At first sight there is no logical inconsistency in the Bayesian agent’s self-assessment, not even with an acknowledgement of everything that can be shown about the failure set of his forecasting. The Bayesian agent assigns probability zero to the failure set while Schervish points out that it is non-denumerably infinite and Belot shows that it is residual. A set of possibilities that is immensely large by measure-independent criteria can have measure zero.
However, this ignores the larger character of probabilism (including Bayesianism), which begins with the view that our opinion is constituted by our probability judgments. We have to imagine the Bayesian who knows all the theorems we have discussed, and must say how he views Belot’s results. How can he do this? What he says must be something of this form:
(1) I am sure that A, but it may not be the case that A
where A says that his own opinion has perfect calibration.
It would not do at all to take that “may” as standing for metaphysical or logical possibility. Belot’s results are about how the actual, presently unknown, future may be, the future that the agent is in the process of forecasting. So (1) would not do at all as a response if it meant that in some other possible world, not this one, A is false.
Therefore (1) must mean
(2) I am sure that A, but for all I know or believe it is not the case that A
But does being sure not mean, or even imply, believing? If it does, as we would naturally take it to do, (2) means or at least implies
(3) I fully believe that A, but for all I know or believe it is not the case that A
which is a rank contradiction.
(With the usual form of semantics, what I fully believe is what is true in all my doxastically possible worlds, and what is the case for all I know or believe is true in at least one of my doxastically possible worlds. On this reading (3) it is certainly an outright self-contradiction.)
I will refer to the above as the Opening Argument, and come back to it later.
It means at least that the logical point, about how large sets can have measure zero, does not lay the issue to rest. The Bayesian could stonewall? Not when living in a world with others …. the total certainty that you are right becomes very uncomfortable when met with others’ incredulous stare!
So I take it that Dawid et al. were right, the Bayesian agent faces a paradox. Paradoxes, however, famously fail to bring a position to its knees. As in other encounters with paradox, we should look into two questions:
(I) What kind of paradox is this? and (II) What would it take to solve, dissolve, or resolve this paradox?
About Question (I): What Kind of Paradox?
For question (I), I submit that the Bayesian’s orgulity dilemma is akin to the Preface Paradox. Bayesian self-assessment is not a case of arrogance, however arrogant it looks at first blush. For anyone, asked for any self-assessment, is in a bind.
This was first brought out in by Makinson, though in a rather naive form. In his example, an author writes a book, and sits down to provide it with a preface. Makinson gives a actual example in the literature, of an author’s bow to humility, with reference to counsel from his colleagues: “the errors and shortcomings to be found herein are not their fault”. At this point, preface plus main content form an inconsistent whole …
There is no excuse for inconsistency. This author went too far in his attempt to be humble. The example becomes more interesting if the author writes “I have taken all care to include only truths, and to be perfectly accurate, but in view of evidence from all previous attempts, I must believe that this too contains errors”. That does not make the book as a whole inconsistent, and yet there is a glaring problem nevertheless. This problem is clearest if the contents of the book is a single statement. In that case, preface + content form an instance of Moore’s Paradox. The point is general, for we can regard the content of any book or factual report as logically equivalent to the conjunction of all the statements it includes. The Preface Paradox is a Moore Paradox.
If the author is constrained to include a self-assessment in the preface, then inconsistency and pragmatic incoherence are avoided only by asserting: “all that follows is true”. How can we call it arrogance, given that it is forced by the need to avoid something much worse than the appearance of arrogance?
I submit that the case of a probabilist agent who has to offer a self-assessment is no different. Challenged to a self-assessment, the requirements of pragmatic coherence dictate that she must see and report her opinion and updating practice to be well-calibrated, e basta!
This is my contention: once we pay attention to the requirements of pragmatic coherence, the Bayesian’s self-assessment is cleared of the charge of orgulity.
But that does not yet mean that the Bayesian is out of the woods. For what I called above the Opening Argument trades on how the central concept of Bayesian epistemology, subjective probability, is to be understood — and it ended in a self-contradiction.
About Question (II): How to Resolve this Paradox?
I used the phrase “Challenged to self-assessment”. As in Moore paradox-type cases generally, we are not dealing with mere fact-stating language, nor with the sort of autobiographical attribution that is in every way on a par with an ostensibly equivalent third-person’s attribution to oneself. The request for a self-assessment is a demand for endorsement, it amounts to the question “Do you stand behind the report you presented, the opinions you expressed? The unspoken accompanying demand is this: “If you cannot endorse them, take them back!”
Returning now to the Opening Argument, how can we mobilize this distinction in order to understand how we can stand behind our judgments, yet with eyes wide open?
When a paradox is presented as a deduction leading to a self-contradiction, there are always ostensible ways out, however painful. In this case, the first would be for the Bayesian agent to stone-wall, sticking by his self-assessment alone: “No, it is not true that for all I know or believe it is not the case that A! On the contrary, I believe that A and it is so.”
Well, stone-walling we have seen as option from the beginning. A somewhat more sophisticated option would be to resist the move from (2) to its rendering (3), based on how subjective probability 1 was equated with or implied by full belief. In the literature on ‘credence’ of the past decade, it is not uncommon to find a ‘pluralist’ approach, in which belief and subjective probability are treated as related yet not identifiable at any level or in any fashion. Lara Buchak provides an original argument in Philosophical Studies 2014; Dan Greco offers an extended critical appraisal of this literature in Philosophical Perspectives 2015. And so we can imagine that the rejoinder could be: “My subjective probability for A equals 1, but it is also true that for all I know or believe it may not be the case that A; knowledge and belief are distinctly different propositional attitudes”.
It does not seem to me that this is a way out, simply because the Preface Paradox pertains equally to each notion of credence, opinion, or belief. Yet there may be a clue in this attempt to separate two doxastic modes, one to express the Bayesian’s ‘orgulous’ self-assessment, and the other to acknowledges the size of the failure set. The former, when made explicit, is clearly expressed in the agent’s subjective probability. What about the latter?
Radicalizing radical probabilism
It is indeed in some such separation that I see the crux of the solution. If the self-assessment is the response to a challenge, the demand to ‘stand behind’ one’s opinions, to ‘have the courage of’ ones opinions, then it is to be viewed quite differently from an autobiographical statement of fact about e. g. one’s propensity to bet one way or another.
I take this apparent paradox that we have been discussing under the heading of ‘orgulity’ to be a clear case supporting the claim that subjective probability judgments are to be regarded very differently from autobiographical statements of fact.
It is surely common sense that I can stand behind my opinions, that I can take such a doxastic stand, while fully cognizant of my fallibility in matters epistemic. The immediate parallel is to judgments expressing intention. Imagine your friend who has been drinking heavily but now seems to be coming to his senses, who then says: “I’m swearing it off for good!” This expresses a firm intention, does it not? But what if he then elaborates:
(*) I’m swearing it off for good, but it does seem likely that I’ll be drunk again next weekend
What is wrong with this? As onlooker I may well think that a moment before he expressed a genuine intention, and also agree to the second conjunct of (*). What I cannot do, now, though, is to hear the first conjunct as an expression of intention, and indeed, I will be hard put now to take it as true.
Nevertheless, there is no doubt that we can form firm intentions, with eyes wide open, not blind to the possibilities of failing to hold them.
So I submit that the separation needed between belief and subjective probability is not a separation between two doxastic attitudes with separate but equal standing. It is a separation between two activities, fact stating and opinion expressing, akin to intention expressing. Confusingly, each of which can be made explicit using precisely the same words (whether “believe” or “seem likely”).
Thus the Bayesian agent can, as his integrity as epistemic agent requires, express the self-assessment which signals that he has the courage of his current opinions, while equally ready to state the fact to which Belot has so saliently drawn attention.
De Finetti proposed to revolutionize the theory of probability, and certainly, he did. Here is the manifesto in his Theory of Probability:
“My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this:
PROBABILITY DOES NOT EXIST
The abandonment of superstitious beliefs about the existence of the Phlogiston, the Cosmic Ether, Absolute Space and Time, . . . or Fairies and Witches was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.”
We must go De Finetti one better, we need to extend this revolution, by recognizing that the subjectivity of our probability judgements has two faces. Just as with intentions, desires, values we can indeed objectively describe a subject’s states, as facts about them. In certain contexts, we can ‘step back from ourselves’, for example on a therapist’s couch, and trot out those facts about ourselves without avowal or endorsement. (“I am leaning more and more to naturalism and materialism, I wonder if it is due to a dietary deficiency?”).
But in the course of practical life the judgements in which we express our desires, values, intentions, commitments, and subjective probabilities play an inalienable, inexpungible part of our practical life, with our integrity at stake. There, despite the use of the very same words, these judgements are not autobiographical attributions but avowals.
NOTE: I argued for this at the end of my “Belief and the Will” (1984), against the contention that the Reflection Principle was a case of intellectual or doxastic arrogance. So for me it is an old theme, and connected with other reflections on Moore paradox-like issues, but I have found little sympathy for it … 🙂
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