...when we have no reason to prefer any one value of a variable p over another in some range R, we should assign equal epistemic probabilities to equal ranges of p that are in R, given that p constitutes a "natural variable." A variable is defined as "natural" if it occurs within the simplest formulation of the relevant area of physics. [§3.3.2, p234.]
To support the validity of the RPOI, Collins appeals to two lines of evidence:
(1) it has a "wide range of applicability" (p235); and
(2) we require some form of the POI as an assumption in order to draw many of the indispensable conclusions we do about the world.
However, it appears to me that both of these claims are false. In the case of (1), he appeals to the common use of uniform distributions when constructing probability models. However, while uniform distributions are certainly required for the RPOI, the converse does not hold. Uniform distributions draw their motivation from various sources, including but not limited to past observations of frequency, and especially pre-existing understandings of physical systems corresponding to the random variables we define. Furthermore, even if (1) were true, it would only show that the RPOI has application, and not that we are justified in applying it. Regarding (2), Collins' aims to show that inductive inferences from past experience are insufficient for developing a robust understanding of how the world works, and so the RPOI must complement induction as a sort of foundational principle. Yet in my survey of the supporting examples he provides for (2) I find that a POI is not indispensable for them as suggested. For instance, he asks us to consider the first balanced twenty-sided die ever produced; certainly, even though we have never encountered one in our previous experience, we ought to assign immediately, if tentatively, a uniform distribution for rolling outcomes. Yet how do we justify this uniform assignment? According to Collins, we must appeal to some form of POI, but is this really so? It seems to me that instead of applying an abstract philosophical principle, we instead draw from what we already know about the world, in this case the behavior of symmetrical bodies, as well as our instinctual expectations (whether gleaned from past experience or evolutionary history) for falling objects and other related physical systems. Perhaps in some cases we appeal to inarticulable intuitions, where those intuitions in turn have experiential (i.e. inductive) support, and in those cases too we have no room for the RPOI. To be fair, in Collins' opinion the clearest and best examples supporting (2) come from the science of physics (note: not "folk" physics), where sadly I cannot follow since I have no high-level training in that field. However it seems obvious to me that at least his examples outside physics don't hold up, and these are the most important for establishing that we ought to use some form of POI. For to the extent that physics is useful, it can be supported inductively apart from any POI; and to the extent that it is not, we are free to leave it behind. Of course there are other possible (though in my opinion similarly impotent) defenses of the RPOI outside (1) and (2); but Collins seems disinterested in them, and so I shall neither give them any attention here.
For these reasons, I don't think Collins has adequately defended the RPOI. However, I want to go a step further and suggest that we have positive reasons for thinking the RPOI is invalid, on account of two problems I see with its application.
The Zero Paradox: Consider a situation similar to the one Collins discusses in Blackwell, where we have a physics model with an unknown parameter p. Let L denote some statement about the physical world. Suppose the range of possible values for p is the interval [0,1], but that L is true if and only if p=0.5. In that case, the range of L-permitting values is a singleton, and hence has (Lebesgue) measure zero, whereas the parameter space has measure 1. Then provided we have no additional relevant knowledge, according to Collins we must apply the RPOI to obtain a probability that L is (possibly) true equal to the ratio of the measures of the respective ranges, which in this case is zero. Hence the epistemic probability that L is false is 1, which is to say that we must take a position of certainty that L is false. However this conflicts with the fact that we know by hypothesis L is possibly true, in particular it is true iff p=0.5. We can toy with this approach to bring it to bear more strongly on our intuitions. For instance suppose that L will be true just in case some term cancels in our physics model, and that this in turn occurs just in case p=1/n for some positive integer n. Then we have countably infinitely many values clustering about zero such that if p takes any one of them, L will be true; and otherwise L will be false. Then the previous argument will hold again, insofar as the RPOI will produce a zero probability that L is true, i.e. complete certainty that L is false, even though there are now infinitely many possible cases for L to be true in our physics model! Moreover, if we let L denote the statement p=x, where x is any value in [0,1], then we can apply the argument to show that for all x in [0,1] we must be certain that p≠x, even though we must also be certain that for some x in [0,1] we in fact have p=x. Since we are dealing with epistemic certainty, this is no mere lottery paradox. Collins, in effect, demands that we place absolute confidence in that which we know to be impossible.
The Continuity Paradox: (This is inspired by the criticisms of the McGrews.) Collins' argument assumes the measures of the "epistemically illuminated" ranges of the (EI) parameter spaces are all finite, even though the spaces themselves are infinite in measure. Suppose, however, that we increase our knowledge so that the EI range for the parameters of our physics models grows larger and larger. Provided that the life-permitting universe (LPU) range remains fixed, i.e. that we do not learn of any additional values which will permit life in the universe, then the probability of LPU will tend to zero. Suppose now that at some point we learn about the whole parameter space (which has infinite measure). According to Collins, we should then decide that LPU has probability zero. In other words, we must again be certain that LPU is false, even though we know that there is a positive-measure range of LPU values for the parameters. This of course re-introduces the zero paradox in an even stronger form. To avoid it, we might posit that there is some nonzero probability on an infinite-measure range. (Note: Collins does not do this, since he denies that the zero paradox poses any problem for us.) However if we assign a nonzero probability to the infinite case, then we have a discontintuity in the limit, which means that there is a point at which expanding the range of the LPU-inconsistent space results in a higher probability for LPU. Though not precisely a contradiction, this situation is clearly intolerable.
In addition to the zero and continuity paradoxes, it appears that the RPOI is inconsistent with Collins proposed rule of inference for the fine-tuning argument, the "restricted likelihood principle" (RLP). Recall that in his fine-tuning argument, he asks us to consider as a conceptual device a disembodied alien observer who witnesses the big bang and, using the information from those first few fractions of a second in the life of the universe, say up to time t*, develops the basic structures of the physics models which, once certain parameters are filled in, will accurately predict its behavior. Observing the ratios of measures between the sets of life-permitting and EI values of those parameters, he applies the RPOI to obtain an extremely low probability for LPU. However, I suggest that we go back further, not to time t*, but instead to the big bang itself. At that time, the alien has no information on which to base a physics model. So if he considers the hypothesis of LPU, then the natural variable is going to be, simply, true or false, i.e. will the universe permit life or not? Applying the RPOI he obtains a probability of 0.5 that the universe will permit life. Now, Collins denotes by k' the background information that the alien has at time t*. Let k'' denote the information in k', except for anything learned between the big bang and time t*. Let NSU denote, following Collins, the naturalistic single-universe hypothesis. Also, let CDL denote the hypothesis that the universe is consistent with the development of life at time t*. Then clearly P(CDL|LPU&NSU&k'') = 1 > P(CDL|(~LPU)&NSU&k''), and using Collins' RLP, we conclude that CDL confirms the LPU hypothesis on the background assumptions of NSU&k'', i.e. P(LPU|NSU&k''&CDL) > 0.5. Besides CDL, the only information involved in k' but not k'' appears irrelevant to the question of whether or not life will develop. Thus we conclude that P(LPU|NSU&k') = P(LPU|NSU&k''&CDL) > 0.5. However, this contradicts Collins' conclusion based on considering time t* exclusively that P(LPU|NSU&k') < 0.5.
In response to this objection, Collins argues that we must "update" the probability obtained from applying the RPOI at the big bang by re-applying it at time t*. Immediately we should note that even if this resolves the problem, it does so on pain of departing from the RPOI as stated in Blackwell, which requires that we have no existing reason to prefer one alternative over another. For when the alien arrives at time t*, according to the RPOI he already has reason to prefer the LPU range over the non-LPU range in the sense of taking it to have a greater probabilistic weight for intervals of fixed measure. Collins would have us modify his RPOI to read, "when we have no reason other than a previous application of the RPOI to prefer any one value...", or something of that sort. Moreover, he must also modify his RLP to ensure that any updates to the RPOI are performed prior to its application. Thus we must continually forget whatever we previously concluded as a result of the RPOI as new information becomes available. In this case, though, one might wonder why we cannot update our information from time t* to the present time. For instance, in the present situation the natural variable for the existence of God seems to me, again simply, true or false, i.e. God exists or not. If Collins wants this variable trumped by the parameter considerations at time t*, then he may be hampered by the additional restriction which we have seen he must add to the RPOI.
I suggest that the reason our intuition will not tolerate us endowing the RPOI with any greater persuasive force than this is that we intuitively understand that the RPOI doesn't give us any reason to prefer one hypothesis over another. So any actual reason to priviledge a particular hypothesis will always trump the RPOI. Indeed, the RPOI is so weak even on Collins' view that previous instances of the RPOI cannot inform its own application. Further, we find that not all of the paradoxes which are so-well known for the unrestricted POI are undercut by the RPOI. Most seriously of all, we don't appear to have any justification for its use in the first place. These problems, in my judgment, demands that we reject, at least tentatively, Collins' formulation of the fine-tuning argument.