also be used to model the example analyzed here.

  23. 23.

    Note that both standard DEL and standard Bayesian update lack this property. Such updates are insensitive to the world of evaluation.

  24. 24.

    One can also construct updated models which are not S5 models, rather only S4. The 5 axioms does not play any crucial rule in my argument.

  25. 25.

    Nothing hinges on this choice of values. For whatever value of margin-for-error we choose, we cannot iterate the Williamsonian reasoning pattern from Sect. 1.1.

  26. 26.

    The models are simplified because the P relation is not drawn.

  27. 27.

    In the terminology of dynamic epistemic logic, the announcement made by the optometrist is not successful, because it is not known after the announcement (see Baltag and Renne 2016). This is an indication that the content of the announcement is context sensitive, in the sense used within dynamic epistemic logic (see, e.g. Holliday 2018).

  28. 28.

    My response essentially appeals to a quantifier shift fallacy. To get a contradictory derivation, the objector needs to assume that there is one type of margin-for-error principle for every way of gaining inexact knowledge. I argue that for every way of gaining inexact knowledge, there is some type of margin-for-error principle. Since I see no reason to assume that the different margin-for-errors have the same structure, I don’t see how one type of margin-for-error can be repeatedly applied to obtain a contradiction.

  29. 29.

    See Dokic and Égré (2009:19) for a response to the ad-hoc accusation.

  30. 30.

    E.g. Elga (2013), Salow and Ahmed (2017).

  31. 31.

    A similar point holds for Bayesian epistemology, in which it is assumed that the posterior epistemic state is transparent to the agent prior to the update (as a prior conditional state).


  1. Baltag, A., & Renne, B. (2016). Dynamic epistemic logic. In E. N. Zalta (Ed.) The Stanford encyclopedia of philosophy (Winter 2016 Edition).

  2. Baltag, A., & van-Benthem, J. (2018). Some thoughts on the logic of imprecise observations. Manuscript.

  3. Bonnay, D., & Egre, P. (2008). Margins for error in context. In M. Garcia-Carpintero & M. Kölbel (Eds.), Relative truth (pp. 103–127). Oxford: Oxford University Press.

    Google Scholar 

  4. Bonnay, D., & Egre, P. (2009). Inexact Knowledge with Introspection. Journal of Philosophical Logic, 38(2), 179–228.

    Article  Google Scholar 

  5. Bonnay, D., & Egre, P. (2011). Knowing one’s limits—An analysis in centered dynamic epistemic logic. In P. Girard, M. Marion, & O. Roy (Eds.), Dynamic formal epistemology (pp. 103–126). Berlin: Springer.

    Google Scholar 

  6. Brueckner, A., & Fiocco, M. O. (2002). Williamson’s anti-luminosity argument. Philosophical Studies, 110, 285–293.

    Article  Google Scholar 

  7. Christensen, D. (2010). Higher-order evidence. Philosophy and Phenomenological Research, 81(1), 185–215.

    Article  Google Scholar 

  8. Cohen, M. (2020). The problem of perception and the no-miracles principle. Synthese,.

    Article  Google Scholar 

  9. Cohen, M. (2020). Opaque updates. Journal of Philosophical Logic,.

    Article  Google Scholar 

  10. Conee, E. (2005). The comforts of home. Philosophy and Phenomenological Research, 70, 444–451.

    Article  Google Scholar 

  11. Das, N., & Salow, B. (2018). Transparency and the KK Principle. Noûs, 52(1), 3–23.

    Article  Google Scholar 

  12. Dokic, J., & Égré, P. (2009). Margin for error and the transparency of knowledge. Synthese, 166, 1–20.

    Article  Google Scholar 

  13. Dorst, K. (2019). Abominable KK failures. Mind, 128(512), 1227–1259.

    Article  Google Scholar 

  14. Dorst, K. (2020). Evidence: A guide for the uncertain. Philosophy and Phenomenological Research, 100(3), 586–632.

    Article  Google Scholar 

  15. Dutant, J. (2007). Inexact knowledge, margin for error and positive introspection. In Proceedings of Tark XI.

  16. Elga, A. (2013). The puzzle of the unmarked clock and the new rational reflection principle. Philosophical Studies, 164(1), 127–139.

    Article  Google Scholar 

  17. Goodman, J., & Salow, B. (2018). Taking a chance on KK. Philosophical Studies, 175(1), 183–196.

    Article  Google Scholar 

  18. Greco, D. (2014a). Could KK be ok? Journal of Philosophy, 111(4), 169–197.

    Article  Google Scholar 

  19. Greco, D. (2014b). Iteration and fragmentation. Philosophy and Phenomenological Research, 88(1), 656–673.

    Google Scholar 

  20. Greco, D. (2015a). Iteration principles in epistemology I: Arguments for. Philosophy Compass, 10(11), 754–764.

    Article  Google Scholar 

  21. Greco, D. (2015b). Iteration principles in epistemology II: Arguments against. Philosophy Compass, 10(11), 765–771.

    Article  Google Scholar 

  22. Greco, D. (2016). Safety, explanation, iteration. Philosophical Issues, 26(1), 187–208.

    Article  Google Scholar 

  23. Halpern, J. Y. (2008). Intransitivity and vagueness. Review of Symbolic Logic, 1(4), 530–547.

    Article  Google Scholar 

  24. Hawthorne, J., & Magidor, O. (2009). Assertion, context, and epistemic accessibility. Mind, 118(470), 377–397.

    Article  Google Scholar 

  25. Holguín, B. (2019). Indicative conditionals and iterative epistemology. Manuscript.

  26. Holliday, W. (2018). Knowledge, time, and paradox: Introducing sequential epistemic logic. In Jaakko Hintikka on knowledge and game-theoretical semantics. Springer.

  27. Horowitz, S. (2014). Epistemic akrasia. Noûs, 48(4), 718–744.

    Article  Google Scholar 

  28. Icard, T., & Holliday, W. (2017). Indicative conditionals and dynamics epistemic logic. In Proceedings of the sixteenth conference on theoretical aspects of rationality and knowledge (TARK XVI).

  29. Lasonen-Aarnio, M. (2014). Higher-order evidence and the limits of defeat. Philosophy and Phenomenological Research, 8(2), 314–345.

    Article  Google Scholar 

  30. Lasonen-Aarnio, M. (2015). New rational reflection and internalism about rationality. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 5, pp. 145–171). Oxford: Oxford University Press.

    Google Scholar 

  31. Lederman, H. (2017). Uncommon knowledge. Mind, 127(508), 1069–1105.

    Google Scholar 

  32. Liu, S. (2020). (Un)knowability and knowledge iteration. Analysis, 80(3), 474–486.

    Article  Google Scholar 

  33. Mott, P. (1998). Margins for error and the sorites paradox. Philosophical Quarterly, 48(193), 494–504.

    Article  Google Scholar 

  34. Neta, R., & Rohrbaugh, G. (2004). Luminosity and the safety of knowledge. Pacific Philosophical Quarterly, 85(4), 396–406.

    Article  Google Scholar 

  35. Roush, S. (2017). Epistemic self-doubt. Stanford Encyclopedia of Philosophy.

  36. Salow, B., & Ahmed, A. (2017). Don’t look now. British Journal for the Philosophy of Science, 70(2), 327–350.

    Google Scholar 

  37. Sharon, A., & Spectre, L. (2008). Mr. Magoo’s mistake. Philosophical Studies, 139(2), 289–306.

    Article  Google Scholar 

  38. Stalnaker, R. (2015). Luminosity and the KK thesis. In S. C. Goldberg (Ed.), Externalism, self-knowledge, and skepticism: New essays. Cambridge: Cambridge University Press.

    Google Scholar 

  39. van-Benthem, J., Gerbrandy, J., Hoshi, T., & Pacuit, E. (2009). Merging frameworks for interaction. Journal of Philosophical Logic, 38(5), 491–526.

    Article  Google Scholar 

  40. Wang, Y., & Cao, Q. (2013). On axiomatizations of public announcement logic. Synthese, 190(S1), 103–134.

    Article  Google Scholar 

  41. Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.

    Google Scholar 

  42. Williamson, T. (2008). Why epistemology cannot be operationalized. In Q. Smith (Ed.), Epistemology: New essays. Oxford: Oxford University Press.

    Google Scholar 

  43. Williamson, T. (2014). Very improbable knowing. Erkenntnis, 79(5), 971–999.

    Article  Google Scholar 

  44. Yli-Vakkuri, J., & Hawthorne, J. (2020). Being in a position to know. Manuscript.

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Versions of this paper were presented in the 2019 Glasgow graduate conference in epistemology and mind, the 2019 Tübingen Masterclass in Theoretical Philosophy with Timothy Williamson, and the Knowledge and its Limits at 20 conference in the University of Geneva. I thank the audiences and commentators for helpful discussion. In addition, I would like to thank Johan van Benthem, Ray Briggs, and Krista Lawlor for the guidance and the many comments they offered in the process of writing this paper. I would also like to thank the anonymous referees who provided detailed comments on the current and earlier versions of this paper.

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Appendix: Deriving an absurdity in the unmarked clock example with dynamic introspection

Appendix: Deriving an absurdity in the unmarked clock example with dynamic introspection

Recall that e abbreviates \(p_{16} \vee p_{17} \vee p_{18}\) and nothing else. We assume the principles:


We further assume that both K and \([\varphi ]\) obey:


First we show that update-knowledge closure holds:

Lemma 1.1

\(\vdash K [\varphi ]K( \alpha \rightarrow \beta ) \wedge K [\varphi ]K \alpha \rightarrow K [\varphi ] K \beta \)



\(\square \)

Now, we show how to derive the problematic conclusion \(K[e]\lnot p_{16}\) (in line 10) from our assumptions. Line 6. establishes what I called earlier the auxiliary assumption.:


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Cohen, M. Inexact knowledge and dynamic introspection. Synthese (2021).

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  • The KK principle
  • Margin-for-error
  • Inexact knowledge
  • Introspection principles
  • Externalism
  • Safety
  • Dynamic epistemic logic