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Two Notions of Epistemic Risk

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Abstract

In ‘Single premise deduction and risk’ (2008) Maria Lasonen-Aarnio argues that there is a kind of epistemically threatening risk that can accumulate over the course of drawing single premise deductive inferences. As a result, we have a new reason for denying that knowledge is closed under single premise deduction—one that mirrors a familiar reason for denying that knowledge is closed under multiple premise deduction. This sentiment has more recently been echoed by others (see Schechter 2011). In this paper, I will argue that, although there is a kind of risk that can accumulate over the course of drawing single premise deductive inferences, it is importantly different to the kind of risk that multiple premise deductive inferences can introduce. Having distinguished these two kinds of risk, I shall offer some reasons for thinking that the kind associated with single premise deductions is, in fact, epistemically benign—it poses no threat, in and of itself, to the knowledge status of a belief. If this is right, then Lasonen-Aarnio’s argument against single premise closure is unsuccessful.

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Notes

  1. Additional qualifications are often placed upon such principles—such as the condition that one not lose one’s knowledge of the premises during the course of the inference (Hawthorne 2004, pp 33–34) or the condition that one not have any evidence defeating the claim that the deduction was competently performed (Schechter 2011). I think such qualifications are apposite, but I omit them here as they are not directly relevant for present purposes.

  2. At a minimum, what a probability function requires, for its arguments, are the elements of a lattice–such as sentences or propositions standing in appropriate entailment relations. The entailment relation serves to constrain the way in which numerical values can be assigned to the elements if the assignment is to count as a probability function. One such constraint is that an entailed element cannot receive a lower value than an element that entails it.

    Strictly speaking, Ent is only guaranteed for the very entailment relation that constrains the probability function in question. There is nothing preventing the stipulation that the arguments of a probability function stand in some further, ‘external’ entailment relation that need not constrain the permissible values of the function. It is not uncommon, for instance, to define a probability function over a set of sentences, relative to the sentential logical entailment relations between them, but to interpret the sentences as, in addition, standing in first-order logical consequence relations that are reflected in special, additional restrictions.

    But one can introduce an external entailment relation without supplying special restrictions. In this case, it would be quite possible for P to stand in an external entailment relation to Q, even though Pr(Q) < Pr(P). Garber (1983) exploits just this sort of possibility in modelling a kind of logical learning within a broadly Bayesian framework.

    Whether this kind of formalism could be useful for Lasonen-Aarnio’s purposes is unclear to me—though it’s not something that I will investigate in any detail here. The formalism is based upon a distinction between an entailment relation that plays the role of constraining the probability function in question and an external entailment relation to which the probability function is effectively ‘blind’. But Lasonen-Aarnio’s worries about single premise deductive inferences don’t seem to support any such distinction—if legitimate they would appear to apply across the board. Another concern is noted in the next footnote.

  3. If we are dealing with subjective or epistemic probability, then the use of an external entailment relation, to which the probability function is blind, can perhaps be motivated as a way of modelling certain sorts of logical ignorance. But if the probability in question is objective, then the use of an external entailment relation seems very difficult to motivate. It’s very difficult to understand how an objective probability function could be blind to any entailment relation. This is further reason for doubting that the kind of formalism described in the previous footnote is a good fit with Lasonen-Aarnio’s concerns about single premise deductive inference.

  4. If my coming to believe that Q were somehow negatively probabilistically relevant to its truth—that is, if Pr(~Q | BQ) > Pr(~Q)—then we could prove only a weaker result—namely Pr(F | BQ) ≤ Pr(~P | BQ). If, in addition, Pr(F | BQ) is sufficiently high as to preclude knowledge of Q, then so too must Pr(~P | BQ) be sufficiently high as to preclude knowledge of P. Since, by stipulation, Pr(~P) is not sufficiently high as to preclude knowledge that P, it must be that my coming to believe that Q serves in such a case to defeat my knowledge that P. Cases of this kind are not, I think, in the running to be counterexamples to single premise closure, as they would be excluded by one of the standard qualifications on the principle—namely, that one’s knowledge that P not be compromised over the course of the inference (see footnote 1). There is, perhaps, more to be said about such cases—but they are not the sorts of cases that Lasonen-Aarnio considers, so I won’t discuss them further here.

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Acknowledgments

I’m grateful to Philip Ebert for very detailed comments on an earlier draft of this paper. Thanks also to an audience at the University of Bristol in December 2010.

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Smith, M. Two Notions of Epistemic Risk. Erkenn 78, 1069–1079 (2013). https://doi.org/10.1007/s10670-012-9376-5

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