Pollock on probability in epistemology
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Abstract
In Thinking and Acting John Pollock offers some criticisms of Bayesian epistemology, and he defends an alternative understanding of the role of probability in epistemology. Here, I defend the Bayesian against some of Pollock's criticisms, and I discuss a potential problem for Pollock's alternative account.
Keywords
Pollock Probability Logic Bayesian EpistemologyJohn Pollock did a lot of interesting and important work on the metaphysics and epistemology of probability over several decades. In Thinking About Acting (Pollock 2006), we find many fascinating and thought provoking ideas and arguments (both old and new) about probability. Owing to limitations of space, I will be confining my remarks to a handful of issues addressed in Pollock (2006) pertaining to probability, logic, and epistemology. First, I will discuss some of Pollock’s arguments against Bayesian Epistemology (BE). Here, I’ll try to defend (BE) from what I take to be less than decisive objections. Then, I will make some critical remarks concerning Pollock’s alternative approach to “probabilistic epistemology”, which is based on his (nonBayesian) theory of “nomic probability” (Pollock 1990). ^{1}
1 Some remarks on Pollock’s critique of Bayesian epistemology
 (1)
prob \((P \mathbin{\&} \sim P) = 0.\)
 (2)
prob \((P \vee \sim P) = 1.\)
 (3)
prob\((P \vee Q) =\)prob(P) + prob(Q) − prob(P & Q).
 (4)
If P and Q are logically equivalent, then prob(P) = prob(Q)
 (5)
For all \(P \in {\mathcal{L}},\) prob(P ≥ 0)
 (6)
prob \((A) = 2\)
 (7)
prob \((\sim A) = 1\)
 (8)
prob \((A \vee \sim A) = 1\)
 (9)
prob \((A \mathbin{\&} \sim A) = 0\)
I think this is highly uncharitable to the Bayesian epistemologist. First, this rests on a misunderstanding of (PC), which only entails that tautologies of \({\mathcal{L}}\) must be assigned a prob of 1. Second, it rests on an implausible assumption about “necessary truths”—that they are all logically equivalent to the simple tautology \(P \vee \sim P\). I’m not sure what Pollock has in mind here, but I don’t see why a Bayesian (or anyone else) should be saddled with such a strong commitment. As a result, it’s unclear what reason Bayesians could have for insisting that all necessary truths be assigned the same probability as a tautology. It seems to me that there are better ways to think about (PC) and (BE).If Q is a necessary truth, it is logically equivalent to (\(P \vee \sim P\)), so it follows from axioms (2) and (4) that every necessary truth has a prob of 1.
 (10)
prob \((B \mathbin{} A \mathbin{\&} C) = 1.\)
 (11)
Any adequate (formal) epistemology must be able to explain why deductive inference from multiple uncertain premises can be expected to preserve justification (and/or warrant).
where, an inference of the form \(P_1,\ldots,P_n\,\therefore\,Q\) is \({\sl probabilistically\,valid}\) just in case its conclusion Q is at least as probable as its least probable premise—that is, iff for all i: prob(Q) ≥ prob(P_{ i }). As it turns out, no deductively valid form of inference with more than one premise is probabilistically valid in this sense. That explains why Pollock thinks Bayesian epistemology cannot satisfy (11). The reason Pollock thinks violating (11) is undesirable is that he thinks violating (11) prevents probabilism from being able to explain how we can reason “blindly” from multiple warranted (or justified) premises, using a deductively valid inference, and expect that the conclusion will also be warranted (or justified). Since “blind deductive reasoning” seems integral to epistemology, this would be a serious shortcoming of (BE)—or, more generally, of any probabilistic epistemology.If degrees of warrant satisfy the probability calculus, then ... we can only be confident that a deductive argument takes us from warranted premises to a warranted conclusion if all the inferences are probabilistically valid.
Strictly speaking, it is true that Bayesianism so construed can’t satisfy (11) in this sense. But, I wonder why one would want to both construe Bayesian epistemology in this way, and understand “probabilistic validity” in this way. It seems clear to me that many contemporary Bayesian epistemologists would neither want to equate prob and \({\sl degree\,of\,warrant}\) (or \({\sl degree\,of\,justification}\), for that matter) nor explicate \({\sl probabilistic\,validity}\) in the way Pollock proposes. Let’s take the second point first. There is quite a long tradition of what is known as \({\sl probability\,logic}\) (PL). In recent years, probabilitylogicians like Adams (1975, 1996) and Hailperin (1996) have done a great deal of work on various notions of “probabilistic validity”. Two important points about (PL) are in order here. First, the notion of “probabilistic validity” that is typically used in (PL) circles is not the one Pollock has in mind. Adams (1975, p. 57) defines a different notion, which I will call probvalidity. I won’t give his definition of probvalidity here, but I will discuss one important consequence of the definition, just to give a sense of how it differs from Pollock’s “probabilistic validity”. Letu(p) = 1 − prob(p) be the \({\sl uncertainty}\) of p. And, consider an inference of the form \(P_1,\ldots,P_n \,\therefore \,Q\). Such an inference will be probvalid in Adams’s sense only if ^{6} the uncertainty of the conclusion is no greater than the sum of the uncertainties of the premises—that is, only if \({\sc{\bf{u}}}(Q) \le \sum_{i = 1}^{n} {\sc{\bf{u}}}(P_i)\). In other words, the uncertainty of the conclusion of a probvalid inference will never exceed the sumtotal of the uncertainties of its premises. Moreover, it is a fundamental theorem of (PL) that all deductively valid arguments are probvalid. So, in this sense, a Bayesian (probabilist) who adopts Adams’s notion of probvalidity, can explain why (in one precise sense) conclusions of deductively valid inferences will never be more unwarranted (or more unjustified) than the premises already were. Of course, this presupposes a different epistemic explanandum than Pollock has in mind in (11). But, in the interest of giving (PC) and (BE) a fair hearing, it is worth noting that other notions of “probabilistic validity” have been investigated by people who are interested in just the sort of deductive inferences from multiple uncertain premises that Pollock is talking about. Putting these alternative (PL)investigations of “uncertain deductive inference” to one side, I want to make a second point about (PL)—that it can be illuminating, even with respect to Pollock’s explanandum [(11)].
 (12)
If prob \((P) > 1  \epsilon\) and prob \((P \supset Q) > 1  \epsilon\), then prob \((Q) > 1  2 \epsilon\).^{7}
 (13)
S is justified (warranted) in believing p iff prob\((p) > 1  2 \epsilon\), for some suitably “small” \(\epsilon\); and, S is highly justified (warranted) in believing p iff prob\((p) > 1  \epsilon\).^{8}
 (14)
If prob \((P \supset Q) > 1  \epsilon\) and prob \((Q \supset R) > 1  \epsilon\), then prob \((P \supset R) > 1  2 \epsilon\).
 (15)
If prob \((P) > 1  \epsilon\) and prob \((P \rightarrow Q)\) = prob \((Q \mathbin{} P) > 1  \epsilon\), then prob(Q) > \((1  \epsilon)^{2}\).

Confirmation as firmness. E confirms_{ f }H, relative to background evidence K if and only if prob\((H \mathbin{} E \mathbin{\&} K) > t\), for some threshold value t (typically, t > 1/2).

Confirmation as increase in firmness. E confirms_{ i }H, relative to background evidence K if and only if prob\((H \mathbin{} E \mathbin{\&} K)\) > prob\((H \mathbin{} K)\).

prob\((Na \mathbin{} Pa \mathbin{\&} K)\) is high (specifically, it’s approximately 9/10).

prob\((Na \mathbin{} Pa \mathbin{\&} K)\) is significantly less thanprob\((Na \mathbin{} K)\).
 (16)
A necessary condition for E’s counting as a reason to believe H (or for it being reasonable to believe H on the basis of E), given background evidence/knowledge K, is that E does not disconfirm _{ i } H, relative to K.
2 Some worries about Pollock’s alternative “probabilistic epistemology”
Pollock rejects (BE), but he still thinks that probabilities (of some kind) are important in epistemology. Pollock’s alternative is what I will call a theory of \({\sl defeasible\,probabilistic\,reasoning}\) (DPR). Pollock’s (DPR) has three main components, each of which differs in important ways from (BE).
The first component of Pollock’s (DPR) involves indefinite probabilities. The probability calculus (and the example we discussed above) involves only definite probabilities—probabilities over closed sentences (i.e., propositions). Pollock’s (DPR) theory involves \({\sl nomic\,probability}\) (Pollock 1990) functions \({\bf{prob}}\), which (formally) take open sentences as arguments. For instance, \({\bf{prob}}(Nx \mathbin{} Px)\) is meaningful in Pollock’s theory, and it denotes “the proportion of physically possible P’s that would be N’s”. So, Pollock is talking about a kind of objective, physical probability, which is indefinite. This differs from the prob’s of (BE) in several respects. First, the prob’s of (BE) are (in some sense) epistemic probabilities. And, while there is disagreement among advocates of (BE) as to whether epistemic probabilities are subjective or objective (see footnote 7), it is clear that prob’s are not physical probabilities. Second, Pollock’s \({\bf{prob}}\)’s are indefinite, while (BE)’s prob’s are definite. This is also important, since both Pollock and the advocates of (BE) want to make inferences about particulars. Pollock will do this via defeasible reasoning from his indefinite, nomic \({\bf{prob}}\)’s (plus definite statements about particulars) to (other) definite statements about particulars. Bayesians will do this via direct appeals to definite probabalistic “facts”. Finally, Pollock’s indefinite \({\bf{prob}}\)abilities formally differ from (PC)’s prob’s in various ways. Pollock has developed a sophisticated formal theory of \({\bf{prob}}\), as well as some ingenious computer programs for calculating and proving general claims about \({\bf{prob}}\)’s. Unfortunately, I don’t have the space to discuss any of that formal work here.^{13} Next, I will illustrate how Pollock’s (DPR) approach differs from (BE) on our example above. But, first, I need to mention the other two components of Pollock’s theory of defeasible probabilistic reasoning.
The second component of Pollock’s (DPR) will require some account of how we can come to know the (true) values of (or, at least, ranges of values of or inequalities involving) salient nomic probabilities. Among other things, this will have to give us some grip on how we might come to know something about the “true proportionality function ρ over nomologically possible worlds”. I put this locution in quotation marks, because I am rather skeptical that there are such proportionality functions, and/or that we can come to know what they are. But, because my space is limited here, I won’t be able to get into the (rather extensive) metaphysical and epistemological worries I have about “proportions of nomologically possible worlds”talk. Pollock does have a lot to say about this second component. And, I refer the interested reader to his 1990 book on nomic probability (Pollock 1990).
 (SS)
If F is projectible with respect to G and r > 0.5, then “\(Gc \mathbin{\&} {\bf{prob}}(Fx \mathbin{} Gx) \ge r\)” is a defeasible reason for believing “Fc”, the strength of the reason depending upon the value of r.
 (17)
Pa is a defeasible reason to believe Na (given what we know about the example in question). Moreover, Pa is a strong (defeasible) reason to believe Na (and we can make it as strong a reason as we like, just by turningup the numbers in our background story about the case).
I wish I had more space to discuss other aspects of Pollock’s (DPR) theory, not to mention his theory of “causal probability” and his new approach to decision theory. There is just a ton of really interesting and novel stuff in this book. And, there is also a lot of neat stuff “under the hood” that isn’t (explicitly) discussed in the book (e.g., some very powerful and ingenious computer programs for calculating and proving general claims about the sorts of probabilities Pollock has in mind). Working through Thinking About Acting was challenging and edifying. I highly recommend it to anyone interested in decision theory, probability, epistemology and/or various other related fields. The only bad thing about this book is that it’s the last one John Pollock had the opportunity to write.
Footnotes
 1.
I regret that I will not have a chance here to discuss Pollock’s theory of “causal probability” (and its application to “causal decision theory”), which is one of the newest (and most exciting) ideas in the book. And, I’m sad that I won’t get to talk to John about any of my queries. I’m sure he would have had many illuminating answers. He always did.
 2.
Strictly speaking, Kolmogorov gives a settheoretic, and not a logical axiomatization of (PC). But, one can give an (extensionally) equivalent logical axiomatization. See Fitelson (2008, Sect. 1) for an axiomatization of (PC) that is along these lines.
 3.
Pollock is in good company here. Skyrms’s (1999, Chap. 6) axiomatization has exactly the same deficiency. I owe this counterexample to Skyrms’s (and Pollock’s) theory to Mike Titelbaum. As Carnap (1962, p. 341) notes, it is surprisingly easy to give equivalentlooking axioms for (PC), which are nonequivalent. This happens a lot in the literature on (PC).
 4.
A bit later in the text, Pollock discusses a related logical impoverishment of (PC), and he complains that it is a shortcoming. On page 108, Pollock rightly points out that (PC) does not say anything (systematically) about probabilities over open firstorder sentences. This is true, of course. But, something much stronger is true—namely that (PC) doesn’t say anything (systematically) about \({\bf \sc {prob}}\)abilities over anything other than sentential languages \({\mathcal{L}}.\)
 5.
Having conceded this point, it is worth mentioning that this problem is far less pressing than the problem Pollock has in mind—which would saddle proponents of (BE) with the commitment to assign probability 1 to all necessary truths. The main point I want to get across here is that proponents of (BE) have the theoretical tools to distinguish various “levels” of ideal epistemic rationality. As such, their framework is not as hopeless as Pollock makes it sound.
 6.
This is only a necessary condition for probvalidity, which is why it is not suitable as a definition (Adams 1975, p. 57).
 7.
I haven’t said anything yet about the interpretation of prob. This is intentional. It seems to me that Pollock’s objections are not restricted to (say) subjective (BE). Rather, he’s taking on just about any kind of probabilistic reduction of \({\sc{dj}}\) or \({\bf{\sc{dw}}}\). I presume this would include nonsubjective probabilists about evidential support, such as Carnap (1962), Williamson (2000), and Keynes (1921), as well as subjective (BE)—ers, such as Skyrms (1999), Joyce (2009), and others. I’ll return to this issue in Sect. 2. But, in the meantime, I will assume that prob is whatever probability function a particular advocate of (BE) has in mind. This will vary, but in a way that is orthogonal to this line of Pollock’s objections.
 8.
Of course, I do not mean to endorse (13), nor do I mean to saddle the proponent of (BE) with it. I am only introducing it here for dialectical purposes—to bring out what I think is an exaggeration in Pollock’s objection to (BE).
 9.
Various commentators have recently come to the view that →MP isn’t even deductively valid (McGee 1985; Kolodny et al. 2009). I will put that controversy to one side here, and I will suppose that modus ponens is deductively valid for the indicative conditional. But, it is worth noting that, if these commentators are right, then “blind deductive →MP reasoning” would not be kosher. I think that would undermine Pollock’s dialectical position visavis (BE). But, I can’t go into that here.
 10.
As I explained in footnote 7, I am remaining as neutral as possible on the interpretation of prob here. I will return to this issue in Sect. 2. In this example, I think the probabilistic “facts” I cite are robust across various interpretations of prob. And, I think I’m not doing any harm here to Pollock’s usage of prob for definite probabilities.
 11.
White (2006, Sect. 5) seems to assume something like (16) in his Bayesian criticism of epistemic dogmatism. Williamson (2000, Chaps. 9 and 10) seems to require some probabilistic relevance in his account of “justification”. And, Shogenji (2009) defends a precise, probabilistic theory of \({\sc {dj}},\) according to which \({\sc {dj}}\) is not a confirms_{ f }function (i.e., not a conditional prob function), but rather a confirms_{ i }–function. I’m inclined to think that that a proper Bayesian theory of \({\sc {dj}}\) (if there be such) will have to be sensitive to both firmness and increase in firmness considerations.
 12.
Note that we can make prob \((Na \mathbin{} Pa \mathbin{\&} K)\) as high as we like, just by fiddling with the numbers specified in K.
 13.
Pollock has made a lot of progress on the formal/computational side of his theory since the book was written. I have had the pleasure of reading a more recent manuscript (Pollock 2009), which develops the formal side in much more detail and generality. I have also benefited from a very edifying email correspondence with John about his quite extensive and impressive computational work on \({\bf{prob}}\), and its relation to my recent computational work on prob (Fitelson 2008).
 14.Here, I mean only to assume some uncontroversial direct inference principle from what we take to be the salient sorts of objective probabilities in the context at hand. One might object that the kinds of (statistical) probabilities at work in the present example aren’t nomic probabilities (in Pollock’s sense). But, one can strengthen the present (statistical) example by adapting it to a case in which one property is nomologically necessary for another. For instance, we could let Px x has stage one syphilis, and Nx x does not develop paresis (Scriven 1959, p. 480). I presume that the salient nomic probabilities in such an example would have the same sort of structure I have in mind for the simpler (statistical) case I am discussing here [and it would more clearly involve a case of (projectible) nomic probabilities].
 15.
Or, in the syphilis/paresis variation of the example (see footnote 14), that the presence of stage one syphilis in a patient is a (arbitrarily strong) reason to believe that the patient will not develop paresis.
Notes
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