Abstract
This paper examines the debate between permissive and impermissive forms of Bayesianism. It briefly discusses some considerations that might be offered by both sides of the debate, and then replies to some new arguments in favor of impermissivism offered by Roger White. First, it argues that White’s (Oxford studies in epistemology, vol 3. Oxford University Press, Oxford, pp 161–186, 2010) defense of Indifference Principles is unsuccessful. Second, it contends that White’s (Philos Perspect 19:445–459, 2005) arguments against permissive views do not succeed.
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Notes
Only virtually certain because in each case there could be measure 0 exceptions.
Although this tracks one use of the term “Bayesian”, this term has been used in a number of different ways. Some have taken Bayesians to only be committed to Probabilism or only committed to Conditionalization. Likewise, Probabilism and Conditionalization themselves have been understood in slightly different ways. For example, some have restricted Conditionalization to agents with probabilistic credences, and Probabilism to agents whose doxastic states are fine-tuned enough to admit of precise degrees.
The distinction between permissive and impermissive Bayesianism mirrors the distinction between subjective and objective Bayesianism. I’ve avoided the subjective/objective terminology because these terms have been used in a number of different ways. And while pretty much every way of making this distinction would classify impermissive Bayesianism as a form of objective Bayesianism, there’s little consensus beyond that with respect to where the line between subjective and objective Bayesianism should be drawn.
I’m assuming here a picture of evidence according to which agents start with no evidence. But nothing hangs on this—those inclined to favor a different picture of evidence can just understand my talk of “evidence” to mean “non-initial evidence”.
So we can ignore agents with infinite pasts or agents who exist during open intervals of time, and ignore cases where cr(A|E) is undefined.
i.e., the range of priors functions held by possible rational agents.
The diagram below should be understood as representing the range of Bayesian views, with different points in the space corresponding to different Bayesian accounts. (Impermissive Bayesianism is not a single point because there are many different versions of impermissive Bayesianism, which differ with respect to what they take the one permissible priors function to be.)
Note that the argument won’t go through if we don’t put aside cases with 0-credence evidence and agents without initial credences. If we don’t put aside such cases, the entailment will only go one way: given Bayesianism, Evidential Uniqueness will entail Agent Uniqueness and Permission Parity, but not vice versa.
Assuming, as in the case above, that the agent initially has a non-extremal credence in both H and E.
Since these objections rely on the plausibility of certain principled constraints on rational belief over and above those imposed by Bayesianism, these objections can be seen as a special case of the impermissive intuitions objections.
These principles have been given a number of different names, including “The Principle of Indifference”, “The Principle of Insufficient Reason”, “The Maximum Entropy Principle”, and “Jeffrey’s Rule”. [See Howson and Urbach (2005), and the references therein.]
Early defenders of Indifference Principles also appealed to empirical evidence to support these principles [see Jaynes (1983)]. For example, it was noted that the degrees of belief suggested by some Indifference Principles matched the real-world frequencies of various thermodynamic phenomena, and it was suggested that this gave us a reason to believe these Indifference Principles were true. But this is now widely recognized by both proponents and opponents of Indifference Principles to be a mistake [see North (2010), White (2010)].
Both of these cases come from White (2010).
(x, y] is the interval between x and y that includes y but not x.
For example, White (2010) takes the moral of the cube factory case to be that there will be some tricky cases, like the cube factory case, in which we won’t know how to apply the Indifference Principle. But, White maintains, this doesn’t make the Indifference Principle useless, as there are plenty of other cases, such as the three door and Monday/Tuesday cases, in which it is obvious how one should be indifferent [see White (2010), pp. 167–169]. I want to suggest, in these other cases, that it’s only obvious how to be indifferent in the same sense that it’s obvious, upon first being presented with the cube factory case, that you should be indifferent with respect to length. While it may at first seem obvious how we should be indifferent, further reflection should make our confidence in these verdicts evaporate.
Note that some assumptions of this kind are already assumed in the initial descriptions of the case. For example, the height and face area descriptions of the case together require space to have at least three dimensions, to be large enough to hold at least a 2 cm cube and a factory for making such cubes, to have features which yield the straightforward relationship between length and face area we’re accustomed to, and so on.
I assume here that the number of particles is finite, so that the number of degrees of freedom in the system is finite. I also assume here that the spatiotemporal extension of the world is finite. (Again, these simplifications make things easier for the proponent of indifference, since it allows them to avoid various infinity worries.)
For example, one might be indifferent in a manner that’s uniform with respect to something like inverse time, assessed with respect to the beginning of the universe. (E.g., let t = the number of seconds since the beginning of the universe, let \(\tau = \frac{1}{t+1}\), and require rational priors to be uniform with respect to τ. (This normalizes because we’re assuming the spatiotemporal extension of the world is finite; see footnote 21.) Since Tuesday is later than Monday, the difference in inverse time between the beginning of Tuesday and the end of Tuesday will be smaller than the corresponding interval for Monday, and Monday will be assigned a higher value.
For example, consider the spatial dimension along which the doors are arranged. Consider a configuration space in which n of the degrees of freedom correspond to the positions of each of the n particles along this spatial dimension, and where one is indifferent in a manner that’s uniform over something like inverse distance along this dimension, assessed with respect to some point to the left of the three doors. (E.g., let x = the number of meters away from the point along this spatial dimension in the relevant direction, let \(\xi = \frac{1}{|x|+1}\), and require rational priors to be uniform with respect to ξ.) Then the credence assigned to the prize being behind each door, as we go from left to right, will decrease.
Note that finding a carving that’s not arbitrary in some respect is not what is required to avoid the arbitrariness horn of the trilemma. What is required is to find a carving that makes the resulting principle epistemically non-arbitrary. One might make a case that some particular carving is more natural than the others in some salient respect—perhaps it lines up with the perfectly natural properties, for example [see Schaffer (2007), Sider (2011)]. But the existence of such a carving doesn’t by itself give us any reason to think that an Indifference Principle should employ it. After all, the existence of carvings that are non-arbitrary (in this sense) is compatible with there being no constraints on rational belief at all.
Of course, a proponent of Indifference Principles who did not appeal to indifference intuitions to motivate the adoption of these principles would not be subject to the second and third worries given above. Instead, they would face the challenge of finding some other compelling reason for adopting an Indifference Principle.
As we will see, it is unclear whether White intends for his principle to be a strong Indifference Principle, and thus unclear whether he takes it to be a principle one could use to support Evidential Uniqueness. Given the understanding of White I’ll suggest, White is neutral with respect to whether his principle is a strong Indifference Principle or not (see Sect. 4.1). But since, as I understand him, he is amenable to it being a strong Indifference Principle, we will need to examine it in order to see whether he has found a way to defend a strong Indifference Principle from the standard objections.
See White (2010), p. 161. White also offers another characterization of evidential symmetry, according to which A and B are evidentially symmetric “for a subject if his evidence no more supports one than the other” [White (2010), p. 161]. I’ve employed White’s “reasons” characterization instead of this one for two reasons. First, if we employ the natural Bayesian understanding of “evidential support” (c.f. Sect. 5.1), White’s principle becomes vacuous. Second, there’s reason to think White is employing a non-standard notion of evidence here (and a fortiriori, a non-standard notion of evidential support), making the content of this characterization unclear. White states that “I mean to understand evidence very broadly here to encompass whatever we have to go on in forming an opinion about the matter. This can include non-empirical ‘evidence’, if there is such” [White (2010), pp. 161–162]. If an agent is deciding what her credence in A should be, then her certainty that the chance of A is 1, her lack of inadmissible evidence with respect to that chance, and the appropriate Chance–Credence Principle are presumably all part of what she “has to go on”. Thus her certainty in the chance, her lack of inadmissible evidence, and the Chance–Credence Principle, appear to all (either singly or jointly) count as evidence in White’s sense, even though they will not all count as evidence in the standard Bayesian sense. In any case, which characterization of evidential symmetry we employ has little bearing on what follows. The dialectic proceeds in precisely the same way if we employ this other characterization of evidential symmetry. Just replace all talk of when an agent has no more reason to suppose A than B with talk of when what an agent “has to go on” no more supports A than B.
Strictly speaking, this argument also assumes that one’s rational credence function cr must satisfy the probability axioms, that the L i s and A i s each form a partition of the doxastic possibilities, and that L 1 and A 1 are equivalent propositions, and so must be assigned the same credence. I follow White in leaving these premises implicit, since both sides will grant these assumptions.
In addition to the assumptions mentioned in the previous footnote, this argument assumes that L 2 and A 2 ∨ A 3 ∨ A 4 are equivalent propositions, and that the evidential symmetry relation is transitive.
White (2010), p. 166.
Likewise, as one would expect, premises (1) and (2) are true given Claim 1, and so the Cube Factory Argument against White’s Indifference Principle succeeds. The highest credence a rational agent could assign L 1 and L 2 is the same—1—so (1) is true. Likewise, the highest credence a rational agent could assign A 1 − A 4 is the same, so (2) is true.
This understanding is suggested by White’s comments on p. 168 of White (2010), and the passage quoted in footnote 35, below.
This point should not be understood as a criticism of White, since it’s unclear how strong White (2010) takes his Indifference Principle to be. For example, on the modest understanding of White I’ve suggested, White is officially neutral about whether his principle eliminates all but one priors function (i.e., is a strong Indifference Principle), eliminates all but a restricted set of priors functions, or eliminates no priors functions at all. (Of course, since he is amenable to it being a strong Indifference Principle, we need to assess it anyway, in order to find out whether he has found a way to defend (what is potentially) a strong Indifference Principle from the standard objections.) In any case, note that the worries for White’s defense discussed in Sects. 4.1 and 4.2 arise regardless of whether we take it to be a strong Indifference Principle or not.
White briefly discusses the trilemma in the following passage:
I suspect that many who are hostile to POI [the Principle of Indifference] view it as trying to do something clearly misguided: taking purely structural features of a space of possibilities as giving conditions on rational credence. The trouble is that there are different structures we can impose on a space. We need something more to tell us which way to cut the pie to get a unique answer. If nothing further is specified our criterion is empty. If all carvings are allowed we get inconsistency. If further criteria are imposed they often seem arbitrary or unmotivated. … POI as I’m understanding it is importantly different. It takes an epistemic input (“having no more reason…”) to deliver an epistemic output (equal credence). This is not open to the same charge of arbitrariness. It is appropriate that facts about the balance of my reasons should put constraints on my credal states. (White (2010), p. 168.)
But again, this claim does not seem sufficient. Without making substantive claims about when an agent has no more reason to suppose A than B, one can’t show that the principle avoids the inconsistency horn, the triviality horn, or the arbitrariness horn.
See Kelly (2012) for a discussion of some of the other arguments that White offers.
See White (2005), p. 458, footnote 12.
For example, one version of the first premise of the Evidential Support Argument (\(\hbox{P}1^{\prime}\) of Sect. 5.1), the first premise of the Flip-Flopping Argument (P1 of Sect. 5.2), the first premise of the Truth-Guiding Argument (P1 of Sect. 5.3), and the first premise of the Practical Deliberation Argument (P1 of Sect. 5.4), would be false if not restricted to some version of permissive Bayesianism that rejects Agent Uniqueness. And the second premise of the Truth-Guiding Argument (P2 of Sect. 5.3) must be restricted to views which accept Permission Parity in order to not be subject to obvious counterexamples
White (2005), p. 447.
Kelly (2012) raises similar worries.
Christensen (2007) seems to offer a similar argument: “There’s something unstable about holding onto my belief while acknowledging that a different belief enjoys equal support from the evidence.” [Christensen (2007), p. 191.] Why unstable? Well, if we assume that the total evidential support relation has only two changeable arguments—one’s total evidence and the proposition in question—then either my total evidence supports A or it doesn’t. And if I believe A while believing that my total evidence supports \(\neg A, \) then I don’t believe what I think I should. In this sense, my beliefs are unstable. The problem with this line of thought is that by assuming the priors argument of the total evidential support relation is fixed, it effectively assumes Evidential Uniqueness from the outset.
I thank a referee for suggesting this understanding.
White (2005), pp. 449–450. Similar considerations are raised in other parts of the paper, such as p. 453 and pp. 454–455.
The argument presented here is somewhat simpler than the argument suggested in this passage. Unlike the argument suggested in the passage, I’ve formulated the argument in terms of what it’s rational to believe, not what one believes it’s rational to believe. One could provide a more accurate reconstruction of White’s argument by formulating the argument in terms of what one believes to be permissible, and adding the premise that in order for an account of epistemic norms to be correct, it must be permissible to believe it. But this merely serves to complicate the argument, and this more complex version of the argument runs into the same difficulties as the simpler argument presented in the text.
What if we instead considered permissive Bayesians who do accept Agent Uniqueness, like Permissive3 Bayesians? Then P1 would be false, so the argument would be unsound.
Thanks to Jonathan Weisberg here for suggesting this understanding.
E.g., a permissive Bayesian might hold that it’s permissible to have a credence of either x or y in A in a tabula rasa case, but also hold that these initial credences are only permissible if one came to have one’s initial credences via the right kind of process.
White (2005), p. 448. White raises similar considerations in other parts of the paper, such as p. 449, pp. 451–452, and p. 457.
See Christensen (1996).
Of course, it’s contentious whether ordinary coin tosses are chancy [see Lewis (1986)]. And among those who think they are chancy, it is contentious whether these chances should be thought of as time-dependent [see Meacham (2005), Hoefer (2007)]. To get around these worries, we could replace the coin toss with the spin measurement of an electron, given an indeterministic interpretation of quantum mechanics like GRW. Likewise, one could restrict the scope of the calibration perspective described below to indeterministic time-dependent chances. (Alternatively, one could generalize the calibration perspective by formulating it in terms of something like conditional chances.)
Note that this question is independent of what evidence one has. We’re considering what epistemic states it’s epistemically better to be in given what the world is actually like, full stop. (Of course, one might flat out reject this kind of “epistemic axiology”. If so, one would be unhappy with both the calibration and the accuracy perspectives described below.)
Of course, Bayesians generally employ a highly idealized conception of the agents in question. So this desiderata might motivate other changes to the standard Bayesian account as well.
Given these notions of truth-guiding, the argument is unsound because P2—that a Permission Parity-satisfying account of epistemic norms cannot be truth-guiding if it takes it to be permissible in some situations to have any one of several doxastic states—is false.
See Hawthorne (2012), and the references therein.
Given these notions of truth-guiding, the argument seems unsound because P3—that the correct account of epistemic norms is truth-guiding—seems false.
White (2005), p. 448.
One might come up with cases in which one’s evidence does fix what the world is like—one might be a deity, say, who receives as veridical evidence the conjunction of every true proposition. But in these cases, Bayesians of any stripe will prescribe true beliefs, so the impermissive Bayesian gains no ground on the Permissive1 Bayesian.
North (2010) raises similar worries.
A similar issue arises with respect to how to weigh the value of prescribing a true belief to one agent at a world versus another.
White (2010), p. 453–454. White raises similar considerations in Sect. 6 of his paper. (The version presented in Sect. 6 is cashed out in terms of what one believes it’s rational to believe, not what it’s actually rational to believe, and has the conclusion that one shouldn’t believe impermissivism is true, not that impermissivism isn’t true. One can turn this into an argument against impermissivism by adding the premise that it must be permissible to believe the correct account of epistemic norms. In any case, this complication doesn’t change the status of the argument; both versions run into the same kinds of difficulties.)
See Zimmerman (2006).
For example, one might consider the objective notion of obligation, which makes an act permissible p iff it maximizes utility. Or one might consider a notion of prudential obligation which makes an act permissible p iff it maximizes expected utility with respect to either one’s actual credences or some permissible e credences. Or one might consider a notion of prudential obligation which makes it’s obligatory p to perform an act that maximizes expected utility with respect to one’s actual credences if they’re permissible e ; and otherwise makes it permissible p to perform an act which maximizes expected utility with respect to any permissible e credence function. But none of these notions will make both P2 and P3 true. (The first makes both P2 and P3 false, the second makes P2 false, and the third makes P3 false.)
It’s worth noting that an impermissive Bayesian could reject the claim that her fortunes are tied to those of Indifference Principles, and instead hold that there are a number of heterogeneous constraints on rational priors, the combination of which leads to a unique permissible prior. This position would avoid many of the problems surrounding Indifference Principles.
For helpful comments and discussion, I’d like to thank Phil Bricker, Maya Eddon, Darin Harootunian, Sophie Horowitz, Tom Kelly, Miriam Schoenfield, Katia Vavova, Jonathan Vogel, Jonathan Weisberg, Roger White, the Northwest Time and Rationality conference, the MIT epistemology reading group, and two anonymous referees.
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Meacham, C.J.G. Impermissive Bayesianism. Erkenn 79 (Suppl 6), 1185–1217 (2014). https://doi.org/10.1007/s10670-013-9547-z
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DOI: https://doi.org/10.1007/s10670-013-9547-z