Here's Day 2, before lunch.

^{*}The day started with a talk by Benjamin Jantzen titled Peirce on Miracles: The Failure of Bayesian Analysis. Benjamin started with a brief explanation of Hume’s criticism, according to which no testimony could be sufficient to justify belief in a miracle, given that the probability of fraudulent or mistaken testimony is always greater than the probability of the miracle occurring. He then considered Hume’s argument as an instance of Bayesian probabilistic inference. The basic idea is that the probability of a miracle having occurred given various testimonies to that effect is computed from the probability that each witness would report accurately given the occurrence of the miracle, the joint probability of the occurrence of such a collection of testimonies, and the antecedent probability of the miracle. Finally, Ben argued that given the Peircean criticism of the Bayesian approach, the probabilistic analysis of this sort is seriously flawed. The main gist of the criticism is that:

- There is no such a thing as the objective veracity of a witness. To apply the Bayesian method we need to know the probability that a witness judged accurately and told the truth in a particular instance. The details of that instance cannot be replicated even in principle, so we have no class of sufficiently similar events to build a sample space (this objection hinges on Peirce’s frequentist account of probability).
- Even if we grant such a thing exists, it doesn’t satisfy the independencies required by the method of balancing likelihoods employed in the argument. What leads one witness into error tends to lead others into the same error.
- History tends to preserve only the positive assertions of the extraordinary and biases the computed posterior probability. When we hypothetize the occurrence of a miracle on the basis of some set of testimonies, we are not rational in using those same testimonies to determine the probability that this hypothesis is true.

The main positive lesson is that after constructing an abductively valid hypothesis we should gather independent data for an inductive phase.

Lydia and Tim McGrew talked about The Reliability of Witnesses and Testimony to the Miraculous. They started with Condorcet’s formula for the probability of an event:

pt/pt+(1-p)(1-t)

where p is the antecedent probability of the event and t is the reliability of truthfulness of the witness. Then, they tracked subsequent changes that led to the formation of Bayes’s Theorem. Indeed, Condorcet’s account seems like a particular instance of B’s theorem, given the similarity between his formula and:

P(H)P(tH|H)/P(H)P(tH|H)+P(~H)P(tH|~H)

In particular, Condorcet’s formula is a special case resulting from three limiting assumptions.

- The witness is equireliable – he is equally likely to tell the truth about H regardless of whether it occurs.
- The witness is forthcoming – he would not be silent on the subject of H had it not occurred.
- Testimony regarding H is restricted to YES or NO.

Lydia and Tim followed the history of the debate surrounding testimony and miracles (Babbage, Reid, Bentham, Hume, Campbell, Venn, Holder, Earman) , showing how it led to the rejection of these assumptions.

Finally, they argued that the move from Condorcet’s formula to Bayesian factors is correct and that the factors should not be construed as modeling witness’ reliability, but rather a function of a wider range of epistemically relevant factors.

^{*}If you're one of the speakers and think your view is misrepresented, drop me a line.
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