Suppose one adopts a closure principle for justification of the following sort:
CLOSURE: If S justifiably believes p and justifiably believes that p implies q, then S is justified in believing q.
Justifiably believing implies believing, in this formulation, but being justified in believing does not.
Suppose one is thinking of the lottery paradox, with n tickets. Suppose also that on your theory you are justified in believing that your ticket will lose. Your theory also endorses the principle that it is impossible to be justified in believing p and at the same time be justified in believing ~p. Finally, suppose you justifiably believe that some ticket will win.
Then consider the following conditional: if ticket 1 loses, then if ticket 2 loses, then . . . if ticket n loses, then it is not the case that some ticket wins. Your ticket is ticket #1, and your evidence is probabilistic. So you come to believe, justifiably, the conditional minus the antecedent about ticket #1. You apply this line of reasoning n times, and get a contradiction.
To save CLOSURE, one might say the following. Once you use your evidence to conclude that ticket 1 loses, the level of confirmation that any remaining ticket will lose is not n-1/n, but rather n-2/n-1. And once you use this evidence to conclude that ticket #2 will lose, your evidence about the remaining tickets gives a confirmation level of n-3/n-2. At some point, the value of n-m+1/n-m is low enough that you it fails to justify believing that the next ticket will lose, so the lottery example isn’t a counterexample to CLOSURE.
Note however that the order of inquiry here makes a difference to what you are justified in believing, if one takes this way out of the problem for CLOSURE. If the tickets are ordered in one way in the conditional with n embedded antecedents, you can come to be justified in believing that ticket #64 will lose. If the antecedents of the conditional are ordered in another way, by the time you get to the question of whether ticket #64 will lose, you’ve already gone far enough that the confirmation level doesn’t justify believing that ticket #64 will lose. So, if you like CLOSURE (or some close cousin of it), and if you like this response to the lottery counterexample to CLOSURE, then you must maintain that the order in which you apply a body of evidence to a set of propositions can make the difference between a claim being justified or not being justified. But how could changing the order of the antecedents in the imagined conditional have this effect? After all, it’s not as if one was unaware prior to drawing the inferences about what one’s body of evidence shows.
So: can one get out of the lottery counterexample without embracing the unusual claim that the order of inference here makes all the difference as to whether a particular belief is justified?