Fitch’s Paradox and Probabilistic Antirealism

Fitch’s Paradox and Probabilistic Antirealism Igor Douven Institute of Philosophy, University of Leuven [email protected] Abstract Fitch’s paradox shows, from fairly innocent-looking assumptions, that if there are any unknown truths, then there are unknowable truths. This is generally thought to deliver a blow to antirealist positions that imply that all truths are knowable. The present paper argues that a probabilistic version of antirealism escapes Fitch’s result while still offering all that antirealists should care for.

1. Antirealism and Fitch’s Paradox. Define antirealism as the thesis that truth is epistemically constrained. While this can be cashed out in various ways, it is commonly taken to imply at least that all truths are knowable, that is,1 ∀ϕ(ϕ → ♦ K ϕ).

(1)

Fitch’s paradox (Fitch [1963]) is generally thought to deliver a blow to antirealism, for the paradox shows that, presuming knowledge both to be factive and to distribute over conjunction, (1) entails the implausible-sounding thesis that all truths are known, that is, ∀ϕ(ϕ → K ϕ). (2) The argument runs as follows: Suppose, towards a reductio, that there is an unknown truth, that is, ∃ϕ(ϕ ∧ ¬ K ϕ). (3) Then let ψ be such a truth, and observe that all steps in the following chain of reasoning are valid, by (1), the distributivity of K, and the factivity of K, respectively: ψ ∧ ¬ K ψ ⇒ ♦ K(ψ ∧ ¬ K ψ) ⇒ ♦(K ψ ∧ K ¬ K ψ) ⇒ ♦(K ψ ∧ ¬ K ψ). But, clearly, ♦(K ψ ∧ ¬ K ψ) ≡ ♦⊥, which contradicts ¬♦⊥ (= ¬⊥), a consequence of the fact that ¬⊥ is a theorem of propositional logic together with the incontrovertible modal principle that all such theorems are necessary. Hence our supposition (3) leads to contradiction and must therefore be false, that is, ¬∃ϕ(ϕ ∧ ¬ K ϕ), (4) 1 “K ϕ” is to be read as “someone at some time knows ϕ.” Here and elsewhere, I use Greek letters as statement variables (though I shall sometimes use Roman capital letters for the same purpose where I believe these to be more suggestive).

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which is equivalent to (2).2 Although not literally a paradox—it is not, or at least need not be, officially part of the antirealist’s view that there are truths that will never be known by anyone—the conclusion is hard to swallow. What can antirealists say in response? Denying that knowledge is factive is not an option. Denying that it distributes over conjunction seems hardly less absurd.3 Furthermore, while in intuitionistic logic (4) is not equivalent to (2), it is a moot point whether it is not already bad enough for the antirealist to be committed to the former (whose derivation is also intuitionistically valid).4 Beall [2000] and Wansing [2002] have proposed to adopt a paraconsistent logic in response to Fitch’s paradox. But whatever the merits of their proposals, the paradox remains of interest to all those who would like to defend a notion of truth that is tied to our cognitive powers without forsaking classical logic. It is widely believed that these philosophers’ best option is to supplant (1) by some other, paradox-free principle which, however, still secures the requisite tie between truth and cognition. This is the option I will be concerned with in the following. A prudent methodology has us first look at alternatives that in some sense are close to (1). Amongst such alternatives are versions of (1) restricted to sentences of a type that does not give rise to paradox or comparable problems. However, proposals in this vein, like those put forward in Tennant [1997, Ch. 8] and Dummett [2001], have met with severe criticisms.5 A different approach is to replace the knowledge operator in (1) by another—if possible, closely related—one. This approach is frequently mentioned in the literature only to be dismissed right away, for—it is invariably said— there seems to exist no reasonable replacement for the knowledge operator that does not also lead to (an analogue of) Fitch’s paradox or, if not to that, then to some other untoward conclusion. By way of example, the suggestion to replace the knowledge operator by one indicating the presence of good evidence is often flagged as being particularly unhelpful.6 In the present paper I will argue that this is a mistake. To be more precise, I will propose as an alternative to (1) a constraint roughly to the effect that if ϕ is true, then a rational person may obtain evidence for ϕ strong enough to make it justifiedly credible to her (if it is not already). As will be seen, given the “right” (probabilistic, Bayesian) conception of evidence, there is every reason for the antirealist to take this option. In section 2 the said constraint is made formally precise. Next it is shown that by adopting the constraint instead of (1) we avoid Fitch’s paradox as well as some 2 In a footnote, Fitch attributes this result to an anonymous referee of an earlier paper which Fitch decided not to publish (Fitch [1963:138 n]). Joe Salerno recently discovered that the anonymous referee was Alonzo Church; see Salerno [2006]. It might thus be more appropriate, as Salerno suggests, to speak of the Church–Fitch paradox. 3 Besides, as Williamson [1993] has shown, it would be rather pointless to deny this, given that even without the distributivity assumption we can still derive some unpalatable consequences from (1). See also Kvanvig [2006, Ch. 4]. 4 See Williamson [1988], [2006], Kvanvig [1995], [2006, Ch. 5], and Tennant [1997, Ch. 8] for a discussion. 5 See, for instance, Hand and Kvanvig [1999], Brogaard and Salerno [2002], Hand [2003], Williamson [2000a], and Kvanvig [2006, Ch. 3]; see Tennant [2001a], [2001b] for responses to some of these critiques. A major type of criticism has been that the proposed restrictions of (1) are ad hoc, motivated only by the wish to block the paradox. In Douven [2005a], I have proposed a restriction that I claimed to be plausible on independent grounds. However, the argument I gave for it depends on an assumption that I no longer endorse (to wit, the assumption that assertion requires knowledge; see section 4). 6 See, for instance, Edgington [1985:558 f], Wright [1987:311], and Williamson [1988:423].

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seemingly related difficulties (section 3). Finally, sections 4 and 5 each address a worry one might have about the resulting position.

2. Probabilistic Antirealism. To present our constraint in a precise fashion some terminology must be introduced. We shall say that a person is rational iff, first, her degrees of belief at all times are coherent in the technical sense that they are probabilities; second, her initial degrees of belief (or probabilities, as we shall often say) are strictly coherent, that is, before she has obtained any evidence, she assigns probability 1 only to necessary truths (and thus probability 0 only to necessary falsehoods); and third, as she receives evidence, she updates her probabilities by dint of Bayes’ rule, that is, for any given sentence ϕ, the person’s new probability after the receipt of evidence ψ equals her earlier probability for ϕ conditional on ψ. We throughout assume that probabilities are defined on the sentences of a language L. By E we shall denote the class of evidence sentences of that language. We may simply think of these as the sentences to which rational persons are, or would be, willing to assign probability 1 as a direct response to their experiences; the other sentences of L can have their probability altered only mediately, because some evidence sentence receives probability 1. No further assumptions need to be made about what distinguishes the evidence sentences from the rest of the language. To say that a sentence ϕ is evidence for some other sentence ψ, relative to a probability function Pr, is to say that Pr(ψ | ϕ) > Pr(ψ); we say that ϕ is evidence against ψ iff Pr(ψ | ϕ) < Pr(ψ); and we say that ϕ is evidentially relevant to ψ iff Pr(ψ | ϕ) ≠ Pr(ψ) (otherwise ϕ is said to be irrelevant to ψ). We also define the notion of t-evidence: ϕ is t-evidence for ψ iff it is evidence for ψ and, in addition, Pr(ψ | ϕ) > t, where t is a value close to 1. We think of t as a threshold value for justification such that if a rational person assigns a probability greater than t to a sentence, she is justified in believing that sentence. For simplicity, we shall say that a person is justified in believing a sentence iff either that sentence has a probability greater than t for her or the sentence follows from a set of sentences all of which have a probability greater than t for her. Here, “follows” admits of stronger and weaker readings—“follows logically,” “is known to follow logically,” and so on—but for our concerns it is unnecessary to decide between those. This account of justification is simplified at least in the respect that one or more provisos will have to be attached to it to keep at bay the so-called paradoxes of rational acceptability (i.e., the lottery and preface paradoxes). It is currently unclear what form these provisos should take,7 but I will assume that what form they take is of no matter to anything that follows in this paper. We next define an operator E, as follows: E ϕ is true iff there is a rational person (with initial probability function Pr) and a set S ⊆ E such that, first, there is a time at which the elements of E that have been ascertained by the person are precisely those V
V
in S; second, Pr ϕ | S > Pr(ϕ); and third, Pr ϕ | S > t. In short, E ϕ iff there is a time at which some rational person has or obtains t-evidence for ϕ. The proposal now is to replace (1) by ∀ϕ(ϕ → ♦ E ϕ). 7 See

Douven and Williamson [2006] for some negative results.

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(5)

So, while it is no longer claimed that every truth is knowable, it is maintained that for every truth ϕ it is possible that a rational person obtains evidence for it that is strong enough to make the sentence justifiedly credible to her, or at least, if she was already justified in believing it,8 makes it more probable to her. It must be immediately noted that (5) needs some qualification. For, as it stands, it excludes that anything could be true to which we already initially assign probability 1. After all, for any Pr, if Pr(ϕ) = 1, then there can be no ψ such that Pr(ϕ | ψ) > Pr(ϕ), and thus there can be no t-evidence for ϕ relative to Pr. Hence (5) implies— absurdly—that no necessary truth (which according to probability theory must receive probability 1 at all times) can be true. The obvious way around this is to restrict (5) to contingent truths. This seems to pose no particular problem, inasmuch as for necessary truths we can accept (1).9 Without the requirement of strict coherence, the suggested restriction would not be enough, for then there might be contingent truths to which all rational persons assign probability 1 so that—again absurdly—(5) would imply that they are not really true. Thus, from here on, when I refer to (5), I take it as being restricted to contingent truths. By way of further comment, let me note that the requirement of strictly coherent initial probabilities is scarcely contestable. That we refrain from assigning extreme probabilities to contingent sentences prior to our obtaining any evidence—which is what the requirement amounts to—seems hardly more than a platitude.10 Strict coherence has even been defended as a general requirement of rationality, that is, as pertaining to all, and not just initial, probabilities; see, for instance, Kemeny [1955], Jeffreys [1961], and Stalnaker [1970]. However, there is a well-known drawback to this position, namely, that it rules out learning by dint of Bayes’ rule, which only applies on the condition that we have become certain of a sentence we previously were uncertain of (and that hence must be contingent). By requiring only initial probabilities to be strictly coherent, so that contingent sentences may get probability 1 as we acquire evidence, we avoid an inconsistency in our definition of rational agency. It is also worth noticing that (5) does not collapse back into (1). This is already so because the antirealist is not committed to the view that justification is enough to turn true belief into knowledge. Even if she were, however, (5) would not entail (1), given that (5) does not imply that the truth at issue would still have been true had one actually obtained t-evidence for it.11,12 8 That her initial probabilities are strictly coherent does not prevent her from assigning an initial probability greater than t to some contingent sentences, of course. 9 It would be wrong to think that then, by Fitch’s argument, all necessary truths are known. A sentence ϕ may be necessarily true yet unknown, but that that is so need not itself be a necessary truth. So the sentence expressing that ϕ is an unknown necessary truth need not be in the purview of (1) when that is restricted as suggested in the text. So then Fitch’s argument is blocked right from the beginning. 10 Some might want to make an exception for so-called pragmatic tautologies such as “I exist” (which are not necessary truths). I see no need to do so, but it can be done without affecting anything of essence in this paper. 11 Thanks to Timothy Williamson here. 12 It might also be thought that (5) fails to entail (1) because there are truths which could be justifiedly credible to a person but could not be justifiedly believed by that person, such as “No one can ever justifiedly believe this sentence.” However, for reasons to be explained later, self-referential sentences such as these are, if indeed they are of the designated sort, likely to qualify as Liar-like paradoxes on antirealist theories of truth satisfying (5). And, as will also be explained, if they do, then they seem amenable to the same treatment as the more typical Liar paradoxes, so that they cannot be justifiedly credible to anyone after all. Much the same seems to hold for the non-self-referential epistemic paradoxes of the kind discussed in Kroon [1993]. From a realist perspective these may seem rather unlike the Liar paradox, and therefore call for a separate treatment. It should be patent, however, that from an antirealist

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Just to have a name for it, we shall call the position which instead of (1) postulates (5) as a constraint on theories of truth probabilistic antirealism. If we assume, with Moore [1962], Unger [1975:83 ff], Williamson [2000b:251], and many others, that knowledge requires probability 1, then probabilistic antirealism clearly places a less demanding constraint on theories of truth than traditional antirealism does. For then—still assuming that rational persons have strictly coherent initial probabilities— (1) implies that it is possible to obtain for every truth ϕ evidence that raises ϕ’s probability not just to some value greater than t but even to 1. The question whether the new position can still be properly regarded as a brand of antirealism will be relegated to section 5. I first want to show that it does not give rise to an analogue of Fitch’s paradox or related problems.

3. Fitch’s Paradox and Other Threats. If the E-operator were factive and distributed over conjunction, then, by reasoning parallel to Fitch’s, (5) would yield ∀ϕ(ϕ → E ϕ), a conclusion hardly more credible than (2). For example, it seems intuitively right to say that for some natural number n it must have been true that there were n leaves on the cherry tree in my garden on July 1, 2000. But neither I nor anyone else took the trouble to count the leaves on that day, and so it is most unlikely that for any sentence of the form “There were n leaves on the cherry tree in ID’s garden on July 1, 2000” anyone will ever have or have had t-evidence. It will be immediately clear that the E-operator is not factive: That we have tevidence for a sentence does not guarantee that it is true. As has frequently been remarked in the literature, however, the factivity of knowledge is only relevant to Fitch’s argument insofar as it licenses the inference of ¬ K ϕ from K ¬ K ϕ. That inference could still be valid even if it were not in general valid to infer ϕ from K ϕ. So we must really ask whether it would be valid to infer ¬ E ϕ from E ¬ E ϕ. It is easy to appreciate that this is not the case either. Suppose we obtain t-evidence for the sentence saying that we never will come to have t-evidence for ϕ. Then E ¬ E ϕ holds true. Contrary to what our current evidence makes us expect, however, t-evidence for ϕ may emerge at a later time. We may thus have both E ¬ E ϕ and E ϕ, whence inferring ¬ E ϕ from the former must be ruled invalid. This suffices to show that there is no analogue of Fitch’s paradox for (5). Yet, for later purposes it is further useful to note that the E-operator does not distribute over conjunction. That we may have E(ϕ ∧ ψ) without E ϕ may be a bit of a surprise to those raised in the Hempelian tradition of hypothetico-deductivism or instance confirmation, the more so since it is surely true that if there is evidence χ for ϕ ∧ ψ which makes that conjunction highly likely, then the conjuncts separately must be highly likely conditional on χ, too.13 But the crucial point to observe is that χ need not be evidence (and hence not t-evidence) for either ϕ or ψ; it may even be evidence against both. Thanks to PrSAT, an add-on for Mathematica developed by J. McKenzie Alexander and Branden Fitelson, it is now easy to find probability models that demonperspective, especially if truth is tied closely to the notion of having strong evidence in the way (5) does, things look quite different. 13 See Tennant [1997:260]: “[Distributivity] holds good even for non-factive interpretations of ‘K’ such as ‘There is good evidence for . . .’ ”; also Künne [2003:438]: “How could there fail to be evidence for either of the conjuncts if there is evidence for the conjunction as a whole?”

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strate this point for all popular values of the threshold parameter t.14 However, the following argument, which I owe to Timothy Williamson, gives a general proof for all values of t. Consider the class of probability models satisfying the following conditions (n ∈ N): ϕ

ψ

χ

Pr

ϕ

ψ

χ

Pr

T T T T

T T F F

T F T F

(2n + 1)/3(n + 2) n/3(n + 2) 0 1/3(n + 2)

F F F F

T T F F

T F T F

0 1/3(n + 2) 3/3(n + 2) 0

One readily checks that, on these models, Pr(ϕ ∧ ψ | χ) = (2n + 1)/2(n + 2) > (3n + 1)/3(n + 2) = Pr(ϕ ∧ ψ); also Pr(ϕ | χ) = Pr(ψ | χ) = (2n + 1)/2(n + 2) < (3n + 2)/3(n + 2) = Pr(ϕ) = Pr(ψ). Further, we can, by choosing n large enough, make Pr(ϕ ∧ ψ | χ) > t, whatever the exact value of t (given that t < 1). Thus, for any value of t, there are probability models satisfying these conditions: (i) Pr(ϕ ∧ ψ | χ) > t > Pr(ϕ ∧ ψ); (ii) Pr(ϕ | χ) < Pr(ϕ); and (iii) Pr(ψ | χ) < Pr(ψ). Even though there is no analogue of Fitch’s paradox for probabilistic antirealism, it might face problems that are as bad, or nearly as bad. In the remainder of this section I consider, and aim to discount, what at first glance may indeed seem to be some obvious problems for it. It is compatible with (5) to suppose that there is a truth for which no one will ever actually have t-evidence. Suppose ϕ is such a truth, so that the following holds: ϕ ∧ ¬ E ϕ.

(6)

♦ E(ϕ ∧ ¬ E ϕ).

(7)

By (5), it then follows that But how could there be t-evidence for (6) that is not at the same time t-evidence for ϕ? And if there is t-evidence for ϕ, then, contrary to our supposition, (6) is false.15 Given what was just said about the non-distributivity of E, it should be easy to understand why the foregoing reasoning is incorrect. First consider that it follows from (7) that, possibly, at some time both (8) and (9) hold for some person with probability function Pr and some set S ⊆ E: V
Pr ϕ ∧ ¬ E ϕ | S > Pr(ϕ ∧ ¬ E ϕ); (8) V
Pr ϕ ∧ ¬ E ϕ | S > t. (9) V
While from (9) it follows that Pr ϕ | S > t, it does not follow from either (8) V
or (9) that Pr ϕ | S > Pr(ϕ); as was shown above, one can have evidence for a

14 Most authors who have written about probabilistic acceptance rules—including Kaplan [1981:308], Moser and Tlumac [1985:128], Kyburg [1990:64], and Douven [2002a:402]—mention .9 as being a reasonable value for t; Harman [1973:118] and Foley [1992:113] mention .99, and Achinstein [2001:156] mentions .5. PrSAT is publicly available via http://www.fitelson.org; see Fitelson [2001:93–100] for an exposition of some of the theory behind it. 15 This is in fact the consideration that led the authors mentioned in note 6 to the conclusion that replacing the knowledge operator in (1) by one to the effect that there is good evidence available would offer no solace to the antirealist. See for a response to it also Kvanvig [2006:20–25]. Kvanvig uses the notion of evidence in an informal way, however, which makes the correctness of some of his claims—in particular those about the interaction of different bodies of evidence—hard to assess.

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conjunction without having evidence for either conjunct. To make this a bit more intuitive, suppose that our initial probability for ϕ already is very close to 1. Then it may well be extremely hard to find any piece of information that will bring our probability for ϕ still closer to 1, and we may come to possess information suggesting that we will never actually find such a piece of information. That information, we may suppose, will increase our probability for ¬ E ϕ. By doing so, it may well increase our probability for the conjunction of ϕ and ¬ E ϕ, and it can be consistently supposed that this probability is either increased to a value above t or remains above t. Yet it may do all that without increasing our probability for ϕ. The next potential problem is a sort of converse of the first: Suppose there is a falsehood ϕ such that E ϕ (this is possible, given that E is non-factive). Then the following conjunction is true: ¬ϕ ∧ E ϕ. (10) Thus, by (5), ♦ E(¬ϕ ∧ E ϕ).

(11)

Again this might seem to be almost as bad as a paradox. For how could there be t-evidence for ¬ϕ that at the same time is t-evidence for the proposition that there is t-evidence for ϕ? But there really is no problem here. Suppose our total present evidence relevant to ϕ is ψ∧χ, and that this is t-evidence for ¬ϕ, but that χ, which was known before ψ became known, on its own constituted t-evidence for ϕ. That this is possible can be seen by considering the class of probability models satisfying these conditions (n ∈ N): ϕ

ψ

χ

Pr

ϕ

ψ

χ

Pr

T T T T

T T F F

T F T F

2/4(n + 2) n/4(n + 2) (n + 2)/4(n + 2) 0

F F F F

T T F F

T F T F

(n + 3)/4(n + 2) 0 0 (n + 1)/4(n + 2)

For these models we have both Pr(ϕ | χ) = (n+4)/(2n+7) > 1/2 = 2(n+2)/4(n+2) = Pr(ϕ) and Pr(¬ϕ | χ ∧ ψ) = (n + 3)/(n + 5) > 1/2 = Pr(¬ϕ); and by choosing n sufficiently large, we can make Pr(¬ϕ | χ ∧ ψ) > t. So it is perfectly possible that, while ψ ∧ χ is t-evidence for ¬ϕ, when we only knew χ we had t-evidence for ϕ. If this possibility obtains, then, having ascertained both χ and ψ, and remembering that at an earlier time we only knew χ, which then constituted t-evidence for ϕ, we can plausibly be assumed to possess t-evidence for ¬ϕ ∧ E ϕ. Another way in which (11) could hold true is if we know both χ and ψ—where this is again all our evidence relevant to ϕ—but also know that some of us who are not presently here are still unaware of the truth of ψ but not of that of χ, which we know to be their total evidence relevant to ϕ. That would plausibly give us t-evidence for the conjunction saying that ϕ is false but that some still have t-evidence for it—which entails (10). The first seeming problem arose from the observation that there could be a truth for which no one would ever have t-evidence. The potential problem now to be discussed has to do with the fact that there could equally well be a truth to which no one

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will ever assign a probability greater than t.16 For expository purposes, I introduce a new operator, O, with the following meaning: O ϕ is true iff there is a rational person (with initial probability function Pr) and a set S ⊆ E such that, first, there is a time at which the elements of E that have been ascertained by the person are precisely those V
in S, and second, Pr ϕ | S > t. Then our problem involves the fact that it should at least be consistent to suppose that there exists a ϕ such that ϕ ∧ ¬Oϕ

(12)

holds true. This may seem problematic because, if (12) is true, then, by (5), so must be ♦ E(ϕ ∧ ¬ O ϕ). (13) And one might wonder how a rational person could ever come to have t-evidence for (12). For—it could be said—if she had such evidence, she would assign a probability greater than t to (12), and hence also—given that she is rational and thus obeys the laws of probability—to ϕ. And this would immediately falsify the second conjunct of (12), which says that no one will ever assign a probability greater than t to ϕ. But then (12) is false, contrary to what we are supposing. Though this third problem is superficially similar to the first, it is unavailing here to point to the non-distributivity of the E-operator. Although by having t-evidence for a conjunction I may not have t-evidence for either conjunct, it must be that on my evidence the probabilities of both conjuncts are greater than t, and the assumption that that holds for the conjuncts of (12) already conflicts with the supposed truth of the sentence. To see that yet (12) is not formally a paradox for probabilistic antirealism, it suffices to note that the fact that (12) would be falsified were we to obtain t-evidence for it does not contradict our assumption that the sentence is true, even though, by (5), that assumption does imply the possibility of finding t-evidence for it. Of course the same assumption also implies that that possibility will never be actualized—but so what? Many possibilities never get actualized. Still, showing that (12) poses no formal problem for probabilistic antirealism is hardly enough by way of response; Fitch’s paradox is, as intimated, not really a formal problem either. Indeed, it might be said that the real puzzle about (12) is that it seems inconsistent even to suppose that someone could have t-evidence for it—even if no one will ever actually have such evidence—for it would seem that if, per impossibile, anyone had such evidence, she would realize that she was in a situation that falsifies (12), so that she could not believe that sentence to a degree above t (or even any positive degree), and thus could not have t-evidence for it. In more detail, the argument for this might go as follows: Suppose (12) is true. Then, according to probabilistic antirealism, so is (13). For (13) to be true, E(ϕ ∧ ¬ O ϕ) (14) must at least be consistent. To see that it is not, suppose it is true. Then there is a rational person with probability function Pr who will at some time have verified precisely the elements of a set S of evidence sentences, and Pr ϕ ∧ ¬ O ϕ |

V
S > t.

(15)

16 Thanks to Timothy Williamson for bringing this problem to my attention. Thanks also to Raf De Clercq and Jan-Willem Romeijn for discussing it with me.

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Suppose this person’s current evidence consists of exactly the sentences in S, and let V Pr* be her probability function updated on S. Then from (15) and the fact that the person is rational, and thus updates by Bayes’ rule, both (16) and (17) follow: Pr*(ϕ) > t;

(16)

Pr*(¬ O ϕ) > t.

(17)

Since the person may be assumed to realize that she herself assigns a probability greater than t to ϕ conditional on her current evidence, we may suppose that
Pr* Pr*(ϕ) > t = 1.

(18)


Pr* Pr*(ϕ) à t > t.

(19)

Given that, by assumption, the sentences in S have been verified by the person, ¬ O ϕ entails Pr*(ϕ) à t. So from (17) and the laws of probability it follows that

But (18) and (19) contradict the assumption that the person is rational. Hence, the supposition that (14) holds true leads to a contradiction, and thus (13) must be false. So to the extent that we have reason to believe that there exist true instances of (12), we have reason to believe that probabilistic antirealism is false. The problem with this argument is that there is no warrant for the step from (16) to (18), or in the preliminary argument the step from “one has t-evidence for ϕ” to “one realizes that (12) is falsified,” both of which assume introspective awareness on the person’s part of her own probabilities. It might at first seem that whereas one can be unsure about an objective probability (say, the chance that a radium atom will decay within the next hour) or about another person’s degree of belief in a given sentence, one cannot be unsure about one’s own degrees of belief; so, for instance,
second-order probability assignments of the form Pr Pr(ϕ) = x = y, with 0 < y < 1 and Pr a person’s degrees of belief function, might seem absurd, or at least irrational. But not only is there nothing in the Bayesian doctrine which implies that one cannot assign a non-extreme probability to any sentence making a claim about one’s own probability function—surely these are not necessary truths or falsehoods—it is, on closer inspection, and at least on some plausible conceptions of probability, quite natural to think that we may be unsure about our own degrees of belief.17 To see this, first recall that according to the familiar betting concept of probability, probability statements express betting dispositions; more exactly, it identifies a person’s degree of belief in a sentence with her fair betting quotient for a bet on that sentence, which for our concerns we may simply think of as the highest price the person is willing to pay for a bet that pays $1 if the sentence turns out true, and nothing otherwise (cf. Savage [1972]). Now suppose that you are offered a bet on a sentence stating that your probability for ϕ is x, but that you are not, and have not been, offered a bet on ϕ itself. Then to determine your fair betting quotient for a bet on ϕ it seems the most you can do is engage in some hypothetical reasoning about how much you would be willing to pay for a bet that pays $1 if ϕ is true, were you offered that bet. But even though you may now feel, on the basis of that reasoning, that you would be willing to buy such a bet for at most $x, you may also believe, or 17 The philosophers referred to in the previous section who regard strict coherence as a general requirement of rationality must even hold that one cannot assign an extreme probability to sentences making claims about one’s own probability function.

9

even know, that your hypothetical reasoning about matters like this is not entirely reliable, and that should you actually be offered the bet on ϕ now, you might be willing to pay more for it. So even if your current probability for ϕ is x, you need not be sure, nor even deem it highly likely, that x is your current probability for ϕ. This means, for instance, that you may be unsure about whether a given sentence is part of your evidence even if that sentence is part of your evidence.18 I am making the point regarding uncertainty about one’s own degrees of belief in terms of the betting concept only because that concept is still widely held. It is not crucial for my point that it is the right concept, however. As is explained in Douven and Uffink [2003:400 ff], on a more realist understanding of probability, which identifies degrees of belief with things that are somehow internal to us (like brain states, for example), it is reasonable to assume that there is a measurement error associated with measuring degrees of belief. So on this conception one may even be uncertain about one’s degree of belief in a sentence after one has bet on it (or after one has measured the degree of belief in some other way, if there is any).19 It is one thing to show that the argument for the inconsistency of (14) is unsound, and it is another to show that good sense can be made of (13), that is, the claim that it is possible to have t-evidence for a conjunction one of whose conjuncts implies that no one will ever assign a probability greater than t to the other conjunct. However, the foregoing remarks on higher-order probabilities suggest a plausible explanation of how someone might actually come to have such evidence for (12). Suppose that over the years a person has been reading a great many reports about other people’s probabilities for ϕ. Since none of the reports indicated that someone assigned a probability greater than t to ϕ, the person on inductive grounds has come to believe rather firmly that no one will ever assign a probability greater than t to ϕ. As it happens, however, she herself possesses t-evidence for ϕ, but, basically for the kind of reason given above, she is inclined to think that this evidence is not really part 18 Skyrms [1980] also argues that “the dispositional sense of belief makes sense of the possibility that someone may not know his own mind with certainty . . .” (p. 115; italics omitted). 19 Two brief comments on this: First, contrary to what some might think, Bayes’ rule does not require us to be certain about what our evidence is. If our initial probability function is Pr and our total current evidence is E, then, whether or not we are aware of what our total current evidence is, our current probability function is Pr updated on E. Things become different if, with Williamson [2000b, Ch. 10], we assume that evidence can be lost. That assumption is incompatible with Bayes’ rule, but not with a generalization of that rule which Williamson offers (which for the purposes of probabilistic antirealism we could unproblematically adopt instead of Bayes’ rule). The second comment concerns Dutch book vulnerability: If, as is customary to do, bookies are assumed to know your current probabilities (and perhaps your update rule), then by being uncertain about some of your probabilities you are vulnerable to a Dutch book; see, e.g., Milne [1991:307 f]. However, at least in the present case such Dutch book vulnerability fails to show anything of significance. After all, anyone who knows something that you do not know, and you do not (fully) believe, is in a position to make money off you. If I have the information that tomorrow it will rain and you do not, and you assign some probability less than 1 to “There will be rain tomorrow,” then it is easy for me to take economic advantage of you. It thus seems that if Dutch books are to expose irrationality, and not ignorance, then the most that can be assumed in such arguments is that bookies know a person’s probabilities insofar as these probabilities are known to the person herself (or, better perhaps, insofar as the person is consciously aware of them). However, I think that ultimately all such considerations are spurious, for in my view Dutch book arguments have no epistemic import; see Douven [1999a] (also Joyce [2004]). Sobel [1987, Sect. 4] argues that a person who is unsure about some of her first-order degrees of belief fails to satisfy an intellectual ideal. However, in the next section of his paper he argues that this ideal can at best be an approachable one, and even ends the paper by counselling that we avoid Dutch book vulnerability in a pragmatic way, by “[looking] out for clever bookies, and [hedging our] bets even to the point of sometimes not betting at all” (p. 75).

10

of her total evidence. So, that she has t-evidence for ϕ does not, or at least not to any significant extent, diminish the degree to which she believes that no one will ever assign a probability greater than t to ϕ. It is thus well conceivable that her total evidence, which comprises the reports she has been reading about other people’s probabilities for ϕ as well as her excellent evidence for ϕ, makes (12) very likely to her, and more likely than it initially was; that is, her total evidence may well constitute t-evidence for (12). The last challenge to probabilistic antirealism I want to consider is this: It would seem that the position should not preclude the possibility that there is a true sentence which is not a conjunct of any conjunction for which we will ever have t-evidence. That is, one would think that it should be consistent to suppose that ϕ ∧ ¬∃ψ E(ϕ ∧ ψ)

(20)

is true for some ϕ. Suppose there is such a ϕ. Then it follows from (5) that
♦ E ϕ ∧ ¬∃ψ E(ϕ ∧ ψ) .

(21)

We could immediately derive a contradiction from this if the E-operator were factive20 —which it is not. However, is it not worrying enough that it is possible to have t-evidence for a conjunction one conjunct of which says that the other one is not a conjunct of a conjunction for which we have t-evidence? For if we had t-evidence for (20), then we would have t-evidence for a conjunction of which ϕ is one of the conjuncts—which would immediately falsify the second conjunct and hence the conjunction, contrary to our assumption that it is true. Our first response to this problem is basically the same as the first response to the previous problem: That (20) would be falsified were we to obtain t-evidence for it is consistent with the assumption that the sentence is true, even if that entails the possibility of getting t-evidence for it. But again more must be said, for here too it might already seem inconsistent to suppose that we could have t-evidence for (20). For—one might argue—according to (21) it is possible that at some time it holds for V some set S of evidence sentences that S constitutes t-evidence for (20) relative to some Pr that satisfies our desiderata. But suppose this possibility obtains. Then there will be t-evidence for a conjunction one of whose conjuncts is ϕ. Surely the person who has the evidence will realize this, which, we may suppose, implies the following: Pr ∃ψ E(ϕ ∧ ψ) | 20 Here

is a proof in natural deduction style:

V
S = 1. 

1


 E ϕ ∧ ¬∃ψ E(ϕ ∧ ψ) ∃χ E(ϕ ∧ χ)


♦ E ϕ ∧ ¬∃ψ E(ϕ ∧ ψ)

(22)


1 E ϕ ∧ ¬∃ψ E(ϕ ∧ ψ) ϕ ∧ ¬∃ψ E(ϕ ∧ ψ) ¬∃ψ E(ϕ ∧ ψ)

∃I

¬∃χ E(ϕ ∧ χ)

⊥

F

∧E

R →E

1 ♦⊥

⊥

The rule ♦⊥ captures the principle that absurd propositions are impossible (cf. Tennant [1997:257]); F is the putative factivity rule for E; R is a rule for renaming variables, subject to the obvious constraint to avoid variable collision (the rule is strictly unnecessary, but permits a more compendious statement of the proof); the other rules are standard (see, e.g., Tennant [1990] or van Dalen [1994]). (The proof is an adapted version of an argument to be found in Williamson [1993].)

11

However, the very fact that

and so, a fortiori,

V

S is t-evidence for (20) implies, among others, that V
Pr ϕ ∧ ¬∃ψ E(ϕ ∧ ψ) | S > t, Pr ¬∃ψ E(ϕ ∧ ψ) |

V
S > t.

(23)

And together (22) and (23) contradict the assumption that Pr is coherent. The response to this problem will already be clear. It simply points to the fact that the contradiction arises because the person is supposed to realize that she has t-evidence for (20), a supposition that, as we explained, may well be false. While this response is sufficient to divert the threat to probabilistic antirealism posed by (20), let me add, albeit more tentatively, a response of a rather different nature which may well work even if we assume that rational persons tend to have a perfect introspective awareness of their degrees of belief. I add this response not because I think the introspective awareness assumption may be reasonable after all; I think it is unreasonable. Rather, the purpose of adding it is to make the more general point that probabilistic antirealism may be liable to paradoxes other than Fitch’s—and even ones that in a sense may be more serious than that—which, however, still do not automatically give us reason to abandon the position. The response starts by arguing that, given the introspective awareness assumption, (20) may well constitute a Liartype paradox for at least some theories of truth satisfying (5). It then continues by noticing that, if (20) is such a paradox, this is not especially worrying, given that there never was reason to suppose that antirealist theories of truth will not be plagued by very much the same paradoxes that are known to plague realist theories of truth. To see that (20) might indeed fall into that category, first consider this: No one will ever have t-evidence for this sentence.

(24)

Lacking currently any fully worked-out antirealist theories of truth, and in particular theories of truth satisfying (5), it is impossible to say for sure whether (24) will constitute a paradox for such theories (or at least for some of them). But that it may do so is suggested by the following informal reasoning: It must be obvious to anyone that if someone were to have t-evidence for (24), the sentence would be false, and that, precisely because this is so obvious, no one will ever have t-evidence for it. However, if that is so, then (24) is true! And the foregoing reasoning in turn is so obvious that it would seem that it must occur to at least some persons, and thus that, as soon as they are allowed to do so—which is as soon as they have started gathering evidence—they will assign probability 1 to it, which would mean that they would have obtained tevidence for it. But if this is so, then (24) is obviously false! So we seem to arrive both at the conclusion that (24) is obviously true—and hence true—and at the conclusion that it is obviously false, and hence false. Because we can keep repeating the reasoning in which we go from obviously true to obviously false—the conclusion that (24) is obviously false seems itself so obvious again that no rational person could possibly ever have t-evidence for it, so that (24) would seem obviously true, and so on—(24) may automatically come out as being truth-valueless (and hence not as contradictory) on some theories satisfying (5). But if it does come out as a genuine paradox for an antirealist theory of truth, then that theory will clearly need some (further) logical regimentation to restore consistency. (Some might think that (24) is a necessary truth and therefore should receive an initial probability of 1. Of course the sentence is clearly not a logical or mathematical 12

truth, but it might seem to be conceptually necessary, following from the concept of t-evidence. But that is not so. What at most could be said to be conceptually necessary is that if we assign (24) an initial probability of 1, then we can never have t-evidence for it. However, if we assign it some other initial probability, then, arguably, the sentence is not even true. So our assigning an initial probability of 1 to (24), if we do, cannot be justified by the claim that it is conceptually necessary. As a result, there seems to be no license for thinking that our requirement of strictly coherent initial probabilities does not imply that (24) ought to be assigned an initial probability unequal to 1.) Next recall Kripke’s [1975:691 f] observation that sentences can be contingently paradoxical. An example of a contingently paradoxical sentence would be ϕ, and (25) is false,

(25)

with ϕ some contingently true sentence. For if ϕ is true, then (25) is true iff its second conjunct is true, that is, iff (25) is false. It seems reasonable to suppose that any adequate theory of truth will say the same about (25) if ϕ is true as it says about the standard Liar sentence (on Kripke’s theory, for instance, both sentences come out as being ungrounded; if ϕ is false, then (25) is false too, as might be expected). If the above informal reasoning was correct, and (24) is a Liar-like sentence for an antirealist theory of truth satisfying (5), then sentences of the form ϕ, and no one will ever have t-evidence for this sentence,

(26)

with ϕ true, would seem to parallel those of the form (25) with ϕ true. The point about (25) then suggests (again, lacking an antirealist theory of truth, no stronger claim can be made) that whatever an antirealist theory of truth will say about (24)— whether it will make the sentence come out truth-valueless, or non-well-formed, or meaningless, or ungrounded, or something else altogether—it will say the same about instances of (26) that have a true first conjunct. Now the final thing to note is that (20) asserts, at bottom, what (26) asserts. By saying something about conjunctions one of whose conjuncts is ϕ, (20) says, like (24) and (25), something about itself, that is, it is self-referential. And what it says about itself is that no one will ever have t-evidence for it. So it expresses whatever ϕ expresses, and in addition asserts about itself that no one will ever have t-evidence for it, which is precisely what (26) also does. It would thus seem realistic to expect that an antirealist theory of truth will also say the same about (20) as it says about (24). Admittedly, should it turn out that (20) is a paradox for any antirealist theory of truth satisfying (5), then it would be pedantic to maintain that it is really only a problem for those theories; in that case it is a problem for probabilistic antirealism as well. But since there is no reason to think that the standard devices for dealing with the semantic paradoxes would not work in this case, it should not be a fatal one. Moreover, because realism has a corresponding problem, it certainly does not provide grounds for preferring realism to antirealism. We have thus seen that probabilistic antirealism is not beset by an analogue of Fitch’s paradox, and that it seems to escape what prima facie look like further threats. It merits remark that, while there is no guarantee that the position will not come to grief over some problem I currently fail to see, the responses to the last two potential problems signalled above do seem to warrant some claims transcending these problems. For the response to the third problem would seem to apply quite generally to sentences which, given probabilistic antirealism, require the possibility of having 13

t-evidence for a sentence that somehow implies that one does not assign a probability greater than t to some sentence implied by the first; and it is not difficult to think of variants of (12) that are of this sort.21 And the response to the fourth problem also seems to generalize, for it may be expected to apply to any sentence that can plausibly be regarded as a Liar-type sentence for antirealist theories of truth satisfying (5). In the following sections we consider two threats of a different kind, namely, that the new position is not a genuine philosophical position at all and, respectively, that it is not deserving of the name “antirealism.”

4. Convergence and Empirical Equivalence. I start by discussing a mathematical result which seems to trivialize probabilistic antirealism. At least prima facie, the result appears to show that we can obtain t-evidence for any truth. If it does show that, then saying that we can obtain such evidence cannot be distinctive of any philosophical position. For now, assume L to be the language of first-order arithmetic with finitely many empirical predicates and function symbols added to it. M is the class of models for this language. All models have the set of natural numbers, N, as their domain; they only differ in their interpretations of the empirical vocabulary, which are assignments of subsets of Nk to k-ary empirical predicates and functions from Nk to N to k-ary empirical function symbols. A class Φ of sentences of L is said to separate a set of models S ⊆ M precisely if for every two different models w , w 0 ∈ S there is a ϕ ∈ Φ such that w î ϕ and w 0 î ¬ϕ (“w î ϕ” means that ϕ is true in w ). Let ϕw be ϕ if w î ϕ, and ¬ϕ otherwise, and let JϕKw stand for the truth value of ϕ in w . Finally, let P be the class of all probability functions defined on the sentences of L; besides the standard axioms of probability theory, these functions are supposed to satisfy the following version of Countable Additivity:

Wn Pr ∃xϕ(x) = lim Pr i=0 ϕ(i) , n→∞

with at most x free in ϕ(x). Then Gaifman and Snir [1982:507] prove this: Theorem 4.1 (Gaifman and Snir) Let Φ = {ϕ1 , ϕ2 , ϕ3 , . . .} separate M. Then for all
Vn ψ ∈ L and all Pr ∈ P, limn→∞ Pr ψ | i=1 ϕiw = JψKw almost everywhere. The expression “almost everywhere” is mathematical jargon and means that the result holds for all w in a set M0 ⊆ M of measure 1, relative to the given Pr. As Gaifman and Snir set up things, this means, more specifically, that their result holds for every Pr

21 One example, discussed in Künne [2003, Ch. 7], is “ϕ, and no one is ever justified in believing ϕ.” Künne would be right that such sentences create difficulties for antirealism if (as he calls it) the common denominator of antirealism were this: “There is no true proposition such that the assumption that it is both true and the content of a justified belief implies a contradiction” (p. 437; italics in the original). But—passing over the proposition/sentence distinction for now—(5) implies no more than that there is no true proposition such that the assumption that it is both true and could become the content of a justified belief implies a contradiction, which is a weaker claim. Künne gives no reason for believing that the antirealist is committed to the stronger principle, and if the arguments to be offered in section 5 of this paper are correct, then the antirealist is not committed to it. To see that the distinction is of crucial importance, let J ϕ mean that someone at some time is justified in believing ϕ. Then for the principle Künne attributes to the antirealist to hold, ϕ ∧ ¬ J ϕ and J(ϕ ∧ ¬ J ϕ) would have to be consistent with one another (supposing that ϕ is a truth that no one ever is justified in believing). But, as Künne shows (p. 439), they jointly entail J ϕ ∧ ¬ J ϕ. By contrast, the “closest” to this we can derive from ϕ ∧ ¬ J ϕ and ♦ J(ϕ ∧ ¬ J ϕ) is ♦ J ϕ ∧ ¬ J ϕ, which is not at all problematic.

14

in all worlds w with the exception of those for which there is a ψ ∈ L with Pr(ψ) = 0 such that w î ψ. Since both probability 0 and probability 1 are, given Bayesian con
Vn firmation theory, “once and for all” matters, we have that limn→∞ Pr ψ | i=1 ϕiw = 0 if Pr(ψ) = 0, even if it should be the case that w î ψ. It will immediately be noted, however, that this really effects no restriction at all if one assumes—as we do—strictly coherent initial probabilities, for that implies that only necessary falsehoods receive an initial probability of 0. If, following Gaifman and Snir’s [1982:507] suggestion, we think of the sentences in Φ as evidence sentences (i.e., Φ = E), then the theorem implies that, if we could just go on checking the truth values of more and more evidence sentences, we would come to have t-evidence for every truth. It thus seems to guarantee that for any truth it is possible to have such evidence. But, naturally, given that the foregoing is a mathematical result, realists and antirealists are both committed to it. So then how could (5) be distinctive of antirealism? Indeed, how could it be a substantive thesis at all, if it is a mere corollary to a mathematical theorem? However, a first thing to note is that we can easily think of senses of “possible” in which Theorem 4.1 does not guarantee the possibility of obtaining t-evidence for every truth. For example, it does not seem to guarantee that this is a practical possibility, and one could argue that (5) ought to be interpreted in terms of this notion of possibility.22 Secondly, and more importantly, the requirement that the set of evidence sentences be separating is not one to be taken lightly.23 Specifically, it denies the so-called Empirical Equivalence thesis (EE), which says that every theoretical hypothesis has at least one empirically equivalent rival. For present purposes, we may simply think of theoretical hypotheses as the contingent elements of L \ E. Generally (and somewhat loosely) characterized, empirically equivalent theories are theories that are accorded the same confirmation-theoretic status in the light of any possible evidence. This may mean different things, depending on which confirmation theory is assumed. But we may suppose that it minimally implies that if hypotheses H and H 0 are empirically equivalent, then H î E a H 0 î E for every E ∈ E. Finally, to say that two theories are rivals is to say that their conjunction is inconsistent. Now, surely there will be hypotheses H ∈ L \ E that are complete with respect to E, so that either H î E or H î ¬E, for all E ∈ E. Then if H is such a hypothesis and EE holds for L, there must be distinct w , w 0 ∈ M such that w î E a w 0 î E for all E. For let H 0 be an empirically equivalent rival of H, and let w î H and w 0 î H 0 . Then w î E a H î E a H 0 î E a w 0 î E (note that w î E ⇒ H î E holds in virtue of the fact that H is complete with respect to E; the same for w 0 and H 0 ). Since H and H 0 are rivals, w and w 0 must be distinct worlds. As a result, E fails to separate the class of models for L.24 Granting EE for the nonce, it must be stressed that the foregoing still leaves open the possibility that some other convergence result does apply even in the face of EE, which might make probabilistic antirealism a corollary to another mathematical 22 See Pacuit and Parikh [2005] for a plausible proposal for an axiomatization of the notion of practical possibility. They also suggest that (1) might be best read in terms of this notion. 23 This has already been noted by Earman [1992:149 ff]. 24 There are explications of empirical equivalence such that EE is incompatible with the assumption of separation even without the assumption that there are any hypotheses that are complete with respect to E (this is for instance so if empirical equivalence is understood in the sense of Earman’s [1993] notion of EI3 theories). But since that assumption is a quite innocuous one, we can content ourselves with the general and simple characterization of empirical equivalence given in the text.

15

result. It is easy to show, however, that given EE and our assumption of strictly coherent initial probabilities, there simply can be no convergence result of the generality of Theorem 4.1. First notice that from the above characterization of empirical equivalence it follows that for the Bayesian the notion of two hypotheses’ H and H 0 being Vn empirically equivalent is to be defined so as to imply that, for all i=1 Ei with Ei ∈ E

Vn Vn 0 25 (1 à i à n), Pr i=1 Ei | H = Pr i=1 Ei | H . But then convergence, in the sense that
Vn limn→∞ Pr H | i=1 Ei@ = 1 for any true H, or even one true theoretical hypothesis H, cannot occur given EE (here, Ei@ is Ei if Ei is true at the actual world, and ¬Ei otherwise).

Vn Vn Quite evidently, limn→∞ Pr H | i=1 Ei@ = 1 implies that limn→∞ Pr H 0 | i=1 Ei@ = 0 for any H 0 that is inconsistent with H, which in turn implies that the ratio between the probability of H and that of H 0 goes either to 0 or to infinity in the limit. However, if H and H 0 are empirically equivalent, then it follows by Bayes’ theorem and the above-stated Bayesian requirement for empirical equivalence that

Vn Vn Pr(H) Pr i=1 Ei | H Pr H | i=1 Ei Pr(H)
=
= Vn Vn , 0 0 0 Pr(H 0 ) Pr H | i=1 Ei Pr(H ) Pr i=1 Ei | H

(27)

Vn for any i=1 Ei with Ei ∈ E (1 à i à n). That is to say, if H and H 0 are empirically equivalent, then the ratio between their probabilities will have a fixed value—namely, that of the ratio between their initial probabilities—however much evidence accumulates. Given our assumption that all contingent sentences have a positive initial probability, this means that that ratio cannot converge to 0 or infinity in the limit. Consequently, given EE, convergence is impossible. In itself, the foregoing does not imply that if EE holds we cannot obtain t-evidence for all true theoretical hypotheses. However, with a very minor additional assumption, it does. The additional assumption is what we may call the Weak Principle of Indifference (WPI). It says that, given (finitely or infinitely many) empirically equivalent rivals H0 , H1 , H2 , . . . , there should be no i ∈ N such that Pr(Hi ) t ! > . Wn 1−t lim Pr j=0 Hj

n→∞

j≠i

So, if t = .9 for instance, then this principle implores one not to assign a probability to any hypothesis that is more than nine times that of the sum of the probabilities assigned to the hypothesis’ empirically equivalent rivals. To see how, given this additional assumption, EE implies the falsity of (5), suppose that H is true and that H 0 is an empirically equivalent rival of it; without loss of generality, assume that it is the only such rival. It then follows directly from (27) that however many evidence sentences we will come to know, the ratio between the posterior probabilities of the two hypotheses will remain the same as that between their prior probabilities. But that means that unless H has a prior probability more than t/(1 − t) times that of H 0 , it can never come to have a posterior probability exceeding t. And so, given that WPI prohibits us from assigning a prior to H that is more 25 Given Bayesian confirmation theory, H and H 0 are guaranteed to have the same confirmation-theoretic status (on what seems to be the natural interpretation of this phrase) only if, in addition to satisfying the requirement given in the text, they have equal prior probabilities; for more on this, see Douven [2005b] and [2007]. But as we shall see, for the present discussion the crucial feature of empirically equivalent theories is that they bestow the same likelihood on all evidence sentences and conjunctions of evidence sentences.

16

than t/(1 − t) times the prior we assign to H 0 , hypothesis H can never come to have a probability exceeding t. In other words, we cannot obtain t-evidence for H, contrary to what (5) implies. So, formally expressed, together EE and WPI imply ∃ϕ(ϕ ∧ ¬♦ E ϕ), which is the negation of (5). In fact, while in the literature it is typically EE that is discussed, it is worth noting that, given WPI, already a weaker version of EE is fatal for probabilistic antirealism. The weaker version claims merely that for some, and not necessarily all, theoretical hypotheses there exist empirically equivalent rivals. For, that some have empirically equivalent rivals leaves open the possibility that some true hypotheses (or even all true hypotheses) have empirically equivalent rivals. And if that is so (and WPI is assumed), then the above argument to the effect that, given EE and WPI, we cannot obtain t-evidence for H, can be repeated almost verbatim (since nothing in it depends on whether other hypotheses besides H and H 0 have empirically equivalent rivals). We in effect can derive ∃ϕ(ϕ ∧ ¬♦ E ϕ) already on the assumption that one true theoretical hypothesis has an empirically equivalent rival (assuming WPI). To be sure, given the weak version of EE we can only suppose that it is possible that some true hypotheses have empirically equivalent rivals and thus also only arrive at the weaker conclusion that it is possible that we cannot obtain t-evidence for some truths. But this is enough to refute probabilistic antirealism, given that, of course, (5) is supposed to hold by conceptual necessity, so that the probabilistic antirealist is really committed to ∀ϕ(ϕ → ♦ E ϕ). And this is the negation of what we said can be derived from weak EE and WPI, namely, that it is possible that there exist truths for which we cannot obtain t-evidence, that is, ♦∃ϕ(ϕ ∧ ¬♦ E ϕ). Thus, the probabilistic antirealist seems committed to denying at least one of EE— even the weak version thereof—and WPI. Since the holding (or otherwise) of either is not a purely logical or mathematical matter, this suffices to make probabilistic antirealism a nontrivial thesis.26 But does it not also make it a hopeless one? Consider the standard version of EE first. This thesis has been accepted as being obviously true for a long time, but more recently it has come to be regarded with suspicion. Indeed, it is no exaggeration to say that, at least among philosophers of science, there is currently a rather large majority who openly doubt the thesis. Here I will not go into the various arguments that have been advanced pro and con EE in the past two decades or so, and refer the reader to my [2007] for an overview. I do want to mention one reason for denying EE, which has actually been around for a longer time, and which has a distinctively antirealist ring to it, namely, that since empirically equivalent rivals have the same empirical content, they have the same content tout court, that is, they really are just notational variants of one another. To be sure, this response to EE has few advocates left. However, that may be largely due to the fact that the equating of a theory’s empirical content with its total content has typically been backed up by reliance on some verificationist theory of meaning of a kind that from a modern perspective looks crassly simplistic. There certainly exists no argument to the effect that all theories of meaning that make what appear to be 26 The alleged proofs of EE offered in Earman [1993] and Douven and Horsten [1998] could easily lead one to think that at least the holding of EE is a purely logico-mathematical affair. However, whether the assumptions these proofs rely on—like that the class of evidence sentences can be defined syntactically— are warranted does not seem to be a purely logico-mathematical question.

17

empirically equivalent rivals come out as being, at bottom, the same theory, must be simplistic or clearly mistaken.27 The weak version of EE is of course harder to tackle. It seems that, where arguments for EE may be said to rely on questionable abstractions and idealizations (see, for instance, the arguments in Earman [1993] and Douven and Horsten [1998]), in order to support the weak version all one has to do is point to some actual examples of empirically equivalent rivals. And here the often-mentioned cases of the special relativity theory versus the æther theory in the Lorentz–Fitzgerald–Poincaré version and, especially, of standard quantum mechanics versus Bohmian mechanics already seem cases in point.28 Note, though, that the just cited meaning-theoretic response to the standard version of EE is, if effective in the case of that version, also effective in the case of the weak version as well. For if the response is correct, then the special relativity theory and the æther theory are mere notational variants of one another, and similarly for standard quantum mechanics and Bohmian mechanics. As for WPI, the only motivated way of discrediting this principle seems to be by arguing that other factors beyond purely empirical ones—such as simplicity or explanatory power—are indicative of truth and therefore should have a say in determining prior probabilities. There are two broad ways in which one can try to do this. First, following the American pragmatists, one might try to argue that there is a conceptual connection between truth and certain nonempirical factors: Explanatory power (or whatever) is indicative of truth because it is (presumably partly) constitutive of it, that is, being a good explanation is (in part) what makes a true theory true. Not many philosophers nowadays will think this is a promising way to go. Second, one can try to show empirically that there is a connection between truth and certain nonempirical factors. Contrary to what is often thought, this is not necessarily a dead end; in particular, no circularity need be involved in such empirical arguments (cf. Douven [2002b], [2005b]). Some authors have already pointed to what seems to be fairly convincing evidence for the existence of the said connection, in particular a connection between truth and explanatoriness (see, for instance, Harré [1988] and Bird [1998:160]). Still, by itself the existence of such a connection does not jeopardize WPI, for it is clearly compatible with our respecting WPI that we let nonempirical factors have some role in determining priors. To let them have a role that is incompatible with our respecting WPI, on the other hand, is to decide on the basis of such factors, previous to having done any empirical research, of a particular one among a number of empirically equivalent rivals (say, T17 ) that if these theories should be empirically adequate, then this theory, T17 , is the one that will eventually become justifiedly credible to us. And making this kind of choice in such an a priori fashion would seem reasonable only if there existed a very strong link between truth and those nonempirical factors, to the extent even that these factors are an unfailing, or at least near-to-unfailing, mark 27 Reichenbach’s [1938, Ch. 1] so-called probability theory of meaning would already seem a counterexample to that claim. The theory has been thought to come to grief over the Quine–Duhem problem (cf. Putnam [1983:31 f]), but while this problem may be a serious one for the early Hempelian theories of confirmation, Bayesians seem able to respond successfully to it; see, for instance, Dorling [1979] and Strevens [2001] (for a discussion see also Fitelson and Waterman [2005] and Strevens [2005]). Nor does it appear to beset the probabilistic version of bootstrap confirmation theory proposed in Douven and Meijs [2006]. 28 Although according to Albert [1992:134] “Bohm’s theory has more or less (but not exactly) the same empirical content as quantum mechanics does . . .” (emphasis added).

18

of truth. And no evidence available to date warrants belief in the existence of such a link.29

5. But is it Antirealism? Even if probabilistic antirealism is a substantive and at least prima facie viable thesis, one may still wonder whether it is antirealism properly so called and, in particular, whether it serves the purposes of those who thought (1) was to be imposed as a constraint on theories of truth. Early reasons for imposing this constraint emanated from the simplistic theories of meaning hinted at previously. But what may well be currently conceived of as the main motivation for (1) is in effect based on a quite sophisticated view of meaning, namely, Dummett’s. Key to the motivation for (1), on this view, is what is sometimes called the manifestability requirement, which demands that knowledge of sentence-meaning be fully manifestable in the practice of speaking. So, if sentence-meaning is identified with truth conditions, as Dummett grants the realist, the question becomes how knowledge of such truth conditions can be manifestable.30 While for some sentences, such knowledge can be manifested by giving an explication of their truth conditions using different, but equivalent sentences, this cannot be the case for all, on pain of going round in circles; for some sentences, knowledge of their meanings must be implicit, and can be manifested only by asserting them when (or if) their truth conditions obtain. To say that some truths may be unknowable is to say that the truth conditions of some sentences may obtain unrecognizably. But if that is so, a speaker will never have warrant for asserting these sentences, whereby the manifestability requirement would be violated. Thus, the manifestability requirement entails (1). Or so Dummett’s argument goes. While much is controversial about this argument,31 the underlying assumption that assertion requires knowledge (recognition of truth) is currently shared by many if not most philosophers.32 However, elsewhere I have argued, against this near29 As I have argued in my [2002b], the evidence mounted by Bird, Harré, and others, for the existence of a link between truth and explanatoriness supports at most the hypothesis that, given a number of empirically equivalent theories that are all in accordance with the available data, the one that explains those data best (supposing there is a unique one) is closer to the truth than any of the others. Indeed, it would seem hard if not impossible to get evidence for anything stronger than this, given that in general there is no guarantee that the truth is among the hypotheses we consider; for more on this, see van Fraassen [1989, Ch. 6]. 30 In early presentations of the present argument (e.g., in Dummett [1976]), it was directed against a truth-conditional view on meaning. But later Dummett came to think that the best way to conceive of it was as undermining theories of meaning formulated in terms of a recognition-transcendent notion of truth, and not truth-conditional theories of meaning per se. See, for instance, Dummett [1978:xxii]: “I should now be inclined to say that, under any theory of meaning whatever . . . we can represent the meaning . . . of a sentence as given by the condition for it to be true, on some appropriate way of construing ‘true’: the problem is not whether meaning is to be explained in terms of truth conditions, but of what notion of truth is admissible”; see also his [1978:xxx, 8, 22 f, 358 f] and [1979:3]. The argument still poses some interpretive problems, though; see, e.g., Kirkham [1992:249 ff]. 31 For a general critique of it, see Devitt [1983]; also Fodor [1998:5 f]. Various authors have raised the specific complaint that Dummett’s argument reflects a very narrow conception of what it is to manifest knowledge of truth conditions; see for instance Prawitz [1977], [1994] and McGinn [1980]. A reason for thinking the complaint is just, though not one noted by the aforementioned authors, is that there seem to be sentences our knowledge of the truth conditions of which we can manifest only by asserting something false. Here one may think of sentences such as “All of us are silent now” (cf. Lewis [1996:238]), whose truth conditions will fail to obtain whenever they are asserted. It seems that in order to manifest our knowledge of the truth conditions of such sentences, we must be able to rely on a readiness of the addressee to be charitable in interpreting our words. This appeal to charity does not seem to commit one in any way to a non-epistemic notion of truth and so seems open to the antirealist. 32 See, among many others, Williamson [2000b], Adler [2002], and DeRose [2002].

19

consensus, that justified credibility is enough to warrant assertion.33 If that is so, then, patently, we do not need (1) as a constraint on truth, but can instead make do with (5) to satisfy Dummett’s manifestability requirement. After all, if we can obtain t-evidence for any truth, as (5) implies, and thus obtain justification for any truth, then no truth is unassertible in principle.34 Consequently, given probabilistic antirealism, knowledge of truth conditions is fully manifestable.35 It it is worth stressing that more follows from this than that probabilistic antirealism serves the antirealist’s (or at any rate Dummett’s) purposes. It also follows that insofar as the motivation for the epistemic constraint on truth is to come from the assumption that grasp of meaning must be fully manifestable, we can at most argue for (5), and not for the more demanding (1).36 We must briefly come back to the potential problems for probabilistic antirealism discussed in section 3. For although we saw there that good sense could be made of claims to the effect that it is possible to obtain t-evidence for true instances of (6), (10), (12), and (20), it might still be implausible to hold that sentences of any of those forms could be assertible. But given the account of assertion I endorse, having t-evidence for any such sentence suffices to make the sentence assertible.37 To forestall misunderstanding, I should begin by emphasizing that my view is not that a sentence is assertible for one only if one has obtained t-evidence for it, nor even that a sentence is assertible only if one assigns a probability greater than t to 33 See Douven [2006]. Sometimes Dummett seems to come close to the position I propose, for instance, when he ties meaning to “the evidence which justifies the assertion of a statement” (Dummett [1978:370]), but it should be emphasized that the notion of evidence involved in this claim is that of conclusive evidence, establishing the truth of the given statement, not the Bayesian notion of evidence or even that of t-evidence. In fact, otherwise his insistence on (1), or at least a version thereof (cf. Dummett [2001]), would be incomprehensible. 34 We are talking here, of course, about contingent truths only, to which (5) is restricted. But that we can obtain justification for any necessary truth follows immediately from our assumption that (1) pertains to them; a truth could not be known without our having justification for it. 35 It might be said that there is still the problem that since according to (5) it is merely possible for any truth that someone at some time will obtain t-evidence for it, there could be true sentences which we will never actually be justified in believing and thus that there could be sentences the grasp of whose meanings we will never actually be in a position to manifest. Note, however, that the original position faces exactly the same problem as regards the knowability of all truths; (1) makes no stronger a claim than that for any truth it is possible that someone at some time knows it. Of course one could strengthen (1) to “Every person will come to know every truth at some time in his or her life,” and similarly strengthen (5). But it does not take anything like Fitch’s paradox to show either to be untenable. There might be a further problem, though, specifically for the combination of probabilistic antirealism and the view that justified credibility warrants assertion. The problem, which was brought to my attention by Timothy Williamson, is that since it follows from probabilistic antirealism that some sentences can be confirmed (in the strong sense that we have t-evidence for them) only when they are false—like (12) for instance—so that, given my view on assertion, some truths would have been warrantedly assertible only if they had been false, it is unclear how the assertibility conditions of such sentences could make their truth conditions manifest. But while I can see how this may seem problematic if it is assumed—what seems to follow from Dummett’s conception of manifestation—that only by making a truthful assertion can one manifest one’s knowledge of the truth conditions of the asserted sentence, it is not clear to me that any problem remains once it is observed, as we did in note 31, that, first, making a false assertion may do—and will sometimes be all we can do—to manifest our knowledge of the truth conditions of the asserted sentence, and second, the antirealist is not committed to Dummett’s strict conception of manifestation. 36 Hand [2003] argues on very different grounds that meaning-theoretic considerations of the kind proffered by Dummett fail to warrant acceptance of (1) (which is not to suggest that Hand would endorse (5)). 37 Well, not quite. A sentence for which one has t-evidence might also on my account still be unassertible, for instance, for Gricean reasons or for more broadly social reasons. Here we can ignore such factors without loss of generality.

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it. If it were the former, then all of the aforementioned types of sentences would be Moore-paradoxical; if it were the latter, then at least instances of (12) would still be Moore-paradoxical. However, on our account of justification one can be justified in believing a sentence for which one has not obtained any t-evidence (and perhaps never will obtain such evidence), for instance because one initially assigned it a probability greater than t and one has received no evidence against it; that sentence will be assertible. Also, a sentence need not be assigned a probability greater than t in order to be assertible; it might follow from sentences that are assigned a probability greater than t, and for that reason be justifiedly credible. Nevertheless, the combination of probabilistic antirealism and the view that justified credibility warrants assertion may still seem to give rise to Moore-paradoxical sentences. Consider the following variant of (12):38 ϕ, and no one will ever be in the epistemic position to assert ϕ.

(28)

While we may be able to make sense of the supposition that someone has t-evidence for (28), the sentence hardly seems assertible. What must be admitted is that an assertion of (28) would sound odd. But this does not necessarily compromise either probabilistic antirealism or the account of assertion I advocate. First, as I have explained in detail in Douven [2006, Sect. 5], the argumentative force of purportedly Moore-paradoxical sentences seems limited at best. The odd-soundingness of such sentences need not betoken any inconsistency or other deep problem, but may simply be due to a lack of exposure to the given sentence, where this lack of exposure may then have some further explanation; for example, there may be good pragmatic reasons for not uttering the sentence—as I argue in the aforementioned paper, it might thwart our attempt to convince our audience—or the world might be such that we find little or no opportunity to use them. Second, for (28) there appears to be an even more direct explanation of why an assertion of it would sound odd, namely, that such an assertion would be odd. For consider that it would be highly surprising if someone asserting (28) would not, at the latest in the course of doing so, come to realize that, given the epistemic position she must be in to be warranted in asserting (28), the second conjunct cannot be true. We would certainly expect that, as a result, she would revise her opinion about (28), and perhaps stop speaking somewhere halfway through the sentence, or at least revoke her assertion of the sentence once she had finished it. One may have t-evidence for (28) without ever having been even consciously aware of the sentence, but it would seem impossible to maintain that one can assert (28), or even just consider doing so, without being consciously aware of it. So if one were then nonetheless to assert it, this would seem to bespeak inattentiveness on one’s part of the grossest sort. In short, one can be in the epistemic position to assert (28); it just strains credulity to suppose that one can remain in that position even while only contemplating asserting the sentence. That, I submit, is what would make an assertion of (28) odd, which in turn explains why we are inclined to deem it unassertible. I end by considering another motivation for antirealism, one of a more directly epistemological nature, which may come from the antirealist’s anti-skeptical intuition which Putnam [1977], [1980] has famously expressed by the claim that An epistemically ideal theory cannot be false,

(29)

38 Given the account of assertion we are supposing, the sentence is actually equivalent to “ϕ, and no one will ever be justified in believing ϕ” (cf. note 21).

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where an epistemically ideal theory is one that satisfies all theoretical virtues and that “does not lead to any false predictions” (Putnam [1980:13]). I am speaking here of an intuition because, although antirealists seem to share a strongly-felt conviction that the negation of (29) makes no sense, or at least that it is too much to believe, one is hard put to locate an argument for this in the antirealist literature;39 and I say that a motivation for antirealism may come from this intuition because I am unaware of explicit attempts to relate (1) to (29). Nevertheless, it will help to further underline the antirealist character of (5) to show that by adopting it as a constraint on truth antirealists automatically honor the intuition. From this it also follows that no stricter constraint than (5) could be motivated by an appeal to (29). The argument presupposes WPI, but, as we saw in the previous section, this may be hard to deny anyway. It assumes that we have arrived at an epistemically ideal theory, TI , but that TI is false, and derives the falsity of probabilistic antirealism from that assumption. By contraposition we then obtain the desired conclusion that probabilistic antirealism entails (29). Since TI is complete—completeness being one of the theoretical virtues (cf. Putnam Vn Vn [1977:125])—it must hold for all i=1 Ei with Ei ∈ E (1 à i à n) that either TI î i=1 Ei

Vn Vn Vn or TI î ¬ i=1 Ei , and hence that either Pr i=1 Ei | TI = 1 or Pr i=1 Ei | TI = 0 for all Pr. Observe that the same must hold for the complete true description of the world, which we may call T Tr . Since, furthermore, TI is supposed not to lead to Vn Vn any false predictions, it must hold that TI î i=1 Ei a T Tr î i=1 Ei , and therefore

Vn Vn that Pr i=1 Ei | TI = Pr i=1 Ei | T Tr , for all evidence sentences and conjunctions of evidence sentences. In other words, TI and T Tr are empirically equivalent.40 Since they are both complete, and T Tr is true and TI is false (by supposition), their conjunction must be inconsistent. Hence they are empirically equivalent rivals. So then, given WPI, we have ∃ϕ(ϕ ∧ ¬♦ E ϕ); as noted earlier, that follows from WPI as soon as one true theoretical hypothesis has an empirically equivalent rival. The final step in the argument is now to cancel the assumption that an epistemically ideal theory may be false, which yields41 ♦∃T EpId(T ) ∧ ¬T




→ ∃ϕ(ϕ ∧ ¬♦ E ϕ).

(30)


∀ϕ(ϕ → ♦ E ϕ) → ∀T EpId(T ) → T .

(31)

And by contraposition, (30) yields

39 Contrary to what some might think, Putnam’s [1977], [1980] celebrated model-theoretic argument is not such an argument. The argument at most shows that the combination of a correspondence theory of truth and a physicalistic theory of reference is incompatible with the supposition that even an epistemically ideal theory might be false. (For reasons given in Douven [1999b], I disagree with the more or less common judgement that the argument is invalid. This is not to say that I believe its conclusion holds true; see Douven [1999c].) That does not explain why antirealists, who typically are committed to neither of the said premises, should believe that an epistemically ideal theory must be true. 40 Or at least—if one thinks a “Bayesian” definition of empirical equivalence should include the requirement that empirically equivalent theories have the same prior probability (cf. note 25)—TI and T Tr are so related that all the results of the previous section relevant to the present argument apply to the theories. 41 I am assuming that the theories over which we are quantifying are finitely axiomatizable. Else we need to replace “¬T ” in (30) by “¬Tr(T )” where Tr is the truth predicate (and make the corresponding substitution in (31)).

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That is, probabilistic antirealism entails that an epistemically ideal theory cannot fail to be true.42 In sum, we appear to have a position that postulates an epistemic constraint on truth strict enough to serve the antirealist’s purposes, and that escapes Fitch’s paradox and other potential paradox-like problems. It seems that nothing should keep antirealists from becoming probabilistic antirealists.

Acknowledgements. I am greatly indebted to Timothy Williamson for extensive and extremely helpful comments on an earlier version of this paper. I am also grateful to Leon Horsten, Marietje van der Schaar, and Christopher von Bülow for valuable feedback. A version of this paper was presented at the 2006 conference of the Dutch– Flemish Society for Analytic Philosophy, which was held in Amsterdam. Thanks to the audience for helpful questions and remarks.

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