Abstract
Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable us to derive a version of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability (as discussed for instance by Bas van Fraassen) need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source.
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Notes
Cf. Moore (1993).
A word of caution. It is not completely obvious that the epistemic version of Moore’s claim preserves the most interesting traits of Moore’s original paradox without turning it into something different. I will comment very briefly on this point below.
Arguably, a plausible version of this principle may require that the ‘K’ operator be read as ‘someone, at some time, knows that’. However, we may well adapt the Knowability Principle so that it could be represented within models that deal with the knowledge of a single (ideally rational) agent. On the other hand, some attempts to solve the paradox would contend that (7) does not express the intuitive idea of the knowability of truth—say, because it fails to include an actuality operator (as in Edgington’s proposal; cf. Edgington (1985)) or because, as it stands, (7) hides the relevant quantifiers, which should be understood as modal indexicals (as in Kvanvig’s account; cf. Kvanvig (2006)). For the most part, in this paper I will not be concerned with possible ways to block the paradox, although I will have something to say about this in the last section.
To avoid quantification over propositions, one might treat (7) and (12) as sentence schemata and thus omit propositional quantifiers; the same applies to (13)–(15) below (thanks to an anonimous referee for pressing this point).
Cf. Kvanvig (2006).
Cresto (2012).
Williamson (2014).
Halpern (2003), chapter 7.
In Sect. 9 I will address some worries on whether this analysis captures what we intuitively demand from a second order probability.
The Reflection Principle discussed in this section is not to be confused with ‘epistemic reflexivity’, most often referred to as ‘epistemic transparency’ or ‘the KK Principle’ (for any proposition \(\varphi :\vdash K\varphi \rightarrow KK\varphi \)). It is interesting to explore how the two senses of reflection interact with each other; I will take up this topic on board explicitly in further sections.
To wit: Assume \(P([\varphi ] \cap P([\varphi ]) < 1) = 1\). Then \(P([\varphi ] \mid P([\varphi ]) < 1) P(P([\varphi ]) < 1) = 1\), which means that both factors are 1. As we will see, ‘\(P([\varphi ] \mid P([\varphi ]) < 1) = 1\)’ is a special case of [RP Failure], as will be stated below.
Actually, van Fraassen (1995) defends [RP] as a modest constraint on diachronic rationality, as opposed to full-fledged Bayesian conditionalization.
Pace van Fraassen, it can be argued that the range of possible opinions I may come to have about E at a later time does not stand for a vague probability, but for a range of possible sharp probability assignments (thanks to an anonimous referee for pressing this point).
This proposition has already been proven by Williamson (2014). By a ‘reflexive structure’ I mean any S with a reflexive R; recall that we need to assume the reflexivity of R anyway in order to account for the factivity of knowledge.
For this part of the proof I follow closely Proposition 1 in Williamson (2014).
Or whatever it is that we think Fitch’s paradox shows. Cf. the last paragraph of Sect. 1.
As a matter of fact we have obtained something stronger, to wit, we have obtained that the sentence stating that [RP Failure] has probability 1 is actually true in w.
Recall that \(P_{w}([\varphi ]) = P_{prior}([\varphi ] \mid R(w))\) is always well defined, due to the regularity of \(P_{prior}\).
This is one of the reasons why Kvanvig (2006) claims that there is no interesting analogy between Fitch’s and Moore’s paradox.
Paraconsistent analyses of Fitch’s paradox (such as Beall (2009)) argue that agents can indeed know epistemic Moorean statements; on similar grounds, a paraconsistent logician may well disagree with the claim that an agent just cannot, as a matter of logic, rationally believe a doxastic Moorean statement.
Actually, Williamson’s own diagnosis is that this particular phenomenon is not due to any putative vagueness related to the concept of knowledge. I agree; just to be clear, although I do think there is an interesting connection with vagueness here, it is not due to the vagueness of knowledge.
See Sect. 9.
Notice that here I am trying to mimic standard proofs of Fitch’s result. Such proofs typically start by assuming that a Moorean statement can be known. Likewise, here I start by assuming that a quasi-Moorean statement (i.e., ‘\(\models _{w} (\varphi \wedge \underline{[P([\varphi ]) < 1])}\)’) can receive maximum evidential probability.
Of course, we cannot demand knowledge of the future on rationality grounds. What can be demanded, however, is that we take active steps to be able to make accurate predictions about our future temporal slices.
Cf. Cresto (2012).
By definition, \({ KR}(w) = \{x \in W\): if xRy, then \(y \in R(w)\), for all y}. Hence \({ KR}(w) \subseteq R(w)\).
It might be contended that agents need not be aware of ‘\({ KR}(w)\)’—in which case they would not know which proposition they should conditionalize on (thanks to an anonymous referee for giving me the opportunity to clarify this point.). However, if there is a problem here, it is not exclusive of the enhanced framework, as similar considerations can be make for the standard setting; to wit, it might be contended that agents need not be aware of ‘R(w)’ either. There are at least two ways out, which relate to two very different interpretations of the formalism. On one hand, we can conceive of the framework as a tool for the theoretician (or the interpreter), who seeks to make knowledge and probability attributions from a third person point of view. She is the one who assesses, to the best of her knowledge, what the agent knows or ignores in each possible situation. On the other hand, we can think of the framework as ‘viewed’ from the inside, as it were, i.e., as structured from the first person perspective. In this case we can take ‘R(w)’ to refer to the information the agent has consciously gathered. ‘\({ KR}(w)\)’ could then capture the subset of R(w) which the agent takes to be the result of extremely reliable research methods, among other possibilities.
In a nutshell, by having the right sequence of knowledge operators we guarantee that statements with evidential probability 1 will be known by the agent.
The proposed solution shares a family resemblance with other attempts to solve Fitch’s paradox with the aid of typed languages, such as Linsky (1986), Linsky (2009), or Paseau (2008). However, the sequence of languages that I have in mind is more restrictive than usual hierarchic proposals, in the sense that ‘\(K^{i}\)’ is only meant to apply to sentences of the form ‘\(K^{i-1}\varphi \)’ or their negations; analogous restrictions apply to higher order probability statements. A rationale for this demand can be found in Cresto (2012).
As suggested by Kvanvig (2006).
Cf. Tennant (1997).
Thanks are due to Wlodek Rabinowicz for this suggestion.
References
Beall, J. (2009). Knowability and possible epistemic oddities. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 105–125). Oxford: Oxford University Press.
Cresto, E. (2012). A defense of temperate epistemic transparency. The Journal of Philosophical Logic, 41, 923–955.
Dummett, M. (2009). Fitch’s paradox of knowability. In J. Salerno (Ed.), New essays on the knowability paradox. Oxford: Oxford University Press.
Edgington, D. (1985). The paradox of knowability. Mind, 94, 557–568.
Edgington, D. (2010). Possible knowledge of unknown truth. Synthese, 173, 41–52.
Elster, J. (1984). Ulysses and the Sirens: Studies in rationality and irrationality. Cambridge: Cambridge University Press.
Fitch, F. (1963). A logical analysis of some value concepts. The Journal of Symbolic Logic, 28, 135–142. Reprinted in New essays on the knowability paradox, pp. 21–28, by J. Salerno, Ed., 2009, Oxford: Oxford University Press.
Halpern, J. (2003). Reasoning about uncertainty. Cambridge MA: MIT Press.
Hart, W., & McGinn, C. (1976). Knowledge and necessity. Journal of Philosophical Logic, 5, 205–208.
Kvanvig, J. (2006). The knowability paradox. Oxford: Oxford University Press.
Linsky, B. (1986). Factives, blindspots and some paradoxes. Analysis, 46, 10–15.
Linsky, B. (2009). Logical types in some arguments about knowability and belief. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 163–179). Oxford: Oxford University Press.
Moore, G. (1993). Moore’s paradox. In T. Baldwin (Ed.), G. E. Moore: Selected writings (pp. 207–212). London: Routledge.
Parfit, D. (1973). Later selves and moral principles. In A. Montefiore (Ed.), Philosophy and personal relations. London: Routledge and Kegan Paul.
Paseau, A. (2008). Fitch’s argument and typing knowledge. Notre Dame Journal of Formal Logic, 49, 155–176.
Rabinowicz, W., & Segerberg, K. (1994). Actual truth, possible knowledge. Topoi, 13, 101–115.
Salerno, J. (2009). Knowability noir: 1945–1963. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 29–48). Oxford: Oxford University Press.
Tennant, N. (1997). The tamning of the true. Oxford: Clarendon Press.
van Benthem, J. (2004). What one may come to know. Analysis, 64, 95–105.
van Fraassen, B. (1995). Belief and the problem of Ulysses and the Sirens. Philosophical Studies, 77, 7–37.
Williamson, T. (1982). Intuitionism disproved? Analysis, 47, 154–158.
Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.
Williamson, T. (2014). Very improbable knowing. Erkenntnis, 79, 971–999.
Acknowledgments
Previous versions of this paper have been presented at the II ALFAN Congress (The Latin American Analytic Philosophy Association, Buenos Aires, August 2012), at the Conference on Probability and Vagueness (The University of Tokyo, Tokyo, March 2013), and at the First Oxford-Buenos Aires Colloquium (The Argentine Society for Analytic Philosophy, Buenos Aires, April 2013). I am grateful to the participants for their questions and comments. In particular, I want to thank Richard Dietz, Paulo Faria, and two anonymous referees for Synthese. Special thanks are also due to Ramiro Caso for his invaluable help at the time of editing the final version of this document.
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Cresto, E. Lost in translation: unknowable propositions in probabilistic frameworks. Synthese 194, 3955–3977 (2017). https://doi.org/10.1007/s11229-015-0884-0
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DOI: https://doi.org/10.1007/s11229-015-0884-0