Abstract
It is tempting to think that multi premise closure creates a special class of paradoxes having to do with the accumulation of risks, and that these paradoxes could be escaped by rejecting the principle, while still retaining single premise closure. I argue that single premise deduction is also susceptible to risks. I show that what I take to be the strongest argument for rejecting multi premise closure is also an argument for rejecting single premise closure. Because of the symmetry between the principles, they come as a package: either both will have to be rejected or both will have to be revised.
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Notes
Cf. Williamson (2002: 117) and Hawthorne (2004) for similar formulations of closure. As should become clear below, my main points don’t depend on this particular formulation of closure.
See, in particular, Dretske (1970) and Nozick (1981). Similarly, to avoid a more recent sceptical paradox exploiting “lottery propositions”, one might maintain that while it is possible to know ordinary propositions, such as the proposition that one won’t be able to afford an expensive safari this year, it is not possible to know lottery propositions, such as the proposition that one’s ticket will lose the lottery. For sceptical arguments exploiting the intuition that subjects cannot know such propositions, see Vogel (1990) and Hawthorne (2004). But note that it is not straightforward that an account of why it is possible to know ordinary propositions but not denials of classical sceptical hypotheses will generalise into an account of why it is possible to know ordinary propositions but not the lottery propositions they entail. For instance, for problems having to do with Nozick’s tracking condition, see Hawthorne (2004: 10–14).
Cf. Kyburg’s (1970) Lottery Paradox. See also Makinson (1965) for the Paradox of the Preface. Note that denying that rational acceptance is closed under multi premise entailment won’t help escape the fact that it will still be rational to accept each proposition in a set of propositions entailing a contradiction.
Indeed, Kyburg’s (1970) own solution to the paradox is to reject the idea that rational acceptance is closed under conjunction, which is a special case of multi premise entailment. If rational acceptance is assumed to be a necessary condition for knowledge, then these considerations provide a rationale for a view that rejects MPC but retains SPC.
For instance, Hawthorne (2004: 47) seems to think that the problem for closure created by the accruement of risks is specific to MPC: ‘Granted, deductive inference from a single premise does not seem like a candidate for risky inference. If p entails q then q must be logically weaker or equivalent to p’.
Assume that P entails Q. Then, P is logically equivalent to the conjunction P & Q. If propositions are individuated in a coarse-grained manner, then these will be one and the same proposition. Any subject who knows P also knows P & Q. If knowledge distributes over conjunction, then any subject who knows P also knows Q, whether or not she satisfies the condition that she believes Q based on competent deduction from P.
Cf. DeRose & Warfield (1998: 13). More formally, using ‘Ks(P)’ for ‘s knows P’; ‘Bels(P)’ for ‘s believes P’, and ‘→’ for strict (logical) implication, the principle reads: For all P, Q, s, if Ks(P) and Ks(P → Q), then Ks(Q). Alternatively, the principle could be formulated as follows: For all P, Q, s (Ks(P) and Ks(P → Q)) → Ks(Q)). Similarly for all of the closure principle I discuss. I won’t fuss over this issue, but prefer to use the potentially weaker, first formulation.
See David & Warfield (forthcoming B).
Cf. DeRose & Warfield (1998: p. 23, note 14) for a principle otherwise similar but without ‘solely’, and David & Warfield (forthcoming A).
Of course, such cases don’t constitute a counterexample to Knowledge of Entailment Closure.
Cf. Carroll (1895).
DeRose (1999: 23, note 14) acknowledges a similar point: ‘Just as I think one can just barely know that P and just barely know that Q, yet just barely fall short of knowing the conjunction of P and Q––even when holding constant the standards for knowledge––because of the accumulation of doubt, so also can one know that P and know that P entails Q, while failing short of knowing that Q, even if one’s belief that Q is based on one’s knowledge of P and knowledge of the fact that P entails Q’.
Cf. Hawthorne (2004: 47–48).
Whether or not the actual world is indeterministic, epistemologists should be wary of allowing the anti-sceptical force of their theories to depend on such empirical matters.
Williamson (2000) defends this idea, which has come to be known as a safety condition on knowledge.
I say ‘normally’ because of certain instances of knowledge of singular, contingent a priori propositions that have low chances. See Hawthorne & Lasonen-Aarnio (forthcoming).
If one thinks that even deterministic worlds have objective chances other than 0 or 1, then the argument need not be stated by looking at subjects in indeterministic worlds.
Pritchard (2005) initially defends such a position, though he then opts for a more demanding version of safety.
This seems to be true generally. Assume that there is a very high chance that I fall climbing a certain route. Even after I have successfully completed the climb and there is no longer any chance of falling, it is still true that I am now lucky that I didn’t fall.
Cf. Hawthorne (2004: 47–50).
The two cases involve different types of defeat: in the MPC case, the subject has evidence for thinking that the proposition she has deduced is false, whereas in the SPC case she merely has evidence for thinking that the proposition she has deduced doesn’t count as knowledge, since she didn’t come to believe it by competent deduction from P. This is an example of overriding defeat, whereas the former is an example of undermining defeat.
Against, for instance, Brueckner (1998).
By ‘being approximately 175.9 cm tall’ I mean: my height is closer to 175.90 cm than it is to either 175.80 cm or 176.00 cm.
Those already convinced by my argument might want to skip this example and move to section 6.
Thanks to Stewart Cohen and Dorothy Edgington for bringing this line of objection to my attention.
Here I am setting aside reasons for questioning this that arise from individuating propositions in a very coarse-grained manner.
Note also that this fix would not solve the paradox for closure arising from Preface-type cases. For even if there is in fact no chance that a subject’s attempted deduction go wrong, the subject might have strong reasons for thinking that there is.
Thanks to Stewart Cohen and Tim Williamson for prompting me think about this issue.
Thanks to Stewart Cohen.
I am grateful to discussions and exchanges with Ville Aarnio, Frank Arntzenius, Stewart Cohen, Dorothy Edgington, John Hawthorne, Daniel Morgan, and above all Tim Williamson.
References
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