Abstract
In “Generalized Conditionalization and the Sleeping Beauty Problem,” Anna Mahtani and I offer a new argument for thirdism that relies on what we call “generalized conditionalization.” Generalized conditionalization goes beyond conventional conditionalization in two respects: first, by sometimes deploying a space of synchronic, essentially temporal, candidate-possibilities that are not “prior” possibilities; and second, by allowing for the use of preliminary probabilities that arise by first bracketing, and then conditionalizing upon, “old evidence.” In “Beauty and Conditionalization: Reply to Horgan and Mahtani,” Joel Pust replies to the Horgan/Mahtani argument, raising several objections. In my view his objections do not undermine the argument, but they do reveal a need to provide several further elaborations of it—elaborations that I think are independently plausible. In this paper I will address his objections, by providing the elaborations that I think they prompt.
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Notes
Here and throughout, I use the expression ‘epistemic probability’ rather than the popular term ‘credence’. This is because ‘credence’ is usually glossed either as degree of belief or as rational degree of belief, and I believe that epistemic probabilities are not degrees of belief. (I also believe that there are no such psychological states as degrees of belief.) Nothing in this paper turns on these claims, but my believing them does motivate me to eschew the term ‘credence’.
This reply to Pust’s objection is neutral about the nature of epistemic probability. In particular, it is consistent with the widely held view that epistemic probability is so-called “credence”—i.e., quantitative degree of belief (or quantitative rational degree of belief), on a zero-to-one scale. If epistemic probability is construed this way, then one’s own preliminary probabilities are naturally thought of as the degrees of belief that would be possessed by a perfectly rational agent whose pertinent evidence exactly matches the pertinent unbracketed evidence that one possesses oneself. My point is that such an agent need not be oneself. In an earlier essay (Horgan 2008), I reply to Pust’s objection a different way: I appeal to my own preferred construal of epistemic probability as quantitative degree of evidential support. On that construal, I argue, it is just irrelevant that Beauty could not be in an epistemic situation in which her total relevant evidence coincides with the unbracketed portion of her actual relevant evidence.
Pust (2012) argues that this claim is a consequence of each of the three major accounts of indexical thought in the philosophical literature. In a similar vein, Pust (2013) argues that the above-described argument for the 1/3 answer to the Sleeping Beauty problem cannot be vindicated by construing epistemic probability (as I do) as quantitative degree of evidential support, because “the most plausible account of quantitative degree of support, when conjoined with any of the three major accounts of indexical thought in such a way as to plausibly constrain rational credence, contradicts essential elements of Horgan’s argument” (p. 1489). Addressing these arguments in detail would take a paper in itself. But, at least as regards the argument in Pust (2013), I think the upshot would be the following. Any adequate account of indexical thought must respect the fact that ‘believes that…’ contexts and epistemic-probability contexts both are intensional. In particular, if two co-referring time-denoting terms are such that one of them is temporally indexical and the other is not, then no adequate account of indexical thought can entail that the two terms are inter-substitutable salva veritate within belief contexts and within epistemic-probability contexts. But I think that the argument in Pust (2013) implicitly assumes that the evidential-support relation is not intensional in this way, and instead allows such substitutions. This I would strongly deny. (For more on the intensionality of epistemic probability, with application to the two-envelope paradox, see Horgan 2000.).
The value and importance of structuring the argument into five distinct stages, and of adding the present section’s subsequent commentary on the argument as so structured, became clear to me from an anonymous referee’s remarks on an earlier draft of the present paper.
It would be better to say that Beauty assigns these preliminary probabilities on the basis of the combination of (i) the strong symmetry of the hierarchical partition-structure, and (ii) what she knows about the objective chances of HEADS and TAILS. Roughly, she knows that HEADS and TAILS each have an objective chance of ½. More accurately, (a) she knows that HEADS and TAILS each have an objective chance very close to ½, and (b) her evidence is indifferent about whether, if the chances are slightly different, the chance of HEADS is slightly greater than the chance of TAILS or vice versa. (For simplicity, hereafter I will ignore the more accurate fact and I will acquiesce in the commonly made assumption that the known chance of an ordinary coin coming up heads when flipped is ½, and likewise for its coming up tails.) I return to this theme in Sect. 4 below.
I am using the locution ‘essentially indexical possibility’, rather than the recently popular locution ‘centered possibility’, because so-called centered possibilities are often described as though they are metaphysical possibilities with “designated centers.” (If one also conceives of a metaphysical possibility as a set of metaphysically possible worlds, then the corresponding centered possibility will be a set that results from the first one by designating one and the same center for each of the metaphysically possible worlds in the original set. I.e., it will be a set of (metaphysically-possible-and-with-designated-center) worlds, all with the same designated center.) It seems to me, however, that a temporally indexical self-locational term like ‘today’ should be viewed as a context-dependent rigid designator: on the day of its usage, it rigidly designates that very day and no other. Thus, on the day of its usage, there are no metaphysically possible worlds in which today is any other day than the day currently rigidly designated by ‘today’. So the essentially indexical possibilities in question in the Sleeping Beauty problem cannot all be rightly regarded as being metaphysical-possibilities-with-designated-centers. Rather, they are epistemic possibilities not all of which are metaphysically possible. Nothing in the current paper turns on these claims, but my believing them does motivate me to eschew talk of “centered possibilities.”
The recipe just described is readily adaptable to various other puzzles in the philosophical literature about probability and essentially indexical possibilities—for instance, to the Vishnu/Brahma self-duplication scenarios discussed in section V of Arntzenius (2003). Halfers about the Sleeping Beauty problem are apt to balk at some of Arntzenius’s claims about the epistemic probabilities that accrue to the outcomes of coin flips in these scenarios; the Horgan–Mahtani recipe can be deployed to bolster those claims.
Although I would prefer to formulate it in terms of essentially indexical possibilities and non-indexical possibilities. Cf. note 6.
And in any event, a proposed definition would have to be tested for adequacy by assessing it for conformity with pre-theoretic intuitive judgments about scenarios that do—and scenarios that do not—exhibit the kind of evidential symmetry that safely makes for indifference-based assignment of probabilities. Proposed theoretical definitions, for philosophically interesting concepts that figure in pre-theoretical reasoning, are accepted or rejected on abductive grounds—with the data for abduction consisting largely of people’s pre-theoretic intuitive judgments about the applicability, or non-applicability, of the pertinent concept in concrete scenarios. In this connection, a referee points out that Carnap, in seeking to develop a systematic theory of confirmation, initially considered a confirmation function c-dagger that was strongly symmetrical, but later gave it up because it did not vindicate what he called “learning from experience.” The confirmation functions he subsequently employed—the function c*, and more complicated successors—did not exhibit strong symmetry. The referee remarks that “allowing for learning from experience is exactly the kind of conformity with pre-theoretic intuitive judgments that the author should be looking for.” By way of brief response, let me say the following. First, I certainly do not mean to suggest that evidential indifference is always grounded in strong symmetry—or even that the only evidential-indifference considerations operative in the Horgan-Mahtani argument are considerations of strong symmetry. (Step 4 invokes the further indifference claim I labeled NDMD.) Second, unlike Carnap, who ambitiously sought a systematic confirmation theory that would assign epistemic probabilities to virtually any proposition an agent might consider, I myself believe that principled epistemic probabilities arise only very rarely, and only under quite special circumstances—e.g., when one knows the objective chance of a proposition and one can rightly treat this known chance as its epistemic probability, or when one confronts a partition of n possibilities (for known n) over which one’s evidence is indifferent (in which case each cell in the partition has epistemic probability 1/n). For the most part, learning from experience is not a phenomenon that justifies the assignment of specific epistemic probabilities at all. Rather, normally such information-gain only justifies certain kinds of qualitiative judgments of epistemic likelihood—e.g., qualitative non-comparative judgments of the form “Proposition p is highly probable,” “Proposition p is improbable,” etc., and qualitative comparative judgments of the form “Proposition p is more probable than proposition q,” “Propositions p and q are equally probable,” etc. So in my view, the principal kinds of intuitive pre-theoretic judgments about the evidential import of symmetry that are pertinent to epistemic probability are intuitive judgments that arise in those relatively rare circumstances in which there is a principled basis for assigning any epistemic probabilities at all.
On some construals of chance, a possible outcome that initially has a chance other than either one or zero takes on either a chance of 1 upon occurring or a chance of 0 upon failing to occur. But the epistemic probability of an outcome can remain equal to its “known chance” even if one knows that by now the outcome has either come about or failed to come about—provided that one’s evidence is indifferent about the outcome.
It might be argued, furthermore, that a rational agent in the envisioned epistemic situation could not have a conditional probability for S that is conditional on any one of these latter statements anyhow, because the fraction-formula expressing such a putative conditional probability would have a denominator that is equal to 0, and hence this fraction-formula would be undefined. (Pust himself argues that way, in the final sentence of the lately quoted passage.) However, Hajek (2003) makes a strong case against treating the familiar ratio formula P(A|B) = P(A&B)/P(B) as a definition of the notion of conditional probability (which is what Kolmogorov did in his classic axiomatization of probability theory), and Hajek also makes a strong case for the meaningfulness of certain conditional probabilities where the condition-statement has probability zero. But I do not think that Hajek’s arguments are directly pertinent here, because (as just argued in the text) there is independent motivation for saying that Beauty should apply generalized conditionalization by invoking preliminary conditional probabilities involving R1 as the condition-statement—a statement whose preliminary probability is ¼—rather trying to apply generalized conditionalization by invoking putative preliminary conditional probabilities involving [R1 & (Pnow(R1) = 1)] as the condition-statement—a statement whose preliminary probability is zero.
References
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Acknowledgments
For helpful comments and/discussion, I thank Don Fallis, Richard Healey, Chris Howard, Justin Lillge, Mark Timmons, Alex von Stein, Sarah Wright, and especially Joel Pust.
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Appendices
Appendix 1: Strongly Symmetrical Hierarchical Partition Structures
I will propose a definition of strong symmetry for hierarchical partition structures, and will comment on its applicability to the partition structure in Table 1 above. First, let a hierarchical partition structure, relative to a space of epistemic possibilities and fully distinct partitions P1 P2,…Pn of that space, be a set {X1, X2, …, Xn} of sets of cells (respectively, the set of first-level cells, the set of second-level cells, …, the set of nth-level cells) such that n > 1 and
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1.
for each i such that 1 ≤ i < n,
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a.
each cell in Xi is filled by exactly one member of a partition Pi,
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b.
each cell in Xi has one or more sub-cells in Xi+1,
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c.
for each cell in Xi and each element e in partition Pi+1, e fills at most one sub-cell of Xi,
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d.
every sub-cell of Xi is in Xi+1, and
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a.
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2.
each cell in Xn has no sub-cells and is filled by exactly one member of partition Pn.
I suggest the following definition. A hierarchical partition structure {X1, X2, …, Xn} is strongly symmetrical just in case:
for each i such that 1 ≤ i < n,
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1.
every cell in Xi has exactly as many sub-cells as every other cell in Xi,
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2.
every member of partition Pi occurs exactly as many times within cells in Xi as does every other member of Pi,
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3.
for every two members z and w of the partition Pi, the number of cells in Xi−1 containing a sub-cell of Xi in which z occurs is the same as the number of cells in Xi+1 containing a sub-cell of Xi in which w occurs, and
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4.
if xi−1,1, xi−l,2, …, xi−1,r are the successive cells in Xi−1 and Yi,1, Yi,2, …, Yi,r are the successive sets of sub-cells in Xi of xi−1,1, xi−l,2, …, xi−1,r respectively, then each member of the partition Pi occurs within exactly the same number of sets from {Yi,1, Yi,2, …, Yi,r} as does every other member of Pi.
The partition structure shared by Tables 1, 2, 3 and 6 above satisfies this definition of strong symmetry (whereas the partition structures in the other Tables above do not). Clause 1 is satisfied by the second level of the partition structure, because each first-level cell has the same number of sub-cells (viz., 2), and each second-level cell has the same number of sub-cells (viz., 3). Clause 2 is satisfied because MON and TUES both occur the same number of times within the second level (viz., 2), and each of R1, R2, R3, and R4 occurs the same number of times within the third level (viz., 3). And clause 4 is satisfied because MON and TUES both occur in the same number of sub-cell sets attached to level-1 cells (viz., 2 such sub-cell sets), and each of R1, R2, R3, and R4 occurs within the same number of sub-cell sets attached to level-2 cells (viz., 3 such sub-cell sets).
Appendix 2: Full Symmetry, Symmetrical Symmetry Breaking, and Strong Symmetry
Let a downward path through an n-level hierarchical partition structure be a set consisting of an element of partition P1 that fills some level-1 cell in that structure, plus an element of partition P2 that fills some level-2 sub-cell of that level-1 cell, plus …, plus an element that fills some level-n sub-cell of that level-(n-1) cell. A hierarchical partition structure is fully symmetrical just in case the following holds: for every combination of one element each from P1, P2, …, Pn, there is a corresponding downward path through the structure.
The hierarchical partition structure in Table 1 above is not fully symmetrical; rather, it contains only 12 downward paths, whereas a fully symmetrical one would have 16 downward paths—one for each combination of one element each from {HEADS, TAILS}, {MON, TUES}, and {R1, R2, R3, R4}. Table 1 thus exhibits a certain kind of symmetry breaking—a breaking of full symmetry. Nonetheless, the manner in which full symmetry is broken is itself symmetrical, in the following way: each of the 4 level-2 possibilities has as sub-possibilities exactly 3 elements of {R1, R2, R3, R4}, and each of the four elements of {R1, R2, R3, R4} is excluded exactly once as a sub-possibility of some level-2 possibility. Although strong symmetry is a more inclusive category than full symmetry, a hallmark of strong symmetry is that it can only deviate from full symmetry by exhibiting a form of symmetry breaking that is itself symmetrical. Strong symmetry—either full symmetry or symmetrically broken full symmetry—is the feature of a hierarchical partition structure that makes for epistemic safety both (a) in equating epistemic probabilities with known chances, and (b) in assigning certain other epistemic probabilities on grounds of symmetry-based evidential indifference.
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Horgan, T. Generalized Conditionalization and the Sleeping Beauty Problem, II. Erkenn 80, 811–839 (2015). https://doi.org/10.1007/s10670-014-9675-0
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DOI: https://doi.org/10.1007/s10670-014-9675-0