Abstract
This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines the various options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis (1976).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It follows that S ∧ S* (i.e. S and S is indefinite) has a non-zero probability of being true, as in general p(A ∧ B) must be non-zero so long as p(A) and p(B) are greater than 1/2. Justification: p(A) + p(B) − p(A ∧ B) ≤ p(A ∨ B). (In classical probability theory, we have the LHS = RHS. But certain non-classical treatments of probability will be relevant later, on which the equality fails but the inequality here still holds). The RHS will be ≤1. So rearranging we have: p(A) + p(B) − 1 ≤ p(A ∧ B).
In the case at hand p(A) + p(B) > 1. Hence, it follows from the above that the conjunction is non-zero. Notice also that if p(A) and p(B) tend to 1, p(A ∧ B) will tend to 1—this will be appealed to in the extended argument to be given shortly.
I’m taking the chances here to be of the sort that quite generally is ascribed to macroscopic processes by statistical mechanics. See Albert (2001) for discussion of the philosophical issues in play here. If one wished to do so, surely one could specify a case where some radioactive decay process was involved in selecting the card, bringing more fundamental physical chances into play.
Typically, we will be able to derive an approximate version of our results. Some cases of the generalized result may be given up altogether: but the approximate results that remain are enough to do damage.
This follows from the argument given in footnote 1.
Evans and Over (2004, pp. 134–138). Igor Douven, in unpublished work, has further explored these issues. He finds that—for a wide variety of cases—there is a strong correlation between conditional probabilities and the probabilities of conditionals.
Some caveats. A high correlation isn’t the same as there being equality or even approximate equality between conditional probabilities and folk probability judgements about conditionals (a fact that Douven emphasizes). But we are not looking to the data to confirm the account in every detail here, it seems to me—we should rather be asking: given that this is the data, what is the best explanation of it? It is hard to imagine what a theory would look like that would explain the strong correlation between folk judgements of conditional probabilities and the probabilities of conditionals, without in some way associating the conditional probability with the conditional statement.
Evans and Over’s work also looked for correlations between judgements of the probability of conditionals and other hypotheses about what the probability should be. (E.g. that the probability of conditionals such as the one with which we are concerned should be the same as the probability of the conjunction of antecedent and consequent. They found some correlation here, but by no means as strong as the correlation with the conditional probability.)
Bennett (2003) is an excellent starting point.
The Kratzer story gives an elegant account of locutions like: ‘if you crash the car and kill Harry, you may crash the car and kill Suzy’. The Kratzer reading would be: (There is a permissible situation x: in x you crash the car and kill Harry)(You crash the car and kill Suzy). This renders the sentence false, as it should (crashing with fatal results isn’t permissible). The same reading is offered for: ‘you may crash and kill Suzy if you crash and kill Harry’.
But notice what happens when we parallel the somewhat artificial locution above: ‘it is permissible for you to make true the following sentence: if you crash the car and kill Harry, you crash the car and kill Suzy’. This seems fine; just by giving Harry and Suzy a lift and driving safely you can make that true. (I owe this example to Brian Weatherson, who used it in a slightly different context.)
Similarly, ‘if you’re male and unmarried, you’re necessarily a bachelor’ is naturally read as false (your bachelorhood isn’t essential to you). But famously there’s an alternative reading of such locutions, which we communicate to first year philosophy students, seemingly unambiguously, by ‘The following sentence is necessarily true: if you’re male and unmarried, you’re a bachelor’. It would be deeply mysterious if probability operators alone among modals didn’t pattern this way.
To illustrate the problems: if the ‘highly likely’ modal is analyzed as having narrowest scope, the above sentence would effectively be asking about whether on each draw, the probability on that occasion of diamond given red is high. But in the case just mentioned the latter is true, whereas as mentioned the sentence is intuitively false.
There are alternatives to this ‘narrow scope’ analysis. But—as far as I can see—they have their own problems. Perhaps with some fancy footwork we can extend the ideas to this case: but it’s not a straightforward generalization.
To anticipate: many of the results in the literature simply do not apply to restrictions of CCCP. Nor, I think, should the failure of the general version be seen as undermining the intuitive data that support thesis 1, since intuitive data never supported the problematic generalizations. The outstanding remaining challenge is that of Hájek (2004), who develops an argument against even single instances of CCCP. Hájek’s argument, however, has to make assumptions about how belief updating functions—it is dynamic rather than static result. And this, I think, gives wriggle room for the friend of instances of CCCP.
Williams (2008a) argues that in some superficially similar cases, there are in fact features to which we can appeal to push worlds further away. If certain worlds are objectively atypical relative to the objective chances, they can be pushed further out. This is intended to show why certain non-zero probability events do not undermine the truth of ordinary counterfactuals. The current setup is not of the form that is amenable to this tactic (allowing sheer improbability to play this role is not a viable option: see Lewis (1979, appendices)).
The alternative is to declare the worlds equally close, and, analogously to Lewis (1979), regard the conditional as false unless the consequent is true at all the closest antecedent-worlds. This is the line taken by Nolan (2003), and extension to the current case has S being false whenever the antecedent fails to hold. Such a theory is already in violation of Thesis 1.
The indefiniteness line is taken explicitly by Stalnaker (1980) and Weatherson (2001). Nolan explicitly flags the possibility of taking this line as a variation on his theory (much of which is independent of this particular issue).
A possible worry. We have to this point formulated the key principles metalinguistically. So even granted that the non-red cases do not differ in ways intuitively relevant to what one might call ‘conditional facts’, we might also wonder whether they vary in ways relevant to whether a certain conditional sentence expresses something which is true.
Perhaps the non-red cases differ over patterns of usage of conditional statements among the wider community (albeit in minor ways). Should such minor variations in usage be counted as relevant to the truth-value of S? It is extremely important to Williamson’s epistemicist that such minor variations in usage do generate differences in what propositions sentences express, and so (sometimes) what truth-value they have. But I take it the incredulous stare that Williamson’s view generates is due to the fact that such minor variations in usage are intuitively irrelevant to the question of what proposition they express. But that may just be an instance of the fallacy underlying sorites reading: we all know that big changes in usage are relevant to what propositions sentences express; and a big change can be nothing but many small changes added together. So it is simply wrong, however appealing, to think that minor changes in usage are irrelevant in this respect.
Nevertheless, in the case at hand, we should stick to our guns. Variation in usage (slightly more people calling a conditional false, for example), might be relevant to whether all non-red worlds are worlds where a conditional S is true. But it’s hard to see how variations in usage could be such as to be relevant to the question of whether a world where I scratch my nose, rather than one where I don’t, is a non-red world where S is truth. So irrelevance can be defended, even if broader principles are more dubious.
This is stated in a strong form. On the subjective probability interpretation, it requires that any possibility one has non-zero credence in, be one that is in this way irrelevant. One could defend this by refining (and idealizing) the setting so that the subject can rule out with certainty any sort of difference that might be relevant to the truth of S. Or one might weaken the principle so that it requires only that the vast majority of probability be invested in possibilities that differ only irrelevantly, and ultimately argue for a slightly weakened form of thesis 2.
We might also consider an extended Agnostic, who is uncertain between options (1), (2) and (3). I set this aside for now.
It is a nice issue how to represent the Agnostic’s evidential probability or credences, for it is consistent with this view that all the non-red situations are in fact cases where the conditional is true. Two options suggest themselves: having some kind of higher-order uncertainty over probability distributions; or enriching a single probability space with an extra element corresponding to different possible meanings of the indicative conditional.
At one point, I thought that phenomena close to those that Hájek (1989) points to could be used to argue that the Molinist was committed to the non-supervenience of the conditional truths on non-conditional truths. The rough idea is that variants of theses 2 should hold in highly idealized settings where there were just not enough no-draw worlds that differed in their non-conditional character, to represent the probability functions of the kind required by thesis 1.
I have come to believe that this will not work, for it would be question-begging to represent the subjective or evidential probabilities of the Molinist via a space that contained only metaphysical possibilities: the Molinist should be allowed to be uncertain about whether a conditional is true or false (while maintaining there is a brute fact of the matter) even when they know exactly what the non-conditional facts are.
The formulation here could do with some tightening up. For example: we shouldn’t allow variation in conditional facts to be a relevant respect of dissimilarity: it is similarity as regards non-conditional facts that we are concerned with.
Notice that Brutalism is substantially stronger than anything we could get by vanilla vagueness. It is true that using classical logic alone and a vague predicate like ‘tall’, sorites reasoning gives us a pair of arbitrarily similar cases, where the first is tall and the second is not. But notice (i) for all we have said, the truth of this existential generalization is compatible with there being no fact of the matter about which pair satisfies this description (cf. supervaluationist treatments of vagueness); (ii) the pair in paradigmatic cases of vagueness do differ in respects which are intuitively relevant to the satisfaction of the predicate. In the case at hand, we have also that the differences are intuitively irrelevant; and this gives extra force to the claimed bruteness.
To begin with, suppose that the Molinist thinks that the conditional has a determinate truth value in each of the non-red cases. (In principle one could adopt the mixed position which combines Molinism with the denial of this—but it seems pretty ad hoc to do so.)
As a recipe for finding such a pair witnessing brutalism, take two non-red worlds C and C′, such that the conditional is true at C, but false at C′. By Irrelevance, the first element of the brutality condition is met. But they may differ markedly in what happens in other ways (perhaps I scratch my nose and jump around a bit in one, and stay perfectly still in the other). Now construct a series of non-red cases which differ only minutely from each other, whose first member is C and whose last member is C′, such that any adjacent pair in the series differs only minutely in the underlying non-conditional facts (consider worlds where I jump around less and less, scratch my nose for briefer and briefer periods of time). Since every pair differs only in ways that are intuitively irrelevant to the conditional, and by construction the differences are minute, the only way they can fail to be a case of the kind needed is if they have the same truth-value. But if this goes for every pair in the series, it follows that the initial two cases we picked must have the same truth-value, contradicting our initial assumption.
Effectively, what we have done is to construct something very like a sorites series. It differs from a sorites series only in that the minor premises are not obviously true and false respectively. Rather, the premises are provided by the Molinist assumption.
The major dialectical change occurs when there is a very high conditional probability of drawing diamonds when one draws reds (and mutatis mutandis for the case where the conditional probability is very small). For in those sorts of cases, the departures from the instance of thesis 1 required by making the conditional true in all the non-red cases, are correspondingly small. For example, suppose that the thesis 1 holds only approximately: we can only say that the probability of the conditional is within 0.05 of the conditional probability. Now consider a case where the conditional probability of diamonds on reds is already very high: say 0.95. Then, consistently with this approximate version of thesis 1+, one may say that all non-red cases are cases where the conditional is true (in violation of thesis 2+). No matter how dominant black draws are over red, at the limit, the probability of the conditional will be 1, which is within 0.05 of the conditional probability.
Nevertheless, approximate versions of thesis 1+ allow us to argue for many instances of Thesis 2+. One might think that this itself is inductive support for the general version. But even if not, suppose that the last instance of thesis 2+ that is established is where the probability of the conditional is below 0.95. Since we still have all the relevant instances of thesis 2 and 3+, we can make ‘it is indefinite whether p’ as high as we like. Overall, we still have an argument that an instance of this form is 0.94 probable. For our purposes, this will be sufficient. Playing with approximations, then, we can expect to weaken the conclusion we can obtain; but we still get striking results.
See Fine (1975), Williamson (1994, ch 5.). Williams (2008c) argues that the reasoning will be successful if we presuppose that the logic of ‘definiteness’ is S5, and we treat ‘definitely’ as a logical constant.
There is a well-known way of defining a formal ‘consequence relation’ for the language that avoids this result: so called local consequence. For arguments against this being truly a consequence relation, see Williamson (1994, ch.5) and Williams (2008c).
If we have these principles, then it will be straightforward to argue from the law of excluded middle to the conclusion that there is no indefiniteness. Suppose for reductio there is indefiniteness: \(\neg Dp\wedge \neg D\neg p\). Then suppose p. An inconsistency follows (\(p\wedge \neg Dp\)); so by reductio, \(\neg p\). But an inconsistency again follows (\(\neg p\wedge \neg D\neg p\)). So we have an overall reductio of our starting point.
I discuss this case in Williams (2008c).
I think this is best construed as a pro tanto rational constraint on credence assignments. Only pro tanto, because I do not wish to rule out that in certain situations, even fully believing something inconsistent might be permitted if the alternatives are even worse. Some, thinking of cases where some highly complex sentence or proposition is inconsistent, might wish to qualify the above claim by requiring only that obvious inconsistencies sustain this rational constraint. That wouldn’t, I think, make much of a difference here, since the case we’re considering is not a case of some hard-to-compute inconsistency.
It is easy to show that the probability-logic link is incompatible with supervaluationist probability behaving in a classical way. In particular, it looks hard for the supervaluationist to avoid assigning probability 1 to a disjunction, each of whose disjuncts are probability 0. There is a non-classical treatment of probability that has this feature: the theory of Dempster–Shafer functions (for an accessible introduction, see Halpern (1995)). The main focus of the question of the tenability of orthodox supervaluationist logic, I suggest, should not be over its treatment of patterns of reasoning allegedly entrenched in practice such as conditional proof, reductio et al; but its implications, via the probability-logic link, for the structure of probabilities such as rational degrees of belief and evidence. It is here we find truly non-classical behaviour, and perhaps can get traction one way or other on the orthodox supervaluationist conception of indefiniteness.
One alternative proposal is that in the context of supervaluationism, we should proportion our credence in p to the proportion of sharpenings on which ‘p’ is true (thanks to Tim Williamson for suggesting this response). In general, this will require in addition to the set of admissible sharpenings, a measure over the sharpenings (it is similar to the theory I will later call ‘degree supervaluationism’, which will also postulate such a measure—but such theorists (a) need not postulate sharpenings in addition to the measure; (b) the measure need not be thought of as a measure of the number or proportion of interpretations, but rather as a measure of the degree of intendedness of the various interpretations; (c) the degree supervaluationist has a natural alternative to the standard supervaluationist’s ‘truth as truth on every sharpening’: namely, that the degree of truth of something is the measure of interpretations on which that thing comes out true.
It is crucial for this that one does not identify rejection with acceptance of negation. The negation of the law of excluded middle will be equivalent to an explicit contradiction, and so one needs to reject the negated version as well as the unnegated version.
One should distinguish the Field view from the ‘degree theoretic’ interpretation of the many-valued setting. Here, we do take the semantic values assigned by the model seriously—the idea being that a sentence assigned value d on the intended model should be regarded as ‘true to degree d’. Rather than the two classical truth-values, truth and falsity, one gets intermediate degrees of truth: perhaps just one intermediate degree; perhaps infinitely many (cf. in particular Van Inwagen (1990) for a degree-theoretic conception that (at least initially) takes place in a three-valued setting). I discuss these settings below.
Field’s logic (at least the non-conditional fragment) is given by 3-valued strong Kleene tables. One conception of those tables is that they describe how the truth, falsity, or truth-value gap status of the elements of a sentence determine the truth value of the whole. This truth-value gap conception is a possible rejectionist account, and susceptible to the arguments here. But it is not Field’s own view—his main interest is in the logic itself, and he can regard the truth-tables as simply useful instruments in describing the extension of the consequence relation. Field is then free to without qualification endorse each instance of the T-scheme—which is notoriously inconsistent with the assertion of truth-value gaps. Many different logics are compatible with this attitude—and there are plenty of LEM-rejecting options that will, by the arguments in the text, count as rejectionist in our sense. Intuitionist logic is a famous example.
One interesting option is a degree theory of vagueness, where the truth values are not exhausted by truth and falsity, but rather come in a variety of intermediate degrees. This can be developed in truth-function terms (‘fuzzy logic’), or in non-truth-functional terms Kamp (1975), Lewis (1970), Edgington (1997).
We would have to supplement each with a story about what the appropriate attitude to have to indeterminate sentences are. It won’t help very much in either case if our credences should be our credence of the sentences being perfectly true—but there may be alternatives. For one approach to this issue in the truth-functional setting, see Smith (forthcoming); and for a similar proposal, see MacFarlane (forthcoming). Smith’s basic idea is to treat our degrees of belief in sentences as ‘expected truth values’.
The non-truth functional setting is similar in spirit to supervaluationism, but instead of identifying a set of precisifications for our language, we posit a measure over the classical interpretations of our language the captures the degree to which they are ‘the intended interpretation’.
One natural suggestion in this setting is that the degree to which a sentence is determinate should act as an expert function for probabilities (so that if one is certain that S is true to degree k, one’s credence or evidential probability of S should be k). If one combines this with a reading of ‘indefinite’ as it figures in thesis 1 as ‘not degree-1 definite’, then if one knows a conditional is determinate to a degree k less than 1, one should invest probability k in the conjunction of the conditional with the claim that the conditional is not perfectly determinate.
This is no more than a sketch of a possible view, and much more work would be needed to defend its details. As in the epistemicist case, it is striking that such a degree theory could be motivated by considerations entirely distinct from the sorites paradox.
A different kind of static result is given by Hájek (1989) (it is sometimes known as the ‘wallflower’ problem). The formal result is that CCCP can’t be satisfied in a finite probability space. This does have some impact in the current setting, raising delicate issues about the space over which evidential or subjective probabilities are defined.
References
Albert, D. Z. (2001). Time and chance. Cambridge, MA: Harvard University Press.
Bennett, J. (2003). A philosophical guide to conditionals. Oxford: Oxford University Press.
Edgington, D. (1986). Do conditionals have truth values? Critica, 52, 3–30. Reprinted in Conditionals, by F. Jackson (Ed.), 1991, Cambridge: Oxford University Press.
Edgington, D. (1997). Vagueness by degrees. In R. Keefe & P. Smith (Eds.), Vagueness: A reader (pp. 294–316). Cambridge, MA: MIT Press.
Evans, J., & Over, D. (2004). If (pp. 134–138). Oxford: Oxford University Press.
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300. Reprinted with corrections from Vagueness: A reader, pp. 119–150, by R. Keefe & P. Smith (Eds.), 1997, Cambridge, MA: MIT Press.
Hájek, A. (1989). Probabilities of conditionals—revisited. The Journal of Philosophical Logic, 18, 423–428.
Hájek, A. (2004). Triviality on the cheap. In E. Eels & B. Skyrms (Eds.), Probability and conditionals. Cambridge: Cambridge University Press.
Hájek, A., & Hall, N. (2004). The hypothesis of the conditional construel of conditional probability. In E. Eels & B. Skyrms (Eds.), Probability and conditionals. Cambridge: Cambridge University Press.
Halpern, J. Y. (1995). Reasoning about uncertainty, revised edition. Cambridge, MA: MIT Press (revised paperback edition published 2003).
Jackson, F. (1987). Conditionals. Oxford: Blackwell.
Kamp, J. A. W. (1975). Two theories about adjectives. In E. Keenan (Eds.), Formal semantics of natural language (pp. 123–155). Cambridge: Cambridge University Press. Reprinted in Semantics: A reader, pp. 541–562, by S. Davis & B. S. Gillon (Eds.), 2004, Oxford: Oxford University Press.
Kratzer, A. (1986). Conditionals. CLS, 22(2). Reprinted in Semantics: An international handbook of contemporary research, pp. 651–656, by A. von Stechow & D. Wunderlich (Eds.), 1991, Berlin: de Gruyter.
Lewis, D. K. (1970). General semantics. Synthese, 22, 18–67. Reprinted with postscript in Philosophical Papers I, pp. 189–229, by D. K. Lewis, 1983, Oxford: Oxford University Press.
Lewis, D. K. (1976). Probabilities of conditionals and conditional probabilities. Philosophical Review, 85, 297–315. Reprinted with postscripts in Philosophical Papers vol. II, pp. 133–152, by D. K. Lewis, 1986, Oxford: Oxford University Press. Also reprinted in Conditionals, pp. 76–101, by F. Jackson (Ed.), 1991, Oxford: Oxford University Press.
Lewis, D. K. (1979). Counterfactual dependence and time’s arrow. Noûs, 13, 455–476. Reprinted with postscript in Philosophical Papers II, pp. 32–51, by D. K. Lewis, 1986, Oxford: Oxford University Press. Also reprinted in Conditionals, pp. 46–76, by F. Jackson (Ed.), 1991, Oxford: Oxford University Press.
Macfarlane, J. (forthcoming). Fuzzy epistemicism. In S. Moruzzi & R. Dietz (Eds.), Cuts and Clouds. Oxford: Oxford University Press.
McGee, V. (1989). Conditional probabilities and compounds of conditionals. Philosophical Review, 98, 485–541.
Nolan, D. (2003). Defending a possible-worlds account of indicative conditionals. Philosophical Studies, 116, 215–269.
Parsons, T. (2000). Indeterminate identity. Oxford: Oxford University Press.
Smith, N. J. J. (forthcoming). Degrees of truth, degrees of belief and subjective probabilities. In: S. Moruzzi & R. Dietz (Eds.), Cuts and clouds. Oxford: Oxford University Press.
Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (pp. 98–112). Oxford: Blackwell.
Stalnaker, R. (1980). A defense of conditional excluded middle. In W. L. Harper, R. Stalnaker, & G. Pearce (Eds.), Ifs: Conditionals, belief, decision, chance and time. The Netherlands: Kluwer Academic Publishers.
Tye, M. (1990). Vague objects. Mind, 99, 535–557. Reprinted with omissions in Vagueness: A reader, pp. 294–316, by R. Keefe & P. Smith (Eds.), 1997, Cambridge, MA: MIT Press.
van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press.
von Fintel, K. (2001). Counterfactuals in a dynamic context. In M. Kenstowicz (Ed.), Ken Hale: A life in language. Cambridge, MA: MIT Press.
Weatherson, B. (2001). Indicative and subjunctive conditionals. Philosophical Studies, 51, 200–216.
Williams, J. R. G. (2008a). Chances, counterfactuals and similarity. Philosophy and Phenomenological Research, 77(2), 385–420.
Williams, J. R. G. (2008b). Conversation and conditionals. Philosophical Studies, 138, 211–223.
Williams, J. R. G. (2008c). Supervaluationism and logical revisionism. The Journal of Philosophy, CV(4).
Williamson, T. (1994). Vagueness. London: Routledge.
Acknowledgements
This paper was written with the support of an AHRC research leave project. Material included in this paper was presented to the Leeds work in progress seminar, the Rutgers philosophy of language group, the Arché Research Centre and at the First Formal Epistemology Festival in Konstanz. I’m very grateful for all the comments and criticism at these events and in other discussion.
Author information
Authors and Affiliations
Corresponding author
Appendix: The Impossibility Results
Appendix: The Impossibility Results
When outlining some reasons to be suspicious of Thesis 1 and 1+, we mentioned the impossibility results arising from Lewis (1976) and the subsequent literature. We here quickly sketch the outlines of this debate, locate Thesis 1+ within the dialectic, and sketch how the considerations of the current paper fit in.
I take it that the impossibility results that arise in the literature that follow Lewis’s original paper are of two kinds: static and dynamic. The static kind argue that even within a single probability function, there are some conditionals whose probability is not equal to corresponding conditional probability. The dynamic kind show that (in precisely defined senses) the equation of probabilities of conditionals and conditional probabilities are unstable under various ways of updating ones credences. (Hájek and Hall 2004, give an excellent survey.)
Most of the static results are not obviously relevant the evaluation of Thesis 1, since there aren’t any static no-go results of this kind that ban instances of CCCP for the conditionals involved in Thesis 1. Indeed, there are tenability results (due to van Fraassen) that assure us Thesis 1 is at least consistent. Footnote 30
One might think, however, that the static no-go results undermine the appeal of Thesis 1 and 1+ indirectly. If the equation of probabilities of conditionals with conditional probabilities can’t be sustained in general, one might think it would be ad hoc to retain it in restricted cases.
This would be a powerful concern, I think, if the data about probability judgements concerning conditionals supported CCCP in general: for we’d then know that such intuitions have to be classed as unreliable indicators of the probability of conditionals, and we would anyway have to search for some way of explaining away the intuitions.
But the data about probability judgements does not support CCCP in general: indeed, it’s hard to see a firm case for it for anything other than paradigmatic simple conditionals (for which we know the instances of CCCP are satisfiable by van Fraassen’s result). This is one moral we can take from Vann McGee’s putative ‘counterexamples to modus ponens’ (McGee 1989). Suppose that you see a creature flapping around on the beach. The likelihood is that it’s either a dolphin or a salmon. However the conditional ‘if it’s a fish, then if it has lungs it’s a lungfish’ seems extremely probable: exactly as probable as the simple conditional ‘if it’s a fish with lungs, it’s a lungfish’. But the probability of ‘if it has lungs it’s a lungfish’ conditionally on ‘it being a fish’ will be low. (The probability of the simple conditional is obviously extremely low, and it’s conjunction with ‘it’s a fish’ will be no higher. We obtain the conditional probability by dividing this the probability of this conjunction by the probability that it’s a fish. If the latter is 1/2, the conditional probability is double the probability of the conjunction. But double something very tiny is still very tiny.) So it’s just not true to say that the intuitive data supports full CCCP.
Given the standard static results don’t undermine Thesis 1, directly or indirectly, attention turns to the dynamic results. I take it that the most powerful dynamic result for our purposes is that of Hájek (2004). He shows that any single instance of CCCP is unstable under a wide array of Bayesian belief update methods, updating on a wide array of propositions. That is, if one takes a probability distribution that vindicates an instance of CCCP, and updates on selected propositions by Bayesian methods (e.g. conditionalization, or Jeffrey conditionalization), one is left with a probability distribution that doesn’t vindicate that instance of CCCP.
To use these considerations to undermine Thesis 1, we’d need to add premises about the stability of Thesis 1 on receipt of various pieces of information, and we’d have to buy into some Bayesian story about belief-updating. Furthermore, updating on any old propositions is not guaranteed to undermine an instance of CCCP—it needs to be appropriately chosen.
The second and third premises here, in particular, seem to me to be hostages to fortune in the argument. There really is a gap, it seems to me, between buying into the probabilistic representation of partial belief at any given time, and the full Bayesian or quasi-Bayesian package of the dynamics of partial belief. And even if we do have an updating-story of the required kind, it seems to me no part of the Bayesian picture that for some arbitrary bit of information A, (no matter how easy and natural to describe) agents can find themselves in situations where they should conditionalize on the information that A. But this sort of issue may well be crucial to the arguments under discussion here.
Suppose, for example, that A is a proposition conditionalization on which would disrupt an instance of CCCP. Now suppose one could systematically describe a proposition A*, conditionalizing on which would give exactly the same results as conditionalizing on A for all non-conditional propositions; and such that conditionalizing on A would not disrupt the instance of CCCP. Then to turn the observation that conditionalizing on A would disrupt the instance of CCCP into an argument against our hypothesis, we’d need to support the claim that A, rather than A*, is at least sometimes the thing we should conditionalize upon.
The upshot is that Thesis 1 and 1+ aren’t knocked out the water by the standard impossibility results. That’s one reason why it’s still interesting to trace their implications. But what we have seen here is that even these highly restricted instances of CCCP are incompatible with certain popular theories of indefiniteness. And so we have a new static argument against these paradigmatic instances of CCCP—if we buy into those standard treatments of indefiniteness.
The current argument therefore does not require us to endorse the theories of belief updating needed to make the dynamic impossibility results damaging—in the terminology above, it is a static argument. True, it has its own philosophical hostages to fortune—several of them. But these are of a radically different kind, involving issues about the nature of semantics facts and of indeterminacy, on which we can hope to get independent traction. So, for one unconvinced or agnostic about some of the assumptions of Hájek’s dynamic argument, the present considerations potentially give us a new angle on the (un)tenability of highly restricted instances of CCCP.
Even one convinced by Hájek’s dynamic arguments against instances of CCCP should, I think, also be interested in this one—from their perspective, it will look like arguments from radically different sorts of premises point in the same direction—and even if one believes one already had a good argument against a thesis, the availability of independent support for the same conclusion can improve one’s epistemological situation.
Rights and permissions
About this article
Cite this article
Williams, J.R.G. Vagueness, Conditionals and Probability. Erkenn 70, 151–171 (2009). https://doi.org/10.1007/s10670-008-9145-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-008-9145-7