Abstract
There is a long-standing debate in epistemology on the structure of justification. Some recent work in formal epistemology promises to shed some new light on that debate. I have in mind here some recent work by David Atkinson and Jeanne Peijnenburg, hereafter “A&P”, on infinite regresses of probabilistic support. A&P show that there are probability distributions defined over an infinite set of propositions {\(p_{1}, p_{2}, p_{3}, {\ldots }, p_{n}, {\ldots }\}\) such that (i) \(p_{i}\) is probabilistically supported by \(p_{i+1}\) for all i and (ii) \(p_{1}\) has a high probability. Let this result be “APR” (short for “A&P’s Result”). A&P oftentimes write as though they believe that APR runs counter to foundationalism. This makes sense, since there is some prima facie plausibility in the idea that APR runs counter to foundationalism, and since some prominent foundationalists argue for theses inconsistent with APR. I argue, though, that in fact APR does not run counter to foundationalism. I further argue that there is a place in foundationalism for infinite regresses of probabilistic support.
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Notes
APR concerns non-circular infinite regresses of probabilistic support. A&P also show that there can be high probability by circular infinite regresses of probabilistic support (or “justification by infinite loops”). See Atkinson and Peijnenburg (2010a).
APR concerns one-dimensional infinite regresses of probabilistic support, that is, infinite regresses of probabilistic support where each node is a single proposition. A&P also argue that there can be high probability by many-dimensional infinite regresses of probabilistic support (or “many-dimensional probabilistic networks”). See Atkinson and Peijnenburg (2012).
Turri (2009, pp. 162–163) argues for a similar thesis. He argues in particular that there is a place in foundationalism for an infinite and non-repeating series of reasons (available to a subject). He does not address APR or A&P’s work more generally, though, and the infinite regresses he has in mind differ in important respects from the infinite regresses I, following A&P, have in mind. See Peijnenburg and Atkinson (2011, Sects. 5 and 6) for discussion of Turri’s argument. See also Herzberg (2013, Sect. 2).
I am not the first to discuss A&P’s work on infinite regresses of probabilistic support. See Gwiazda (2010), Herzberg (2010), and Podlaskowski and Smith 2014. But my discussion is very different than Gwiazda’s, Herzberg’s, and Podlaskowski and Smith’s discussions. See Peijnenburg (2010) for a response to Gwiazda (2010). See Atkinson and Peijnenburg (2010b) for a response to Herzberg (2010).
The extant literature on the regress problem is vast. See Cling (2008) for references (and for helpful discussion).
A&P typically assume IP/IP* in their work on infinite regresses of probabilistic support. Peijnenburg and Atkinson (2014) is an exception. There they assume HIP/HIP*.
The assumption that \(\hbox {Pr}(p_{i} {\vert } p_{i+1}) = \alpha \) and \(\hbox {Pr}(p_{i} {\vert } \lnot p_{i+1}) = \beta \) for all i is inessential. A&P show that the target proposition in an infinite regress of probabilistic support can have a high probability even if it is not the case that \(\hbox {Pr}(p_{i} {\vert } p_{i+1}) = \alpha \) and \(\hbox {Pr}(p_{i} {\vert } \lnot p_{i+1}) = \beta \) for all i. See, for example, Atkinson and Peijnenburg (2009), Peijnenburg (2007), and Peijnenburg and Atkinson (2008, 2014).
See Fumerton (2010) for relevant discussion.
Cornman (1977, p. 291) defends a claim to this effect by appeal to the thesis that a proposition is justified for a subject only if it is more reasonable for that subject than is its denial.
IJ1 stands in contrast to:
Infinitism on Justification 1* (IJ1*)p is justified for S only if p is the target proposition in a regress of probabilistic support RPS such that (a) \(\hbox {Pr}(p) > \mathbf{t}\), (b) RPS has an infinite number of nodes, and (c) none of RPS’s nodes is an ancestor of itself.
IJ1 gives a putative sufficient condition for justification whereas IJ1* gives a putative necessary condition. The latter, unlike the former, is correct. In fact, it is trivially correct. Any p such that p is justified for S is also such that there is regress of probabilistic support RPS where p is the target proposition, \(\hbox {Pr}(p) > \mathbf{t}\), the number of nodes is infinite, and none of the nodes is an ancestor of itself. See Cling (2004, Sect. 2, p. 103) for a closely related point.
See Roche (2012b) for discussion of a coherentist theory similar in relevant respects to IJ2.
It is important to note that (a) in IJ3 is not redundant. Let \(p_{1}, p_{2}, p_{3},\ldots , p_{n}, \ldots \) be a regress of probabilistic support RPS such that RPS has an infinite number of nodes, none of RPS’s nodes is an ancestor of itself, and the conditional probabilities involved are empirically credible for S. Suppose, consistent with this, that \(\hbox {Pr}(p_{i} {\vert } p_{i+1}) = 2/100\) and \(\hbox {Pr}(p_{i} {\vert } \lnot p_{i+1}) = 1/100\) for all i. Then by (9) it follows that \(\hbox {Pr}(p_{1})\) is roughly equal to 0.010 and thus is less than t. See Herzberg (2014, Sect. 7) for related discussion.
IJ3, understood as IJ3* below, is similar to but importantly different than “(PBPIJ)” in Herzberg (2014, p. 714). IJ3 gives a putative sufficient condition for justification whereas (PBPIJ) gives a putative necessary and sufficient condition for justification.
See also Peijnenburg and Atkinson (2011, p. 124).
There would be a similar point if N2 were broken up into an infinite number of nodes: N2.1: \([\hbox {Cr}(p_{1} {\vert } p_{2}) = 0.99]\), N2.2: [\(\hbox {Cr}(p_{1} {\vert } \lnot p_{2}) = 0.04\)], N2.3: [\(\hbox {Cr}(p_{2} {\vert } p_{3}) = 0.99\)], N2.4: [\(\hbox {Cr}(p_{2} {\vert } \lnot p_{3}) = 0.04],\ldots , \mathrm{N}2.2_{n-1}: [\mathrm{Cr}(p_{n-1} {\vert } p_n) = 0.99], \mathrm{N}2.2_{n-2}: [\mathrm{Cr}(p_{n-1} {\vert } p_{n}) = 0.04],\ldots , \) would need to be modified accordingly. Then the point would be that if FJ* so modified were assumed, if the credences in N5 were not non-inferentially justified for S, and if the case were otherwise the same, then the credences in N2.1, N2.2,...would not be justified for S and so, despite the fact those credences together fully determine \(\hbox {Cr}(p_{1}) = 0.8, \hbox {Cr}(p_{1}) = 0.8\) would not be justified for S.
There is a place in foundationalism for infinite regresses of probabilistic support. The same is true with respect to coherentism and infinitism.
If, instead, \(\hbox {Cr}(p_{1} {\vert } p_{2}) = 0.98, \hbox {Cr}(p_{1} {\vert } \lnot p_{2}) = 0.99, \hbox {Cr}(p_{2} {\vert } p_{3}) = 0.98, \hbox {Cr}(p_{2} {\vert } \lnot p_{3}) = 0.99,\ldots , \hbox {Cr}(p_{n-1} {\vert } p_{n}) = 0.98, \hbox {Cr}(p_{n-1} {\vert } \lnot p_{n}) = 0.99,\ldots \), then \(\hbox {Cr}(p_{1}) \approx 0.980\).
If, instead, \(\hbox {Cr}(p_{1} {\vert } p_{2}) = 0.98, \hbox {Cr}(p_{1} {\vert } \lnot p_{2}) = 0.99, \hbox {Cr}(p_{2} {\vert } p_{3}) = 0.98, \hbox {Cr}(p_{2} {\vert } \lnot p_{3}) = 0.99,\ldots , \hbox {Cr}(p_{99} {\vert } p_{100}) = 0.98, \hbox {Cr}(p_{99} {\vert } \lnot p_{100}) = 0.99\), and \(\hbox {Cr}(p_{100}) = 0.1\), then \(\hbox {Cr}(p_{1}) \approx 0.980\).
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Acknowledgements
Thanks to David Atkinson, Jeanne Peijnenburg, Adam Podlaskowski, Tomoji Shogenji, Joshua Smith, and two anonymous referees for helpful comments and/or discussion.
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Roche, W. Foundationalism with infinite regresses of probabilistic support. Synthese 195, 3899–3917 (2018). https://doi.org/10.1007/s11229-016-1289-4
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DOI: https://doi.org/10.1007/s11229-016-1289-4