Robin Collins' argument in Blackwell is invalid on two counts
I want to post this more or less for reference, because even though it's just a simple observation, it has serious implications for the argument of Robin Collins. Usually, when a professional philosopher publishes an argument in a peer-reviewed journal or book, certain things are taken for granted, like the validity of any deductive arguments given in it. So it's a pretty straightforward matter to make sure that one's central argument, if it's intended to be deductively valid, is *in fact* deductively valid. But Robin Collins' argument is not.
Here's an excerpt from the book:
"(1) Given the fine-tuning evidence, LPU is very, very epistemically unlikely under NSU: that is, P(LPU|NSU & k') << 1, where k' represents some appropriately chosen background information, and << represents much, much less than (thus making P(LPU|NSU & k') close to zero).
(2) Given the fine-tuning evidence, LPU is not unlikely under T: that is, ~P(LPU|T & k') << 1.
(3) T was advocated prior to the fine-tuning evidence (and has independent motivation).
(4) Therefore, by the restricted version of the Likelihood Principle, LPU strongly supports T over NSU."
Now, by the "Likelihood Principle" (hereafter, LP) he means this (again, quoting directly):
"Let h1 and h2 be two competing hypotheses. According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1 than h2. Put symbolically, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) represent the conditional probability of e on h1 and h2, respectively. Moreover, the degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2; specifically, it is proportional to P(e|h1)/P(e|h2)."
He goes on to hedge this principle, thusly:
"The restricted version limits the applicability of the Likelihood Principle to cases in which the hypothesis being confirmed is non-ad hoc. A sufficient condition for a hypothesis being non-ad hoc (in the sense used here) is that there are independent motivations for believing the hypothesis apart from the confirming data e, or for the hypothesis to have been widely advocated prior to the confirming evidence."
This argument is invalid on two counts.
COUNT #1: Collin mixes up the roles of LPU and the fine-tuning evidence (call it FT) in his application of the restricted likelihood principle (call it RLP). Indeed, we read in premise (3) that it's FT from which T has independent motivation, where in RLP it's e from which h1 has independent motivation. So in order to apply RLP we need e=FT and h1=T. However, FT is not mentioned anywhere else in the argument. Instead, he concludes that LPU, not e, strongly supports T over NSU. That is to say, in the conclusion (4) he wants e=LPU, h1=T and h2=NSU. By mixing up the roles of LPU and FT in his argument, he leaves his conclusion as a nonsequitur.
Luckily for Collins, the Likelihood Principle is true regardless of whether it is restricted or not. Indeed, it's just a consequence of Bayes' Theorem, together with some mild assumptions about the probabilities involved being nonzero. Recall that on Bayes' Theorem (with nonzero probabilities) we have
P(h1|e) / P(h2|e) = [P(h1) / P(h2)] * [P(e|h1) / P(e|h2)].
So the invalidity in count #1 has a more or less easy fix. Or at least it would, were it not for this next bit:
COUNT #2: Unfortunately, the "strong support" in the conclusion (4) is not guaranteed by either LP or RLP. Instead, the strength of the evidence depends on the prior probabilities P(h1) and P(h2). Namely, if the prior probability P(T) of theism is too small relative to the other probabilities, then the evidence will be far too weak to matter. So, it's actually false that the strength of the evidence is proportional to the quantity [P(e|h1) / P(e|h2)], as Collins mistakenly claims, because the would-be constant of proportionality is actually the function [P(h1) / P(h2)].
The invalidity in count #2 has no easy fix, and in fact it seems to doom the argument from the outset.
"(1) Given the fine-tuning evidence, LPU is very, very epistemically unlikely under NSU: that is, P(LPU|NSU & k') << 1, where k' represents some appropriately chosen background information, and << represents much, much less than (thus making P(LPU|NSU & k') close to zero).
(2) Given the fine-tuning evidence, LPU is not unlikely under T: that is, ~P(LPU|T & k') << 1.
(3) T was advocated prior to the fine-tuning evidence (and has independent motivation).
(4) Therefore, by the restricted version of the Likelihood Principle, LPU strongly supports T over NSU."
Now, by the "Likelihood Principle" (hereafter, LP) he means this (again, quoting directly):
"Let h1 and h2 be two competing hypotheses. According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1 than h2. Put symbolically, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) represent the conditional probability of e on h1 and h2, respectively. Moreover, the degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2; specifically, it is proportional to P(e|h1)/P(e|h2)."
He goes on to hedge this principle, thusly:
"The restricted version limits the applicability of the Likelihood Principle to cases in which the hypothesis being confirmed is non-ad hoc. A sufficient condition for a hypothesis being non-ad hoc (in the sense used here) is that there are independent motivations for believing the hypothesis apart from the confirming data e, or for the hypothesis to have been widely advocated prior to the confirming evidence."
This argument is invalid on two counts.
COUNT #1: Collin mixes up the roles of LPU and the fine-tuning evidence (call it FT) in his application of the restricted likelihood principle (call it RLP). Indeed, we read in premise (3) that it's FT from which T has independent motivation, where in RLP it's e from which h1 has independent motivation. So in order to apply RLP we need e=FT and h1=T. However, FT is not mentioned anywhere else in the argument. Instead, he concludes that LPU, not e, strongly supports T over NSU. That is to say, in the conclusion (4) he wants e=LPU, h1=T and h2=NSU. By mixing up the roles of LPU and FT in his argument, he leaves his conclusion as a nonsequitur.
Luckily for Collins, the Likelihood Principle is true regardless of whether it is restricted or not. Indeed, it's just a consequence of Bayes' Theorem, together with some mild assumptions about the probabilities involved being nonzero. Recall that on Bayes' Theorem (with nonzero probabilities) we have
P(h1|e) / P(h2|e) = [P(h1) / P(h2)] * [P(e|h1) / P(e|h2)].
So the invalidity in count #1 has a more or less easy fix. Or at least it would, were it not for this next bit:
COUNT #2: Unfortunately, the "strong support" in the conclusion (4) is not guaranteed by either LP or RLP. Instead, the strength of the evidence depends on the prior probabilities P(h1) and P(h2). Namely, if the prior probability P(T) of theism is too small relative to the other probabilities, then the evidence will be far too weak to matter. So, it's actually false that the strength of the evidence is proportional to the quantity [P(e|h1) / P(e|h2)], as Collins mistakenly claims, because the would-be constant of proportionality is actually the function [P(h1) / P(h2)].
The invalidity in count #2 has no easy fix, and in fact it seems to doom the argument from the outset.
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