Probable But Still Unjustifiable
I take it as a truism that a subject may not justifiably believe a set of inconsistent propositions which she recognises to be inconsistent. My argument will take the form of a reductio beginning with the assumption, “S may justifiably believe that her ticket, t1, will lose”, and concluding with the negation of the aforementioned truism. Assuming that the first premise is the least plausible of all the premises in my argument, then my argument should establish that my first premise ought to be rejected. I would greatly appreciate any feedback concerning the structure, validity or soundness of my argument, or questions regarding any of my assumptions or steps. My reductio runs as follows:
(1) S may justifiably believe that her ticket, t1, will lose.Prima facie, (1)-(7) only shows that a subject is not justified in believing something she recognises to be inconsistent. Such cases fall under the umbrella of what Jonathan Sutton has dubbed, "known unknowns"—namely, instances in which the subject is aware that she does not have the knowledge in question. But this argument seems ineffective against certain types of "unknown unknowns"—i.e., cases in which the subject does not know that she does not know. Specifically, (1)-(7) does not seem to apply to cases in which the subject fails to recognise that a certain set of her beliefs are inconsistent. In such cases (6) would fail to apply. Thus, for all that has been shown, (1) may be true in cases in which the subject does not recognise her beliefs to be inconsistent. (Moreover, once we have dispatched with the tendentious Cartesian notion of the transparency of the mental, a subject's failure to recognise such an inconsistency in her beliefs becomes a live possibility.)
(2) If S may justifiably believe that t1 will lose, then she may also justifiably believe that t2 will lose, she may justifiably believe that t3 will lose ... she may justifiably believe that ticket tn will lose.
(3) S may justifiably believe that tickets t1, t2 ... tn will lose. [from (1) and (2)]
(4) S may justifiably believe that either t1 will not lose or t2 will not lose ... or tn will not lose.
(5) Propositions of the following form comprise an inconsistent set: (a) p1, p2 ... pn, either not-p1 or not-p2 ... or not-pn.
(6) S recognises that propositions of the following form comprise an inconsistent set: (a*) t1 will lose ... tn will lose, either t1 will not lose ... or tn will not lose.
(7) S may justifiably believe a set of inconsistent propositions that she recognises to be inconsistent. [from (3), (4), (5), and (6)]
At least two points should be noted in reply. For starters, we may widen the domain of known unknowns to include beliefs that a subject is in a position to know (say, via reflection alone). Since S is in a position to recognise that the propositions are inconsistent, assuming she is rationally competent, (6) still applies. Alternatively, we may simply note that the failure on S's part is a rational one, which (on even the narrowest J-internalist reading) would ex hypothesi render her belief unjustifiable. Given these considerations, the conclusion of the argument seems generalisable to all cases of belief based on evidence that renders the belief likely with a probability of <1. † See Dana Nelkin's paper “The Lottery Paradox, Knowledge and Rationality” for a discussion of the lottery paradox regarding knowledge and justifiably held belief.