I just posted the paper here. Forthcoming in Oxford Studies in Metaphysics.
Abstract: I shall argue that considerations about frequency-to-chance inferences make very plausible some a localized version of the Principle of Sufficient Reason (PSR). But a localized version isn’t enough, and so we should accept a full PSR.
I’m a bit worried by the application of Bayes to nonmeasurable events. Applying Bayes over an interval could make sense (with caveats) where the interval relates to a definite but unknown probability. (For example, under one hypothesis, a coin might have a bias that lies in some specified interval.) This is not the case for nonmeasurable events: there is indeed an interval, but there is no definite probability.
ReplyDeleteI'm thinking that Bayes works for interval-valued probabilities as follows: we look at the family of all the prior probability functions that fit with the intervals, apply Bayes with each one, and then take the smallest interval containing all the posteriors.
ReplyDeleteYes, you can plug a range of numbers into Bayes. But what does the calculated interval mean?
ReplyDeleteStrictly, there is no posterior probability. It seems reasonable that you should not adopt a posterior credence outside the calculated interval. It’s not so clear what can be said about posterior credences inside the calculated interval.
An example. You have two N-sided dice, each with faces marked 1 to N. They look the same. You know that one is fair but you know nothing at all about the other. You model it by a complete range of probabilities for each side, subject only to the constraint that each set of probabilities adds up to 1. You choose one die on a fair coin toss. What is the probability that it is the fair one? 1/2, of course. You roll it and note the outcome. What should be your new credence that is it the fair one? Applying Bayes as you have suggested gives the range [1/(N+1), 1]. This is the “Bayesian divergence” of your paper. But surely the setup is symmetric and the outcome, whatever it was, gave you no useful information. So shouldn’t you stick with 1/2?.
I wonder what this implies for the nature of explanation? Because even a saturated nonmeasurable set (SN) that implies no probability of being hit or missed still has explanations in the scenario you bring up; there's an explanation for how the target and SN subset exists or came to be, why we're throwing the dart at it, why it hits or misses the set, etc.
ReplyDeleteSo I wonder what difference having an explanation brings in this specific case, as every part of the set-up and result has an explanation even though hitting or missing the SN set has no probability of itself. What would become different or worse if every part of the set-up was also a brute fact?
I think you can get a kind of probabilistic explanation for why you hit a specific point on the target, even though the probability of hitting the point was zero. But if zero-probability is consistent with explanation, so is SN.
ReplyDeleteThis may make some trouble for my arguments in the paper.
1) Could it be said from this that explanations could exist without probability - or that the concept of explanation doesn't necessarily include having any probability?
ReplyDelete2) If SN sets could exist in concrete reality, would they have to be constructed in some way by making the set via some sort of realization of Axiom of Choice? Or would they have to exist as a built-in part of a Real line?
3) Say hitting a SN target has no probability, and say you don't like the idea of SN and are unlikely to throw a dart to actualise the lack of probability or you don't create the SN set in the first place - in that case it would seem extrinsic circumstances also present probability bounds even for SN.
If the external preconditions were unlikely to co-operate, this also makes the whole no-probability aspect unlikely to be actualised. What do you think?
Ad 1: I think theists are already committed to it if they do not think that probabilities can be assigned to divine action.
ReplyDeleteAd 2: I suspect they'd have to be constructed through some essentially infinitary process that violates some metaphysical constraints like causal finitism. See the chapter on the Axiom of Choice in my infinity book.
Ad 3: Well, in that case, hitting the target is not an SN event. One gets an SN event by conditionalizing on throwing the dart, but without that conditionalization, the event is not SN. For by definition if there are non-trivial probability bounds, there is no SN.
I tend to agree that classical probability doesn't apply to higher things like qualities, minds or libertarian agents, but can't we still attribute some sort of likelihood even to the divine act? For example, we can say it's fitting and more likely that something would exist given the goodness of there being a creation and that God would create it - we can even say it's fitting for there to be all the possible grades of being ranging from elemental to angelic.
ReplyDeleteSo it seems we can say that God might prefer such possible worlds to worlds with less types of being, though He could also have created only those. As for why He created this world specifically, even then we might point to some factors that God considered that make it understandable or likely why He would create this world.
What do you think?
Also, given that explanation is compatible with there being no probability / SN, what do you think would be the difference between brute facts and SN explanations? Both would lack probability, so lacking any explanation seems to be even more unconstrained than SN in some way due to that. How would that be?
ReplyDeleteWhat do you think?
"For each possible answer, arbitrary
ReplyDeleteor not, to the question of the probability that the coin lands heads conditionally on the no-explanation hypothesis, there is a possible stochastic process hypothesizing which yields the same prediction as the no-explanation hypothesis does."
This is useless, as we only test the hypothesis, not craft others around the results, or test upon new methods.
An atom with a prob of 70/20/10 for staying and 2 disintegrations modes has a 30/30/30 in your exemple. Something empirically falcifiable.
And if you assign the real results as the no explanation ones to degenerate the theory with pure randomness, the same theory will give others numbers for another element, it would be very unlikely that again by chance the two matches, or you will have to again use the theory number as your own arbitrary... So the theory can be tested to give real stats about causeless events ( disintegrations or not).