Let us define the support that the hypothesis 
receives from the evidence 
as the increase in its probability: 
. Popper & Miller [1987, p. 574] prove that this support function may be split into two terms:
\begin{equation}\label{one}s(h|e)=s(h\lor e|e)+s(h\leftarrow e | e).\end{equation}
It is easy to evaluate these terms, which shows that the first term is always positive (or 0):
\begin{equation}\label{dedsup} s(h\lor e|e)= 1-p(h \lor e),\end{equation}
while the second term is always negative (or 0):
\begin{equation}\label{indneg} s(h\leftarrow e | e)=- \left ( 1-p(e) \right)\left ( 1-p(h|e) \right).\end{equation}
From here Gillies [1986, p. 111] is lured into concluding the refutation of induction as follows:
"Since 
follows logically from , 
must represent purely deductive support. So, if there is such a thing as inductive support in the Bayesian sense, it must be contained in the term 
. However ... this term is always negative. It therefore follows that there cannot be inductive support of the kind that the Bayesians postulate."
But to say that " must represent purely deductive support" is naive. As Townsend [1989, pp. 493f.] points out, it is clear from (2) that such a stipulation has absurd consequences, namely that any hypothesis is deductively supported by any evidence (unless 
). In particular, a contradictory hypothesis 
receives deductive support, as does a hypothesis which is directly contradicted by the evidence.
Popper and Miller are not as naive as Gillies. They never explicitly identify with deductive support. How, then, do they conclude their proof that there can be no inductive support? They are actually alluding to two different ways of completing the proof. Let me begin with the first of these, which is related to Gillies'. In fact, while this approach appears more sophisticated, I shall argue that it is still susceptible to the same criticism.
Note first that "unless happens to be deductively independent from , the value of 
is deductively contaminated." (This and the following quotations are all from Popper & Miller [1987, p. 574]. They are insubstantially edited.) That is to say: if and have common consequences, then will be supported by having these consequences confirmed. But this is deduction, not induction: the evidence does not point beyond itself. But is presumably inductively contaminated as well. How are we to disentangle one contaminant from the other? Gillies' simplistic approach will not do, as we know. Nevertheless, one feels that 
is the key. "If there is such a thing as pure inductive dependence at all, there seems nothing for it but to measure it by something like 
," which is not deductively contaminated because nothing (nontautological) can be derived from both and 
, so there can be no deductive support in this case. We already know that is negative by (\ref{indneg}), so it is tempting to jump to the conclusion that inductive support is negative. But this would be too hasty: how do we know that the inductive contamination of is negative simply because is negative? To solve this problem Popper and Miller make this very convenient assumption:
(A) "If there were to be some genuinely inductive dependence [i.e. inductive contribution to 
] between a hypothesis and some evidence , it could hardly change if were replaced by some hypothesis 
equivalent to (given , or equivalent to in the presence of )."
Indeed, is equivalent to in the presence of , so we have the desired result: "Inductive dependence is counterdependence," because it is so in the pure case and thus also in all other cases by (A).
But isn't (A) just Gillies in disguise? Let us investigate the case where the evidence refutes the hypothesis, which was so embarrassing to Gillies. If we take 
for our hypothesis we find that 
and 
. Since in general 
, this means that 
. According to (A) the inductive support should be the same in both cases, so apparently the difference must be due to some positive deductive support for , which is absurd. This refutation of (A) is simpler than the only other refutation of (A) that I have found, namely that of Rodriguez [1987, pp. 356f.]. It seems so simple, indeed, that I cannot help but suspect that I am missing something in my interpretation of (A). But I can find no indication of this in Popper and Miller. They also follow the statement of (A) by saying "This much has recently been argued at length by Levi [1986]." This is an extremely leisurely piece without a single formula in it which indeed seems to embrace the simple interpretation of (A) that I have used (esp. p. 136).
It seems that Popper and Miller were themselves aware of the dubious status of (A), for later they offered a second (more famous) proof which avoids it. The idea of this proof is:
(B) 
is the part of that goes beyond .
Here I am using the shorthand of identifying with its content, i.e., the set of all its (nontautological) logical consequences. From (B) Popper and Miller claim to prove their theorem roughly as follows: inductive support means support that goes beyond , but the part that goes beyond always has negative support by (3). QED.
Popper and Miller justify (B) thus: "
is just what needs to be added to 
, without duplicating anything already there, in order to yield the system 
." The problem is that in general most of is in neither nor : such propositions can be derived only when the two are pooled. And yet such propositions clearly seem to go beyond , so it seems that (B) patently fails to capture the part that goes beyond . Indeed, (B) "has been almost uniformly rejected" (Howson & Franklin [1994, p. 452]), primarily for this reason, and a number of convincing refutations of (B) using such propositions have been given. These examples span the range from simplistic logical ones ( itself), to ravens, to actual scientific ones.
Such examples are summarised in Popper & Miller [1987, p. 580]. Their reply is revealing. They begin by admitting that "the facts are of course as stated." But then they insist that "the objection is devoid of merit" (p. 581). And why? Because "the objection misses its mark: it fails to show that positive probabilistic dependence can be achieved in the absence of some degree of deductive dependence"! Of course there can be no positive probabilistic dependence without some degree of deductive dependence. That is an immediate corollary of (1). Popper and Miller are trying to defuse a correct rebuttal of (B) by claiming that "its mark" was not (B) at all but rather a completely different theorem, and one with an elementary proof.
A second diversion they employ is to try to shift the burden of proof onto their critics by challenging them to characterise the excess content of in a better way: "that, we feel, is a problem for those who believe in induction, not for us." (Miller [1990, p. 151]; cf. Popper & Miller [1987, p. 581]. This challenge has been taken up by Mura [1990] and Elby [1994].) Perhaps, but this does not change the fact that Popper and Miller's "proof" that deductive support is countersupport is only as good as their characterisation of excess content. One must conclude that they use these rather dishonest tricks because they have no convincing arguments in defence of (B). (Even their own disciple, Gillies [1986], apparently wished to avoid (B), which was his motivation for giving the foolish argument discussed above.)
It may seem obvious that (B) is the weak point in the Popper-Miller argument, but actually quite a few attempted refutations mistakenly charge at other parts of it. I want to discuss the attempt of Good [1990]. Note first that it is not necessary to identify with the part of that does not go beyond . To do so is a natural complement to (B), especially in light of (1), and Popper and Miller do indeed use this mode of expression. But as I outlined above one may prove that inductive support is countersupport from (B) alone. Thus it is not quite fair to formulate the Popper-Miller argument as if it depends crucially on this additional assumption, and then go on to reject it for this reason. And yet this is what Good [1990] does, for example. Here is what he calls a "suspicious feature" of the Popper-Miller argument:
"Whatever may be, whether it is supported or completely refuted by , or if it has nothing at all to do with , it remains true that a 'part' of , namely , is deducible from ."
Besides missing the core of Popper-Miller, this argument does not appear to me to be as "suspicious" as it first looks. If and are unrelated, then nothing nontrivial can be derived from both and separately, which means that nothing can be derived from , which means that the content of is in effect empty. Thus there is nothing suspicious about this case. And why is it suspicious that part of is deducible from even when refutes ? Take the simple case 
and 
. The part of deducible from is of course 
. In terms of content: 
. Hardly very suspicious.
Good [1990] continues:
"[If consists of these two parts,] Does this then put us under an obligation, whenever refutes , to express this refutation by saying that refutes 
, thus removing from the part that supports? I don't believe that Popper and Miller would use such an absurd mode of expression."
But this mode of expression is not absurd. It would in fact be perfectly sensible if (B) was correct. Consider an example: my hypothesis is Euclidean geometry , where I have divided its assumptions into the parallel postulate 
and the other axioms . Suppose the evidence tells us that space is non-Euclidean, that is: . Now is refuted, but this does not mean that we should throw out all of . On the contrary, 
remains intact. To say that what is falsified is the part that goes beyond is to say that what is falsified is precisely the part of Euclidean geometry that can only be proved using the parallel postulate . The remaining part 
, which supports, includes the first handful propositions of Euclid's Elements. It should indeed be preserved. The problem for Popper-Miller is again (B): Pythagoras' theorem, for example, is a consequence of that clearly goes beyond ; and yet it is obviously not in 
. Thus Good should have attacked (B) instead of this "mode of expression," which would in fact be correct if (B) was correct.
REFERENCES
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