The infinitist insists that a given belief is justified iff that belief is supported by an infinite set of non‐repeating justifiers. One standard complaint about this proposal is that it puts justification out of our reach because – given the kind of cognitively limited creatures we are – we simply don't have an infinite set of justifiers available to support our beliefs. If this is right, infinitism doesn't provide us with a non‐skeptical response to the regress problem. Here's the argument in outline – that is, the Argument from Finite Minds:
Argument from Finite Minds
P1. You have a finite number of beliefs.
P2. Nothing could be a justifier that supports your beliefs at any given time unless it is itself a belief.
C1. You have a finite number of justifiers that support your beliefs at any given time.
P3. Infinitism tells us that a belief is justified only if supported by an infinite set of non‐repeating justifiers.
C2. Infinitism implies that none of your beliefs could be justified.
An initial line of response the infinitist could offer here would be to contest the relevance of P1 to the truth of infinitism. As this line of thinking goes, infinitism is a thesis about epistemic justification “in and of itself” and not merely about justification that's accessible to the kinds of minds that humans happen to have.17 Accordingly, what infinitism says about epistemic justification is not beholden to anything we might discover about human minds and their distinctive limitations. Leaning on this point, though, would not be very good strategy for the infinitist, all things considered. After all, even if the reply is granted, a counterreply awaits: infinitism still implies that no human beliefs can be justified. Thus, infinitism is not a non‐skeptical response to the regress problem when it comes to the very kind of justification we're most interested in: the kind we humans (with whatever limitations we have) are capable of attaining.
1.26 A more promising line of response to the Argument from Finite Minds is that it rests on a mistaken picture of justifiers. While a justified belief might be a perfectly good justifier, maybe – and contrary to P2 – not every justifier is another justified belief.18 For instance, if we understand justifiers as justified beliefs or available reasons that an agent could cite if challenged, the set of justifiers might be larger than the set of beliefs. It might (as the infinitist says) be infinite even if what we actually believe is not.
1.27 This might seem like a decent line for the infinitist to press, though it invites a new kind of problem. Consider that to respond to the Argument from Finite Minds the infinitist has had to broaden the class of justifiers to include things that aren't now believed but are nevertheless available in some sense. Perhaps these are things that we would accept or believe and cite in support of our beliefs if challenged or asked for a justification.19 If this line of reply could be made to work, then it would free the infinitist up to deny that justification requires having more beliefs than we really have.
1.28 At this same time, though, if the set of justifiers that would justify the beliefs you hold now could consist not entirely of things that moved you to form the beliefs in the first place, then a new kind of worry surfaces, which is that the infinitist would have no way to distinguish justification from mere rationalization (e.g. citing reasons that weren't your actual reasons, but which you might nonetheless cite if pressed).
1.29 One strategy in response to this kind of worry that has been pursued by Peter Klein (2005, pp. 135–136) is to clearly differentiate the infinitist's demands on two distinct kinds of epistemic justification: propositional justification (roughly, the kind of justification you have for a proposition if it would be reasonable for you to believe it) and doxastic justification (roughly, the kind of justification you have for beliefs held on the basis of reasons you're propositionally justified in believing20). For the infinitist, the proposition expressed by B1 (i.e. rental prices will continue to increase in London in the coming year) is propositionally justified for you just in case there is available to you at least one infinite non‐repeating series of propositions (or reasons) such that R1 is a good reason to believe B1, R2 is a good reason to believe R1, R3 is a good reason to believe R2 … and so on.21 But to be doxastically justified in believing B1, the mere fact that B1 is propositionally justified for you is not sufficient. A further necessary condition is that you must have appropriately provided, as Turri (2013, p. 792) puts it, “enough reasons along at least one of the infinite non‐repeating series of reasons, in virtue of which [sic. your belief] is propositionally justified for you, to satisfy the contextually determined standards.” Regarding “contextually determined standards”: in short, the idea is that providing several reasons (from the infinite series that is available to you) may suffice for doxastic justification in an epistemically friendly setting, whereas in the context of a hostile interrogation, say, doxastic justification might require that you provide (say) 10 or 35 of them. But you are not required to provide an infinite series of reasons.
1.30 By drawing this distinction – in particular, by insisting that infinitist propositional justification is not sufficient for infinitist doxastic justification – the infinitist looks initially to be in a position to sidestep the kind of rationalization worry noted above. After all, once the distinction is drawn in the way Klein draws it, your actually citing certain reasons (from the relevant infinite non‐repeating series) is doing some of the heavy lifting in accounting for your belief's doxastic justification.
1.31 However, it's hardly the case that the infinitist is in the clear simply by drawing the distinction noted. To appreciate this point, consider the following two infinite series:
Series 1: R1, R2, R3, R4, R5 …
Series 2: R1, R3, R2, R4, R5 …
Assume that these two series contain the same infinite number of reasons, but in a different order; while there is no reason in Series 1 that does not occur in Series 2, and vice‐versa, Series 1 might suffice to justify your belief while Series 2 does not. (After all, R2 might be a good reason to believe R1, but R3 might not be.22)
1.32 As Podlaskowski and Smith (2011) have argued, this fact – in short, that “ordering matters” – turns out to be relevant to whether Klein's strategy of response to the Argument from Finite Minds ultimately works.23 Podlaskowski and Smith's objection proceeds as follows (pp. 521‐2): even if it is granted that we have an infinite series of beliefs available to us, we don't have such a series available to us in the right order!
1.33 This is initially a perplexing point. Why, exactly, would they think this? Their reasoning has two steps. The first is to point out that Klein is by his own lights committed to thinking about the notion of “availability” present in the infinitist's account of propositional justification in terms of dispositions24 – viz. that every reason in the infinite chain is one that you'd be disposed to cite at an appropriate point. While Klein doesn't think you have to actually possess (for some reason in the infinite chain, R) the first‐order disposition to cite R at the appropriate point, you must at least possess the (second‐order) disposition to form
