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In defence of single-premise closure


It’s often thought that the phenomenon of risk aggregation poses a problem for multi-premise closure but not for single-premise closure (either with respect to knowledge or with respect to justified belief). But recently, Lasonen-Aarnio and Schechter have challenged this thought. Lasonen-Aarnio argues that, insofar as risk aggregation poses a problem for multi-premise closure, it poses a similar problem for single-premise closure. For she thinks that, there being such a thing as deductive risk, risk may aggregate over a single premise and the deduction itself. Schechter argues that single-premise closure succumbs to risk aggregation outright. For he thinks that there could be a long sequence of competent single-premise deductions such that, even though we are justified in believing the initial premise of the sequence, intutively, we are not justified in believing the final conclusion. This intuition, Schechter thinks, vitiates single-premise closure. In this paper, I defend single-premise closure against the arguments offered by Lasonen-Aarnio and Schechter.

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  1. The lottery paradox and the preface paradox are often thought to pose problems for MPC (or versions of MPC) while leaving SPC (or versions of SPC) intact. For more on the lottery paradox, see Kyburg (1970); for an attempt to solve it, see Nelkin (2000). For more on the preface paradox, see Makinson (1965); for an attempt to solve it, see Ryan (1991).

  2. Following Lasonen-Aarnio (2008) and Schechter (2013), this paper focuses on the problem that the phenomenon of risk aggregation is supposed to pose for SPC. One might wish to deny SPC (or versions of it) for other reasons—for instance, to try to solve the problem of external world scepticism. See, for example, Dretske (1970) and Nozick (1981).

  3. In discussing Lasonen-Aarnio’s arguments, I’ll focus on objective chance—as she does—but Lasonen-Aarnio (2008) holds that similar points can be made with respect to other notions of probability (159).

  4. Williamson (2009) has challenged the claim that knowledge that P is incompatible with there being a low chance that P is true. But I’ll assume that the claim is true. This assumption is unproblematic for the purposes of this paper—if anything, it makes SPC harder to defend.

  5. Lasonen-Aarnio (2008) is right that ‘competence doesn’t require infallibility’ (159).

  6. My distinction between the two senses in which there might be a chance of failure bears some similarity to Smith’s (2013) distinction between an unconditional error risk and a conditional error risk. Suppose a subject S competently deduces Q from P. According to Smith (2013), unconditional error risk has to do with the probability that S’s reasoning process will generate a false belief whereas conditional error risk has to do with the probability that such a process will generate a false belief given that it generates a belief that Q (1073). Further, Smith argues that the former kind of risk is epistemically benign, whereas the latter risk is no greater than the risk of P being false (ibid.). I’m hugely sympathetic to Smith’s defence of SPC. But it is incomplete in certain respects. First, Smith (2013) argues for his claim that unconditional error risk is benign by way of analogy with other cases that do not involve SPC (1075–8). He also thinks that his argument is ‘somewhat speculative’ (1076). But while I’ll consider an analogy too, I’ll offer an example in which it’s intuitively clear that a subject may gain knowledge via a competent single-premise deduction even though the chance of her performing an incompetent deduction and ending up with a false belief is very high. This gives us direct evidence for thinking that the chance of a subject performing an incompetent deduction—or what Smith calls ‘unconditional error risk’—is benign. Second, Smith thinks that his ‘objections to Lasonen-Aarnio’s argument [against SPC] apply equally to the argument that Schechter mounts’ (1072). But he does not address Schechter’s (2013) intuition that, after we have performed a long sequence of competent single-premise deductions, we should think it likely that we have made a mistake somewhere in our chain of reasoning. Schechter takes this intuition to support his claim that SPC (or a version of it) is false. To defend SPC thoroughly, we need to resist the intuition or show that, pace Schechter, it does not undermine SPC. I adopt the latter strategy in Sect. 3.

  7. Perhaps, even though she should be reasoning by modus ponens, she chooses to reason by some incorrect rule. Or perhaps she correctly chooses to reason by modus ponens but ends up performing such reasoning poorly. Schechter (2013) calls the distinction between applying an incorrect rule and misapplying a correct rule the ‘competence/performance distinction’ (431).

  8. This intuition seems robust whether the chance of the chip being activated is 90% or as high as 99.9%.

  9. An anonymous reviewer suggested that, perhaps, the former is not akin to the latter—perhaps there is a sense in which the chip is external Beth’s mind, and when the chip is activated, it is not really her who is doing the reasoning. If that’s right, then one might grant that in the example above, Beth does come to know Q by competently deducing it from P—but the chance of her performing an incompetent deduction was never 90% to begin with, since the chip’s activating 90% of the time does not affect the reliability of Beth’s reasoning.

    Even if the chip is external to Beth’s mind, it seems to me that when it activates and causes Beth’s epistemic capacities to misfire, it is still Beth who is doing the reasoning. After all, our glasses and hearing aids are external to us, but whether they improve or impair our perceptual abilities, we are still the ones who are doing the perceiving. Now, perhaps things are different when long-term and widespread changes in our ability to reason occur. And perhaps, talk of an implant in the brain tends to bring up images of brainwashing. Let us thus modify the example above slightly. Suppose there’s a pill that has a 90% chance of making Beth perform a deduction incompetently when she wakes up, and suppose that the effects of the pill are one-off. Suppose also that Beth accidentally takes the pill and is unaware that she has done so. As luck would have it, the pill doesn’t work, and when Beth wakes up, she competently deduces Q from P, as she would have done normally. Intuitively, the case is one in which Beth may come to know Q by competently deducing it from P, despite the high chance of her performing an incompetent deduction. Granted, pills and drugs are also external to our mind. But we are prepared to think that some of these substances boost, while others impair, our cognitive performances without being inclined to think that they turn us into another person—at least given that the effects are short-term and local.

  10. To be more careful, we might hold that the employee’s knowing that he’ll be fired is compatible with the claim that he is lucky to form the belief that he’ll be fired, where the belief is properly based on the evidence in question. For suppose he ignores the evidence and forms a belief at random. If, by luck, he happens to form the belief that he’ll be fired, then since this belief isn’t properly based on the relevant evidence, it does not amount to knowledge.

  11. Smith (2013) also observes that when the chance of performing an incompetent deduction is high, but a person goes on to perform a competent deduction, the person is doxastically epistemically lucky.

  12. Lasonen-Aarnio (2008) discusses a version of the argument briefly (171). I focus on Schechter’s version since he discusses the argument in greater detail than does Lasonen-Aarnio.

  13. Cf. Burge (1995). To be clear, I’m not claiming that successful deductions must appeal to premises about one’s memory. As Burge writes, ‘[i]t is one thing to rely on memory in a demonstration, and another to use premises about memory’; and while ‘[a]ny reasoning in time must rely on memory ...not all reasoning must use premises about memory or the past’. (ibid., 276–7)

  14. One might think that, even as Sue is performing a single-premise deduction competently, and the deduction is fresh in her mind, her having reason to believe that she is fallible means that she has reason to believe that there’s a possibility that she’ll make a mistake this time, which in turn means that she should entertain some doubt as to the competence of her deduction. One might then take this to be a problem for SPCJ*—one might take such doubt to show that Sue can have enough justification for believing a premise without having enough justification for believing the relevant conclusion. Note, however, that the foregoing argument is crucially different from the long sequence argument. The latter depends largely on the intuition that, at the end of a long sequence of competent single-premise deductions, one is not justified in believing the final conclusion. The failure of SPCJ* is then taken to be the best explanation of this intuition. The current argument, however, takes aim at SPCJ* directly. Now, a discussion of the argument’s merits will take us beyond the aim of this paper, which is to defend SPC and SPCJ* against specific arguments from Lasonen-Aarnio and Schechter. But it is at least worth noting that the relevant intuition with respect to the long sequence argument is reasonably strong, whereas it is less obvious whether the following claim is true: Even while one is performing a deduction competently, merely having reason to believe that one is fallible should deprive one of some justification for believing the conclusion of the deduction. In fact, one might think that such a claim is precisely what the long sequence argument aims to support.

  15. What if, instead of considering a subject with perfect memory, we consider human beings with better or worse memories? (Thanks to an anonymous reviewer for raising this question.) It seems that no matter who we pick, we’ll have the intuition that if a relevant sequence of competent deductions is long enough, he or she she will not be justified in believing the final conclusion. Does this show that an appeal to memory does not help explain the relevant intuition after all? I don’t think so. For each human being with a less than perfect memory, there will be a sequence of competent deductions that is long enough that he or she will not be justified in believing its conclusion. But it seems that, all things being equal, the worse (or better) a person’s memory, the shorter (or longer) such a sequence will need to be.

  16. For more discussion of SPC (that has a different focus from this paper), see Hawthorne (2004), Wedgwood (2012), and Williamson (2009).


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Thanks to an anonymous reviewer and to the audience at the 2015 Australasian Association of Philosophy Conference for valuable comments. I’ve also benefitted from discussion with Michael Pelczar, Ben Blumson, and the honours students in my epistemology class in 2015. Thanks, in particular, to Bernadette Chin for feedback on a draft of the paper.

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Correspondence to Weng Hong Tang.

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Tang, W.H. In defence of single-premise closure. Philos Stud 175, 1887–1900 (2018).

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  • Single-premise closure
  • Multi-premise closure
  • Deduction
  • Knowledge
  • Justification
  • Reasoning