Probabilistic justification and the regress problem
Consider a countable infinite sequence \((p_n)_n\) of propositions. The Regress Problem is the following foundationalist challenge to infinitism: If one's belief in some proposition \(p_0\) is justified by one's belief in some other proposition \(p_1\), which in turn is justified by one's belief in yet another proposition \(p_2\), and so on \textit{ad infinitum}, then there is no epistemic justification for one's belief in any of the propositions \(p_n\). Consider next a probability measure \(P\) and a countable infinite sequence \((S_n)_n\) of events -- identified by the authors with propositions. In the terminology of the article under review, \(S_{n+1}\) \textit{probabilistically supports} \(S_n\) if and only if \(P(S_n|S_{n+1})>P(S_n|\neg S_{n+1})\). The Probabilistic Regress Problem is now the following contemporary counterpart of the traditional Regress Problem: If one's belief in some proposition \(S_0\) is probabilistically supported by \(S_1\), which in turn is probabilistically supported by \(S_2\), and so on \textit{ad infinitum}, then there is no probabilistic justification for one's belief in any of the propositions \(S_n\). The (Probabilistic) Regress Problem is typically used by foundationalists to challenge infinitism through a two-pronged attack: Firstly, they argue that \textit{finite minds} are incapable of drawing infinitely many conclusions in finite time [cf., e.g., \textit{R.~Fumerton}, Epistemology. Oxford: Blackwell (2006)]. Secondly, foundationalists claim that inference from \(S_{n+1}\) to \(S_n\) cannot originate, but only \textit{transfer} justification [cf. \textit{J.~Dancy}, Introduction to contemporary epistemology. Oxford: Blackwell (1985)]. On a number of occasions, \textit{P. D.~Klein} [e.g. in: \textit{M.~Steup} and \textit{E.~Sosa} (eds.), Contemporary debates in epistemology. Oxford: Blackwell, 131--140 (2005)] has dismissed both of these objections to infinitism by arguing that they essentially amount to begging the question. The authors of the article under review present another line of argument against both the ``finite-minds objection'' and the ``transfer objection'' (to infinitism), by providing an example where the (unconditional) probability \(P(S_0)\) can be explicitly computed, with a non-zero result, from a sequence of conditional probabilities of the form \(P(S_n|S_{n+1})\) and \(P(S_n|\neg S_{n+1})\). (A different, more complicated, example has already been provided previously by the first author [\textit{J.~Peijnenburg}, Mind 116, No.~463, 597--602 (2007)]). Indeed, the authors show that if there were a sequence of events \((S_n)_n\) such that both \(P(S_n|S_{n+1})=1-\frac{1}{n+3}\) and \(P(S_n|\neg S_{n+1})=\frac{1}{n+3}\) for every \(n\), then by an iteration argument, exploiting various cancellations, one would arrive at \(P(S_0)=\frac{1}{2}\). However, it would have been helpful if the authors had proven also that there actually does exist such a probability space and a sequence of events \((S_n)_n\) which support these conditional probability assignments. The reviewer is considering a reply.
scientific article; zbMATH DE number 5529385