Philosophical Perspectives, 22, Philosophy of Language, 2008 BETTING ON BORDERLINE CASES 1

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1 Philosophical Perspectives, 22, Philosophy of Language, 2008 BETTING ON BORDERLINE CASES 1 Richard Dietz University of Leuven Arché Research Centre, University of St. Andrews 1 Introduction Opinion comes in degrees. One may be more or less confident in a proposition. Not every opinion is reasonable (in other words, rational); e.g., it does not seem reasonable to be more confident in the propositional content of contradictory statements than in the propositional content of logical truths. If we can be confident to higher or lesser degrees regarding the contents of some sentences, so can we also be regarding the contents of some sentences that we regard as potentially indefinite in truth-value in other words, as vague in truth-value or borderline. 2 E.g., suppose we are confronted with two cylindrically shaped glasses of water, the first one being 100% filled, the second one, being a borderline case of being full, only to 50% plausibly, our degree of belief for the proposition that the first one is full ought to be higher than our degree of belief for the that the second is full. What is less evident is whether vagueness in our language has any impact on the normative constraints on degree of belief. In his (2000) paper Indeterminacy, degree of belief, and excluded middle, Hartry Field argues for answering the question in the positive. He makes a case for two radical theses. For one, he contends that the standard Bayesian approach to subjective probability is inappropriate for the everyday case that we consider some sentences of our language as potentially indefinite (in truth-value). Standard Bayesianism has it that the degrees of belief of a rational agent obey the classical probability calculus. That is, as defined for sentences in a language PL of propositional logic, probability functions (1) take values within the unit interval of reals [0,1] for each sentence, (2) take the maximum value 1 for tautologies, and (3) take for disjunctions the sum of the probabilities of their disjuncts, if the latter ones are jointly inconsistent (finite additivity). As a joint consequence of (2) and (3), for any sentence s, the probabilities of s and of its negation sum up to 1. Field does subscribe to the classical calculus, as far as precise languages are concerned, that is languages where every sentence is treated as definite in

2 48 / Richard Dietz truth-value in other words as precise in truth-value. He contends though that the classical calculus is inadequate, insofar as we consider also vague languages, that is languages where some sentences s are considered as potentially indefinite, in the sense that we have a positive partial belief that s is indefinite. 3 Specifically, on Field s suggested non-classical calculus, for any sentence s that is not taken as definite, the probabilities of s and of its negation do not sum up to 1. His revisionism about probability is not motivated by any qualms against classical logic for vague languages. Starting himself from classical logic for precise as well as for vague languages, he subscribes not only to the standard constraint (1), but also to (2) in general. However he abandons the finite additivity constraint (3) for vague languages: if we do not consider a sentence s as definite, the probabilities of a sentence and its negation do not sum up to the value of their disjunction. For another, Field contends a reductionist thesis regarding the probabilities of indefiniteness statements that goes beyond what is to be expected on classical probability. According to this, if a sentence is considered as potentially indefinite, this amounts to the case that we should have beliefs in the sentence and its negation to degrees that are structured in a distinctive way. In effect, this idea seriously challenges the received line in philosophical discussion. The latter has it that for giving a full account of what it means for a sentence to be considered as potentially indefinite, we need first of all an account of what it means for a sentence to be indefinite. Field suggests that this view is misguided. In more recent papers (2003a and 2003b), Field takes a position that essentially departs from his (2000) paper in that it abandons a classical logic in favor of Strong Kleene logic for vague languages. To discuss the further epicycles of his theory of vagueness would go beyond of the scope of this paper. I focus here on Field (2000) as an account in its own right. If not otherwise qualified, page references are to be understood elliptically as references to Field (2000), and Field abbreviates Field (2000). 4 My aim is two-fold. In the first place, it is suggested that Field s reductionism about indefiniteness is only tenable on logical provisos that are problematic in a classical semantics, but not so in a supervaluationist semantics. Second, I am outlining an argument in favor of Field s theory, which is more forceful than Field s own argument. Contrary to his suggestion, the standard Bayesian argument for classical probability, where probabilities are modelled as fair betting quotients, works also for vague languages, assuming a classical logic for vagueness. On the other hand, it may also be employed for motivating the Fieldian calculus, provided that (a) we do not select a classical but a supervaluationist logic for vague languages, and that (b) the standard account of degrees of belief in terms of bets is appropriately generalized. In summary, my arguments suggest that Field s model of subjective probability has some motivation in a supervaluationism but pace Field, it is ill-motivated on a classical framework for vagueness, and by no account does it challenge the Bayesian approach to subjective probability. To begin with, Field s formal theory of subjective probability is set out ( 2). It is noted that for Field s reductionism about indefiniteness to be viable, his favored normal modal logic for definite truth is to be strengthened to

3 Betting on Borderline Cases / 49 S5 logic ( 3). I show that if we refer to S5 logic, Fieldian probabilitities are representable as classical probabilities ( 4). Finally I consider arguable motivations for Fieldian probability, the conditional argument from expectations, and Dutch-Book arguments. It is suggested that if such arguments work for precise languages, then they do so also for vague languages. In fact these argument strategies can be employed also in support of Field s revisionism if we refer to a supervaluationist logic instead of a classical logic. I am going to give a Dutch-Book argument for Field s non-standard calculus in both directions ( 5). Field s theory of degree of belief for vagueness overlaps with various other central problems in probability theory and the theory of vagueness, which are not at issue here. This is the right place for some disclaimers. (1) What are degrees of belief? What should they go with? Evidently opinion comes in degrees. Less evident is what degrees of belief more exactly are, and what structural rules they should obey. Bayesian arguments for classical probability (a) start from an account of what degrees of belief are and (b) provide an argument that those things should obey the classical probability. In Dutch-book arguments, degrees of belief are assumed to be either definable, or at least measurable in terms of betting quotients considered as fair; the aim of such arguments is to show that fair betting quotients are susceptible to a pragmatic self-defeat if they violate the classical calculus. 5 A different famous probabilistic argument from utility theory shows that any rational ordering of preferences can be represented as maximizing expected utility as calculated from a utility and a classical probability function which, if we think of degrees of belief in a functionalist fashion as the things that as multiplied with utilities represent expected utilities, suggests that degrees of beliefs obey the classical calculus. 6 My discussion will focus on Dutch-book arguments and Field s suggestion that they are rejectable for vague languages. There are numerous well-known objections to this kind of argument, as well as against the argument for classical probability from utility theory both against (a) the underlying account of degree of belief and against (b) the argument that those things obey classical probability. 7 For all those objections, the distinction between fully precise and vague languages is irrelevant. In fact, Field does not voice any doubt on Dutch-book arguments, as far as fully precise languages are concerned. What is at issue here is whether the Dutch-Book argument strategy works for vague language, granted that it works for fully precise languages. What is not at issue here is whether this argument strategy effectively works for fully precise languages. For the purposes of our discussion, we can therefore follow Field s line leaving aside more general concerns about Dutch-book arguments that apply equally to fully precise languages. (2) Logic for vague languages. I am concerned with the question of whether Field s revisionism about the structure of subjective probability and his reductionism about indefiniteness are viable options on his suggested logical framework for vagueness i.e., a classical logic. I am not concerned though with

4 50 / Richard Dietz arguments in favor of classical logic or the popular objections against it. In view of the less well-known but as serious problems of the standard contenders (e.g., supervaluationist logic, fuzzy logics, intuitionist logic), it seems premature not to consider a classical logic as an arguable option. 8 The same consideration applies to a supervaluationist logic for vague languages, which I will finally suggest as a better match with Field s position. It is not my aim to defend the latter type of logic. (3) Degree of belief, belief, and borderline cases. If it is clearly indefinite whether P, can we reasonably claim (i.e., believe or assert) that P, or alternatively claim that not P? Field contends that if a sentence is considered as indefinite, one ought to reject any (assertoric) use of the sentence where rejection is meant to be weaker than the corresponding claim of falsity and stronger than the corresponding claim of truth. According to Field, the relevant qualitative notion of rejection can be cashed out in quantitative terms of the degree of belief. Be that as it may, I will not discuss this arguable application of Field s theory of subjective probability. 9 (4) Diachronic coherence. Field keeps the focus on synchronic coherence. I will do the same. (5) Links with Schiffer and Edgington. Three guiding ideas in Field s theory are (i) the idea that subjective probability should depart from the classical probability for vague but only for vague languages; (ii) the idea that subjective probabilities should have the structure of associated classical probabilities of definite truth (rather than the structure of classical probabilities simpliciter); (iii) the rejection of Bayesian preference-based accounts of degree of belief that suggest some sort of correspondence between degree of belief and betting behavior. As for the circumscribed revisionist thesis (i), Field agrees essentially with Stephen Schiffer s recent theory of degree of belief, as presented in his The Things We Mean (2003). In contrast to Field, however, Schiffer does not offer a unified formal model for degree of belief that covers both precise and vague languages: whereas degrees of belief for precise languages of propositional logic are supposed to obey classical probability, for vague languages of propositional logic, they are supposed to obey the truth-value tables that are familiar from Łukasiewicz s infinitely-many valued fuzzy semantics (i.e., sentences take reals in the unit interval [0, 1] as values, v( A) = 1 v(a), v(a B) = min{v(a), v(b)}, v(a & B) = max{v(a), v(b)}.) 10 According to Schiffer, strictly speaking, there is not one kind of belief attitude, where associated degrees are supposed to obey either these or those structural rules. Rather, depending on whether we take a sentence as definite or consider it as vague, we have different kinds of attitudes. 11 I agree with Pagin (2005) and Field (forthcoming a), who dismiss both the suggested dualism about belief and fuzzy-type constraints on degree of belief for vagueness as being weird and ill-motivated results. Even worse, as MacFarlane (2006) observed, Schiffer s constraints rule out a wide range of cases that should be considered as possible. 12 In view of these points, it is legitimate to put Schiffer s theory aside. As for the other two mentioned ideas (ii) and (iii),

5 Betting on Borderline Cases / 51 there are links between Field and Dorothy Edgington s position, as formulated in her Vagueness by degrees paper (1996). Edgington offers constraints for degrees of closeness to definite truth, which she calls verities. Unlike Field, she does not mean to make a case against a classical probability calculus for degree of belief. In fact, she does not address the issue of subjective probability for vague languages. Her point is rather that verities do not have the functional role subjective probabilities have in practical reasoning, if we consider vague languages specifically, we cannot reckon in general then that as multiplied with utilities, verities represent expected utilities. This sounds close in spirit to Field s position that the degrees of belief do not correspond with fair betting quotients, as far as vague languages are concerned. Also Edgington s interpretation of verities seems akin to Field s intended interpretation of subjective probabilities as measuring classical probabilities of definite truth. Unlike Field s subjective probabilities, however, Edgington s verities have a classical probabilistic structure also for vague languages. I will not go any further here into links between Edgington s and Field s positions. 2 Field on degree of belief and definite truth Field derives his calculus for subjective probability from the classical probability calculus and some further assumptions. I will come back later to his way of motivating his calculus ( 5). At this stage, only the formal theory is relevant. As Field notes, there is a close connection between his theory and Glen Shafer s calculus of belief functions (also known as Dempster-Shafer theory). As far as the language of propositional logic (PL) is concerned, there is in fact no difference between Field and Shafer. I therefore start with introducing the classical calculus and Shafer s calculus for PL ( 2.1). In the next step, I will turn to Field s calculus of probability for a language of PL in which indefiniteness is expressible ( 2.2). 2.1 Shafer s calculus of probability for the language of propositional logic (PL) Field formulates his calculus for probability functions over languages, but in the further discussion, it will turn out to be helpful to consider also probability functions over spaces of possible worlds. Insofar as subjective probability should reflect the epistemic situation of an agent, it is most natural to think of the relevant modal space W as a set of epistemically possible worlds. Subsets of W are standardly referred to as propositions which on an epistemic reading of the relevant modal space, we can understand only in a transferred sense, considering that the term proposition is typically reserved for sets of metaphysically possible worlds or entities that fix such sets. For the general case that we want to make allowance for uncountable modal spaces, it is typically required that

6 52 / Richard Dietz probability is defined on a family F of subsets of the space W of possible worlds that is a σ -field, i.e., it includes W and is closed both under complementation with respect to W and under countable union: 1. W F; 2. For all p F, the absolute complement of p is an element of F; 3. For any countable set Ɣ of members of F, the union of members of Ɣ is an element of F. Shafer s original calculus is only designed for the algebra 2 W on a modal space W. 13 It can be easily generalized, so that the domain is an arbitrary algebra over W. A function S on a σ -field F on W is a Shafer function just in case for any elements p, p 1,..., p n of F: (S1 set ) S(p) 0; (S2 set ) S(W) = 1; (S3 set ) S( ) = 0; (S4 set ) S(union of p i, for all i = 1, 2,...n) i {1,...n} {I {1,...n}: I =i} ( 1)i+1 S( j I p j ). (S4 set ) is just the classical inclusion-exclusion rule with = replaced by. If we consider only finite spaces, we can characterize Shafer probability for any space W in terms of a mass function (or basic probability function) on W. A mass function of W is a function m: 2 W [0, 1], satisfying the conditions: M1 : m( ) = 0; M2 : U W m(u) = 1. Given a mass function m, the Shafer function based on m, S m, is defined as: S m (U) = U U m(u ). 14 Shafer interprets mass functions as measurements of evidential support: m(p) measures the given evidential support for exactly p. Shafer probabilities for propositions, according to this, measure the direct and indirect evidential support: S(U) measures the given evidential support for U and any proposition entailing U. 15 As a consequence, we can distinguish between the case that A lacks evidential support (i.e., the case where (1 S(A)) is high) and the case that A has high evidential support (i.e., the case where S( A) is high). Here is a translation of Shafer s calculus for the language PL: a function S on a collection C of sentences (closed under standard truth-functions) of PL is a Shafer function just in case it meets for classical entailment CL, for all sentences A, A 1,..., A n, B in C the conditions:

7 Betting on Borderline Cases / 53 (S1 PL ) 0 S(A) 1 (S2 PL ) If CL A, then S(A) = 1 (S3 PL ) If CL A, then S(A) = 0 (S4 PL ) S(A 1... A n ) i S(A i ) i< j S(A i & A j ) ( 1) n+1 S(A 1 &...&A n ) (S5 PL ) If A CL B, then S(A) S(B). It is to be noted that (S5 PL ) is not implied by (S1 PL )-(S4 PL ) (see Appendix C, proposition 1). Every classical probability function is a Shafer probability function. On the other hand, some Shafer functions are not classical probability functions, e.g., consider a set W = {w 1, w 2 } with a mass function that assigns to each nonempty subset of W the value 1/3. The associated Shafer function then assigns to the proposition {w 1 } the value 1/3 and the same for its absolute complement {w 2 }, whereas their union takes the value 1. That is, we have a counterinstance to finite additivity Field on probability for the language of propositional logic + definite truth (PL+) Being vague as applied to sentences is an intuitive notion, familiar from everyday speech and thought. In this respect, it is nothing like Borel set, neutrino or Turing machine. There are numerous suggestions of formalizing the concept, standardly as a sentence operator for indefiniteness ( ), which is definable in terms of a sentence operator for definite truth (D) where it is indefinite whether P just in case it is neither definitely true that P nor definitely true that not-p. Field favors a normal modal logic for definite truth analogous to necessity (p. 294). 17 On the standard axiomatization, the minimal normal modal system K amounts to classical propositional logic (i.e., tautologies, universal substitution and modus ponens), the (K)-schema D(P Q) (DP DQ), saying that D distributes over implication, and the definitization rule, saying that if P is provable, so is DP. Other systems of normal modal logic are formed by adding further axioms to this base, with the result being closed under the three transformation rules. Famous other proponents of a normal modal logic for definite truth are Tim Williamson (1994), who suggests an epistemic interpretation of definite truth, and Vann McGee and Brian McLaughlin (1995), who interpret definite truth in semantic terms. All named authors endorse at least T logic, which is obtainable from K logic by adding the (T)-schema DP P, saying that definite truth is factive. Williamson seems to favor B logic, which is obtainable from the system T by adding Brouwer s axiom P D D P, saying that truth entails definite falsity of definite falsity. 18 Field endorses S4 logic, which results from adding to the system T the (S4)-schema DP DDP, saying that definite truth iterates. Field does not make much effort to motivate

8 54 / Richard Dietz this logic by some interpretation of D. Rather, he treats D like a conceptual primitive that is unanalyzable. 19 What is of interest here is not the motivation for his primitivist stance about definite truth, but rather his claim that there is more to the semantic meaning of definite truth than its contribution to truthconditional content. Specifically, he claims that once we enrich our language by a notion of definite truth, this imposes a constraint on subjective probability for the language that is not simply derivative from the logic of definite truth. Call PL+ the language of propositional logic plus a D-operator that is closed under logical combination. We start from a standard semantic relation of logical consequence for S4, S4, on PL+, which suggests the following generalization of the Shafer calculus for the language PL+: (S1 PL+ ) 0 S(A) 1 (S2 PL+ ) If S4 A, then S(A) = 1 (S3 PL+ ) If S4 A, then S(A) = 0 (S4 PL+ ) S(A 1... A n ) i S(A i ) i< j S(A i & A j ) ( 1) n+1 S(A 1 &...&A n ) (S5 PL+ ) If A S4 B, then S(A) S(B). In effect, Field s suggested calculus can be reconstructed as the suggestion to replace Shafer s constraint (S4 PL+ ) by (S4 PL+ ) S(DA DB) = S(A) + S(B) S(A & B). 20 Two things are to be noted here: (1) Field s constraint (S4 PL+ ) is strong enough to entail jointly with (S5 PL+ ) Shafer s constraint (S4 PL+ ). For (S4 PL+ ) generalizes to S(DA 1... DA n ) = S(A i ) i< j S(A i&a j ) ( 1) n+1 S(A 1 &...A n ). And as (DA 1... DA n ) S4 (A 1... A n ), by (S5 PL+ ), S(DA 1... DA n ) S(A 1... A n ). (2) (S4 PL+ ) goes beyond what is to be expected even on classical probability (see Appendix C, proposition 2). The resulting calculus is hence not only in some respect weaker than classical probability (finite additivity does not hold) but also in some respect stronger ((S4 PL+ ) holds). To clarify the extent to which Field s model of the structure of subjective probability is revisionist: for any normal modal logic for definite truth that is at least as strong as T logic, Fieldian probability functions are classical, just in case they treat every sentence as definite, in the sense that associated statements of definiteness receive the maximal value (see Appendix C, proposition 3). That is, exactly in the everyday case where some sentence is not treated as definite, a departure from classical calculus is demanded. Not treating a sentence of our

9 Betting on Borderline Cases / 55 language as definite and letting one s probability depart from a classical structure, according to this, are two sides of the same coin. Field s revisionism about degree of belief is conjoined with a reductionism about definite truth and its cognates. More precisely, the probabilities for statements of definite truth, definiteness and indefiniteness (for any sentence s) are supposed to be all fixed by associated probabilities for truth and falsity (of s), as follows: (D Red ) Any rational degree of belief for the proposition that it is definitely true that P is given by the associated rational degree of belief for the proposition that P: d(dp) = d(p). 21 ( Red ) Any rational degree of belief for the proposition that it is definite whether P is given by the sum of the associated rational degree of belief for the proposition that P and the associated rational degree of belief for the proposition that P: d( P) = d(p) + d( P). 22 ( Red ) Any rational degree of belief for the proposition that it is indefinite whether P is given by the extent to which the associated rational degree of belief for the proposition that P and the associated rational degree of belief for the proposition that P do not sum up to 1: d( P) = 1 (d(p) + d( P)). 23 In effect, this suggests that one may consider a sentence as definitely true, definite or indefinite without having acquired these concepts. Field however, does not go so far as to also propose a reductionist account of subjective probability for all embedded occurrences of definite truth and its cognates and indeed, it is hard to see how the suggested reductionist theses could be extended to an across-theboard reductionism about definite truth and its cognates. 24 E.g., consider embedded occurrences of D of the form (1) DA B and the form (2) DA DB, where in either case, A and B are atoms. On any plausible generalization of Field s reductionist theses to a full scale reductionism, the probabilities of (1) and (2) should be identical. But Field s calculus allows for counterexamples to this natural requirement (see Appendix C, proposition 4). In fact, there is even a problem with Field s modest reductionism, as far as indefiniteness is concerned.

10 56 / Richard Dietz 3 Field s reductionism about indefiniteness and the logic of definite truth Field s reductionist constraints on definite truth and definiteness ((D Red ) and ( Red )) both follow from his calculus (see Appendix C, propositions 5 and 6 respectively). Things are different for the reductionist thesis about indefiniteness, ( Red ). One can easily give counterinstances, where the Fieldian probability for indefiniteness is lower than one minus the Fieldian probability for definiteness. Even stronger, the values for indefiniteness are by no account recoverable from associated values for truth and falsity (see Appendix C, proposition 7). 25 On the other hand, one can show that for the thesis to be valid on S4 logic, any definiteness statement must itself be definite. Starting from a logic that is at least as strong as S4 logic, however, ( Red ) is valid just in case our logic is no weaker than S5 logic (see Appendix C, proposition 8) i.e., the logic that is obtainable from T logic by adding the S5 principle DP D DP. There is a notorious objection to S5 logic from higher-order vagueness. 26 On S4 logic, we can still coherently assume cases of higher-order vagueness, that is cases where statements of definite truth or its cognates are themselves indefinite. 27 On S5 logic, however, any statement of definite truth or its cognates is definite, that is, we have S5 DA, S5 A and S5 A. That is, cases of higher-order vagueness cannot be consistently assumed on this logic. But if there is evidence for the existence of first-order vagueness, there seems to be equally evidence for higher-order vagueness. Consider colored patches that form a sorites series of redness. If adjacent patches are sufficiently similar in shade, there will not be only cases where it is borderline whether we have still to do with a case of redness or already with a case of non-redness. There will be also cases where it is borderline whether we have still to do with a definite case of redness or already with a case that is not definitely red that is, we will have also borderline cases of being a borderline case of redness. 28 The scope of what we are allowed to state on S5 logic thus appears to be implausibly narrowed down. 29 Is there any way of deflating this point? It seems that there are only two ways: either we grant that there is higher-order vagueness but deny that any object-linguistic operator can fully express the concept of definite truth. Or we argue that there is no higher-order vagueness. On the first option, it is suggested that we need a hierarchy of operators of definite truth to capture fully our talk of vagueness. To express the vagueness of D within the object-language, we would need another operator D and so forth. Let me mention just one serious concern here: a hierarchy of operators of definite truth is faced with the same problem as a Tarskian hierarchy of concepts of truth it appears that our informal uses of true and definitely true pick out concepts outside a hierarchy, e.g., when we say that every proposition is either definitely true or definitely false. 30 On the second option, what we would need is an argument against higherorder vagueness that is at least compatible with S5 logic for definite truth. It is not my aim here to give a comprehensive account of possible no-need or no-use arguments of the required type. It suffices to note that, notwithstanding

11 Betting on Borderline Cases / 57 the intuitive force of the assumption of higher-order vagueness, there is also space for reasonable doubts about the explanatory value of the thesis of higherorder vagueness. 31 Indeed, on some philosophical accounts of definite truth, the case against higher-order vagueness seems more promising than on others. Notably, if we follow Williamson s epistemic interpretation of definite truth, it is suggested that the relevant accessibility relation is some type of indiscriminability relation on interpretations, which fails to be transitive in analogy to the modeltheoretic argument for the failure of the KK principle (if it is known that P, then it is known that it is known that P). As a consequence, S4 logic, and hence S5 logic is not sound. On semantic interpretations of definite truth as a type of semantic determinacy (in the vague sense of McGee and McLaughlin (1995)), one may as well suggest that the relevant admissibility relation between interpretations fails to be transitive, and/or that it fails to be symmetric. But this move on the semanticist part seems far less well motivated and more ad hoc than the epistemicist account of definite truth. E.g., consider the following suggestive consideration for the conclusion that definite truth is a sort of semantic determinacy but subject to S5 logic. Interpretations in the relevant sense are precisifications of a language. But how to cash out the supposed non-epistemic sense of admissibility that is not an equivalence relation? To put the same point in another way, if we can give a space of interpretations of a language that exhausts all precisifications to be considered for providing a full account of vagueness in the language, is not higher-order vagueness ultimately a rather elusive phenomenon? What plausible account of semantic admissibility tells us that the relation fails to be an equivalence relation if not the question-begging requirement that higher-order vagueness is to be accommodated? Obviously, this is not a conclusive argument in favor of S5 logic but it highlights a problem with semantic theories of vagueness, which favor a normal modal logic for definite truth weaker than S5. Let us take stock: on any plausible normal modal logic for definite truth weaker than S5, Field s reductionism about indefiniteness is not a viable position. S5 logic is faced with a serious problem insofar as the assumption of higher-order vagueness has intuitive force. On the other hand, in view of reasonable doubts about the theoretical value of the thesis of higher-order vagueness, it seems fair not to dismiss outright S5 logic on the ground that it does not countenance higher-order vagueness. A further note: even if Field s reductionism about indefiniteness is not sustainable in its original version, it ought to be noted that Field s probability calculus supplies sufficient means of maintaining a weaker interesting claim about indefiniteness after all. For it follows that if we consider a sentence as indefinite (to some positive degree), we do not treat the sentence as fully definite, which amounts to the case that the degrees of truth and falsity do not sum up to 1. In this respect, though the Fieldian calculus is too weak to characterize the conditions on which we rationally consider a sentence as indefinite, it is strong enough for characterizing a necessary condition for doing so.

12 58 / Richard Dietz 4 Representing Fieldian probabilities as classical probabilities Field s argument for his theory of subjective probability factorizes into two points. (1) He shows that for any classical probability function on PL+, the probabilities of definite truth for sentences behave like a Fieldian probability function on PL+. (2) He suggests some arguments for dismissing classical probability functions in favor of their associated Fieldian functions whenever they are distinct. Before turning to his arguments (in 5), I show that Field s first point can be significantly strengthened: for one, on S4 logic, for every Fieldian function S on PL, there is a classical probability function P on PL+, where the S-values for sentences A agree with the P-values for the associated definitizations of the form DA. That is, we can represent Fieldian probabilities on PL as classical probabilities of definite truth. For another, on S5, we can prove the same also for every Fieldian function on PL+. Field (pp ) gives a straightforward way of recovering Fieldian probability functions from classical probability functions. Take any classical probability function P on PL+, and define an associated function Q as follows: (#) Q(A) = df. P(DA). For any classical probability function P (on PL+) call the function that is obtainable from P by way of (#) the Q-function for P. As Field observes, on S4 logic for definite truth, for any classical probability function P, the Q-function for P is a Fieldian function (see Lemma 1). Can every Fieldian function be recovered from some classical function P as a Q-function for P? If the answer is positive, Fieldian probabilities are representable as classical probabilities of definite truth. Field does not discuss the question. He raises, though without answering, the following more special question (p. 299): can every Fieldian function on the D-free fragment of PL+ be recovered as a Q-function for some classical probability function? In fact, one can prove that the answer is positive (see Appendix C, proposition 9).,32 How about Fieldian functions on PL+? Can they be recovered the same way from classical probability functions? We can prove that the answer is positive provided that we adopt S5 logic for PL+. 33 Where are we now? Field observes that on S4 logic, classical probabilities for definite truth have the structure of Fieldian probabilities for plain truth. I observed that on S4 logic, Fieldian probability functions on PL can be represented as a classical probability function for definite truth, and on S5 logic the same for PL+. In the light of the first result, we can put Field s revisionism about subjective probability in the form of the following slogan: for sentences of PL, degree assignments should have the structure of classical probabilities for Just recently, I learnt that a proof of this result was also sketched in Brian Weatherson s unpublished notes (2002: 73-4). The discussion of Field (2000) in these notes will deserve further consideration.

13 Betting on Borderline Cases / 59 definite truth. According to the second result, on S5 logic, Field s revisionism suggests the analogue result for the language PL+. But why should we let subjective probability go with classical probabilities of definite truth instead of classical probabilities of plain truth? To put the point in another way: if as far as definite truth is concerned, we should follow the classical model of probability, why should not we do so, as far as truth is concerned? 34,35 5 Fieldian vs. classical probability In his case against classical probability, Field mentions basically two points, the first one is an argument for adopting his alternative model, the second one is a concern about Dutch-book arguments for classical probability. To begin with, I will scrutinize Field s points and argue that they lack any force ( ). In the next step, I am going to argue that starting from a supervaluationist framework, one can indeed reasonably argue for Field s alternative calculus ( 5.3). 5.1 Field s point in support of Fieldian probability revisited Why should we adopt the Fieldian model of degree of belief? Field does not motivate his calculus in terms of a philosophical interpretation of degree of belief. Rather he gives just a couple of principles of obtaining genuine subjective probability functions P from classical probability functions P (p. 294): (1) If S4 A, then P (A) = 1; (2) If P (A) = 1, then P (DA) = 1; (3) If P(DA) = P(A) and P(D A) = P( A), then P (A) = P(A) and P ( A) = P( A); (4) If (P(DA) P(A)) or (P(D A) P( A)), then (P (A) P(A)) or (P ( A) P( A)). On S4 logic, these constraints are indeed met by functions that are obtained as Q-functions from classical probability functions. But this result alone hardly provides a motivation for Field s account of probability. For one, the given constraints (1) (4) do not entail Fieldian probability. Take any classical probability function P on PL+. Let P be defined as follows: for any sentence X of PL+, if P(DX) = P(X), then P (X) = P(X), otherwise P (X) = 0. On S4 logic, P meets all constraints (1)-(4), but it fails to be a Fieldian function, for if P(DX) P(X) and P(DX) > 0, then P (X) P (DX). As importantly, the given constraints lack any clear intuitive force and they are not underpinned by any further theoretical explanation. Field fails to motivate his non-classical model of probability. Nor does he vindicate even the weaker thesis that for vague languages, some departure from

14 60 / Richard Dietz the classical model is called for. However, he voices a concern that highlights an arguable problem with Dutch-Book arguments for classical probability, as far as vague languages are concerned. Let us turn to this point. 5.2 Does vagueness undermine the motivation for classical probability? Arguing for classical probability from expectations Before turning to Dutch-Book arguments, consider first a different kind of argument, which is not considered by Field. Here is a well-known conditional argument for classical probability on the language PL+ from classical probability on sets of propositions. One can easily generate classical probability functions on PL+ from classical probability functions on modal spaces. If PL+ has a classical interpretation relative to any possible world in a modal space, sentences of PL+ are random variables that take a value 1 or 0 relative to possible worlds. The expected truth-values of sentences are then given by the classical probabilities of the propositions picked out by the sentences. It is easy to see that the expected truth-value function on PL+ is a classical probability function on PL+. 36 Take any modal space W, and any classical interpretation function v of PL+. For any sentence A of PL+, A v is then a random variable that takes value 0 or 1 relative to worlds in W. Let F be any algebra over W, where for each sentence s of PL+, the proposition picked out by s, i.e., {w: s v (w) = 1}, is a member of F, and P any probability function on F. Let P be the function that takes for each sentence of PL+ the expected truth-value of s relative to P, in other words the classical probability of the proposition picked out by s. P is then a classical probability function on PL+: as P takes always values in [0, 1], P does so as well. P assigns to any tautology the maximal value, as (i) the P-value for W is the maximal one and (ii) on any classical interpretation, tautologies pick out the universal set. P also satisfies finite additivity, by (i) finite additivity for P, (ii) the fact that on any classical interpretation, jointly contradictory sentences pick out disjoint propositions, and (iii) the fact that the propositions picked out by disjunctions are the unions of the propositions picked out by their disjuncts. Assume our degrees of belief with respect to sentences should be the expected truth-values in view of our evidence. Our evidence is representable by some classical probability measure on some space of epistemic possibilities. For any epistemically possible world, the semantic properties of PL+ with respect to that world are representable by some classical valuation of PL+. Thus for any distribution of degree of belief on PL+ to be rational, there must be some classical probability function P and some classical interpretation function v for PL+ on some epistemic modal space F such that the degrees for sentences are the expected values relative to F, v and P. Consequently, degrees of belief on PL+ must be classical probabilities. It is easy to see that by parity of reasoning, one can give a conditional argument for classical probability functions on PL+

15 Betting on Borderline Cases / 61 that respect S4 logic for we are free to assume that interpretations that respect S4 logic are classical valuations. The argument is obviously only conditional, since it already takes for granted that the epistemic situation with respect to sentences is representable in terms of expected truth-values. If Field is right, there must be a way of blocking the argument, at least as far as vague languages are concerned. Indeed the argument seems oversimplifying in that it treats PL+ like a precise language. It hinges on the assumption that for any epistemic modal space, to compute our expectations regarding the truth-values of sentences, we need to consider only one interpretation of the language with respect to that space. If our language is vague, however, it seems to admit of more than one way of classifying some epistemically possible worlds where admissibility in the relevant sense may be specified in epistemic or semantic terms. E.g., suppose that what we know does not rule out that a particular glass is filled to 50% (we may be in another room and have only unspecific testimonial knowledge about that glass). We can represent this as the case where the relevant epistemic space includes a world where the glass is filled to 50%. How to classify this world in terms of the vague sentence The glass is full? It seems admissible to classify it as a world with respect to which this sentence is true, but equally admissible to classify it as a world with respect to which this sentence is false. Or so one may argue. This suggests that we need to take into account sets of interpretations rather than a single interpretation with respect to the relevant modal space. I cannot fully treat the matter here, but let me point out that the suggested refinement will not make any difference. One can reasonably argue that we end up again with a classical measure on PL+, irrespective of whether we interpret definite truth in epistemic or semantic terms (see Appendix A) Reconsidering the Dutch-Book argument for classical probability Dutch-Book arguments suggest that we can make a much stronger case for classical probability, which does not already invoke the classical probabilistic notion of an expectation. One can describe bets on a hypothesis A as contracts whereby the bettor pays the sum ps to the bookmaker in exchange for the payment of the sum S if A is true, and nothing if it is false. p is called the betting quotient (on A) and S the stake. The payoff for the bettor is hence (1 p)s if A is true, and ps if A is false, where the stake is positive. 37 Representing truth by 1 and falsity by 0, we can rewrite this as G x = (x p)s,

16 62 / Richard Dietz where x ranges over values {0, 1} for A. We can obtain the corresponding payoff for the bookmaker just by taking the reversed sign for the stake. The idea is now to interpret degrees of belief (with respect to A) as betting quotients (on A) considered as fair, i.e., as being such that one sees no advantage in betting one way (on A with the quotient p) rather than the other (against A with the quotient (1 p)). The point of Dutch-Book arguments is to show that betting quotients p 1,..., p n for a set of hypotheses A 1,..., A n cannot be consistently considered as fair if they violate the classical probability calculus. Specifically, if anyone were to accept such betting quotients, he would be susceptible to a sure loss contract (a Dutch Book), i.e., a contract where the payoff is under any circumstances negative (see Appendix B). On Field s account, our degrees of belief should violate the classical calculus whenever we do not treat some sentence of our language as definite; in which case finite additivity is not satisfied. But it is hard to see why the shift from precise to vague languages should make any difference as to the soundness of the Dutch-Book argument for classical subjective probability. Thus, if Field is right, there must be a way of blocking the Dutch-Book argument for finite additivity as far as bets are concerned that involve sentences that are considered as potentially indefinite. In fact, Field vaguely suggests (p. 303) that there may be such a way, without going into any details. So what exactly is supposed to block the argument as unsound? Let us be clearer about the barebones of the argument and try to sort out the potential trouble spots. In order to sidestep notorious issues regarding the supposed connection between degree of belief and betting quotients for monetary bets, I refer in an idealizing way to agents who value money positively, in a linear way, and nothing else positively or negatively I call them (following Christensen (2004)) simple agents. The Dutch-Book argument may be then reconstructed as factorizing into the following components: 38 (1) Sanctioning: An agent s degree s of belief sanctions as a fair monetary bets at quotients that match his degrees of belief. (2) Defective bets: For a simple agent, a set of bets that is guaranteed to leave him monetarily worse off is rationally defective. (3) Defective beliefs: If a simple agent s belief sanctions as fair each member of a set of bets, and that set of bets is rationally defective, then the agent s beliefs are rationally defective. (4) Dutch-Book Theorem: If the betting quotients considered as fair by an agent violate any axiom of classical probability, then there is a set of monetary bets with these betting quotients that will guarantee a monetary loss on the agent s part. From (1) (4), it follows that: (5) Simple Agent Probabilism: If a simple agent s degrees of belief violate classical probability, they are rationally defective.

17 Betting on Borderline Cases / 63 The argument is valid, so if the argument is not sound for beliefs involving sentences that are considered as potentially indefinite, one of its premises must be false. Theorem (4) holds independently of assumptions regarding the precision/vagueness of sentences involved. This limits down the space to (1)-(3). It is hard to see why the precision/vagueness distinction should be any relevant to (2) or (3). I take here only (1) as a potential problem. Consider a coin, where the heads side is clearly green and the tails flip side a clear borderline case of being green. The coin is going to be tossed, and our reliable informant tells us that the coin is manipulated to fall always heads. However as we cannot rule out that our informant is this time wrong, we cannot rule out that the coin is in fact not manipulated and that it will fall next time tails. In this case, the hypothesis (#) The outcome of the next coin toss will be a green side is highly likely to be definitely true, but owing to a small chance say 0.1 that the informant is wrong after all, we should consider the hypothesis as potentially indefinite after all. Suppose that given the coin is manipulated we are 100% confident that it will fall tails; and furthermore that given that the coin is not manipulated, we are 50% confident that it will fall tails. Then we should be 95% confident that (#) is definitely true, and (as definite truth entails truth) at least to the same degree confident that (#) is true. In that case, it would be unreasonable to accept a bet against (#) at a betting quotient higher than 5%; that is a betting quotient lower than 95% regarding (#) could not be considered as fair. More importantly, there are also betting quotients that may be plausibly considered as fair, and that (by the foregoing reasoning) should fall into the interval [0.95, 1]. Insofar as this reasoning is sound, it supports (1), also with regard to sets of hypotheses some of which are considered as potentially indefinite. 39 Consider this objection: The plausibility of the foregoing argument draws on the particular example, where we are almost fully confident to have to do with a definite hypothesis. Things are different if we consider the converse type of case. Take the limiting case where we have to do with a hypothesis of which we are fully confident that it is indefinite e.g., replace the above coin with a different coin where both sides are clear borderline cases of being green and reassess the hypothesis (#). It seems that we are not confident to any positive degree that (#) is true, nor that (#) is false. That is, we lack any confidence that we might be winning a bet regarding (#), whether it is on or against it. But then bets regarding (#) seem as pointless as bets regarding meaningless sentences. In particular, the normative issue of fixing a fair betting quotient seems to be then entirely beside the point. We may have zero-degreed beliefs with respect to (#) and its negation, but these degrees of belief do not sanction any betting-quotient regarding (#) or its negation as fair. Or so one may argue. I grant that this objection has some intuitive force. Insofar as it has, the case for classical subjective probability for vague languages seems prima facie in trouble and Field s calculus seems in this

18 64 / Richard Dietz regard better off, since his account accommodates the idea that clear borderline cases and their negations should receive any positive degree of belief. Having said that, the above reasoning is dubious if the background assumption is that also for vague languages the semantics is classical. On classical semantics, the modal space of possible outcomes can be partitioned into cases where (#) is true and cases where (#) is false there is no third possibility, even if (#) is clearly borderline. But if we are committed to assess the situation this way, how can we at the same time be justified to have zero-degreed belief in (#) and its negation? Furthermore, if the outcomes are bound to be either cases where bets on (#) are winning or cases where bets on (#) are losing, how can bets on or against (#) be pointless? More specifically, how can the question of what betting quotient regarding (#) is fair be pointless? The intuitive idea that clear borderline cases do not allow for positive degrees of belief and the associated idea that bets regarding clear borderline cases are pointless have some intuitive force. But in the absence of a proper theoretical interpretation of degree of belief that accommodated the zero-degree intuition, these intuitions do not deserve much credit. In particular, the intuitions lack any force in combination with a classical semantics for vagueness. I discussed two ways of motivating classical constraints on subjective probability, the conditional argument from expectations and Dutch-Book arguments. According to Field s account, there should be ways of blocking effectively both types of arguments as far as vague languages are concerned. Field does not offer any such counterarguments. On the contrary, one can reasonably argue that if the arguments work out for precise languages, they do so also with appropriate adjustments for vague languages. 5.3 Arguments for Field s calculus from supervaluationism Arguing for Fieldian probability from expectations On S5 logic, Fieldian probabilities are representable as classical probabilities of definite truth. On this logic, however, definite truth cannot be plausibly interpreted as truth: D is factive (the inference rule DA/A is valid) but from the assumption that A, it does not follow conversely that DA. The classical probability calculus therefore allows the degree for some sentences A to differ from the associated definitizations of the form DA. In view of this, the suggestion to let probabilities be modelled on classical probabilities for definite truth seems ill-motivated: if classical probabilities for definitized sentences are a reliable guide for having coherent degrees of belief on the associated undefinitized sentences, why not model our degrees of belief directly on classical probabilities for the undefinitized sentences themselves? Field does not provide a satisfactory response. I do not aim to resolve the problem. What I suggest here is rather an

19 Betting on Borderline Cases / 65 arguable way of dissolving it. The basic idea is to change the logic for vague languages such that definite truth that obeys S5 logic can be reinterpreted as truth. On standard possible-worlds semantics for S5 logic, logical consequence is defined as preservation of truth relative to any valuation point in any model, where valuation points in models are associated with classical interpretations. A sentence may be true, relative to a valuation point in a model without being definitely true at that point in other words, without being true at all points in that model. If we define truth in a model as being true relative to all valuation points, what we get is a notion that is standardly referred to as supertruth (in a model) analogously, falsity on all valuation points is also known as superfalsity (in a model). The resulting valuation in terms of supertruth and superfalse is not two-valued sentences may be neither true nor false. It is a well known fact that if we keep to S5 models but redefine logical consequence in terms of preservation of supertruth in models, the resulting logic mirrors S5 logic in the following way: S5 Dα iff SV α, where is obtained from by attaching a D-operator to every member of. 40,41 We can exploit this connection between S5 and supervaluationist logic for reinterpreting any classical probability of definite truth that obeys S5 logic as an expectation of supertruth, as follows: take any classical probability measure m on any frame <W, R> for S5, where W is the domain and R an equivalence relation on W. 42,43 Let f be a mapping from atoms into classical propositional functions, i.e., mappings from W into {0,1}. By standard valuation rules for connectives and D, we get from f a mapping [ ] from the whole language PL+ into classical propositional functions. For each sentence A of PL+, [A] represents the proposition that A. The following mapping then seems the natural way of representing for each A the associated proposition that it is supertrue that A: for any sentence A of PL+, let [A] SV = df. [DA]. Like [ ], [ ] SV takes for sentences twovalued random variables as values but whereas in the former case, the values 0 and 1 represent classical truth-values, in the latter case, they represent the supertruthstatus, with 1 representing supertruth and 0 representing lack of supertruth. 44 This suggests two corresponding measures on PL+. The first measure is defined as follows: for any sentence A of PL+, P 1 (A) = df. m([a]). P 1 measures the expected classical truth-value of A, in other words, it gives the probability that it is true that A. We can easily obtain from this another measure by saying: for any sentence A of PL+, P 2 (A) = df. m([a] SV ).

20 66 / Richard Dietz P 2 measures the expected supertruth-status of A, in other words, it gives the probability that it is supertrue that A. From this, by definition of [ ] SV, it follows that P 2 and P 1 are related as follows: P 2 (A) = P 1 (DA). As P 1 is a classical probability function that respects S4 logic and P 2 is obtainable as a Q-function for P 1, it follows that P 2 is a Fieldian function. What have we gained here? Recall that we could prove that Fieldian probabilities on PL+ for S5 logic are representable by classical probabilities of definite truth on PL+ for S5 logic. The observed link between S5 logic and supervaluationist logic allows us to infer from this that Fieldian probabilities on PL+ for S5 logic are representable as expectations of supertruth. This is not the place to give an appraisal of the arguments in favor or against a supervaluationist framework for vague languages. I do not want to offer new arguments with regard to this controversial issue. 45 The point I want to make is rather conditional: if we model subjective probabilities for any sentences A on classical probabilities that A is true, assuming a classical semantics, we should have degrees of belief with a classical structure. On the same premise, assuming a supervaluationist semantics, however, we should have degrees of belief with a Fieldian structure A Dutch-Book argument for Fieldian probability We can do even better by showing that assuming a supervaluationist semantics, we can also make use of Dutch-Book arguments for justifying Fieldian probability. Call bets for a classical framework, where the win/loss conditions are specified the standard way, classical bets. In a classical semantics, a sentence is not true just in case it is false, or (if the language contains negation) just in case its negation is true. In non-bivalent frameworks such as standard supervaluationism, the latter principle fails a sentence may be untrue without being false. This gives rise to the question of how to generalize the winning/losing conditions for bets on hypotheses in a supervaluationist framework. Here are two natural suggestions: (a) Bets on a hypothesis are only conditionally holding. That is if the hypothesis turns out to neither true nor false, the bet is cancelled, if it turns out true, one wins, and if it turns out false, one loses. Or (b) bets are specified as unconditional: One wins a bet on a hypothesis if it turns out true, and one loses it if it turns out to be untrue. Call for a supervaluationist framework, conditional bets conditional supervaluationist bets, and unconditional bets unconditional supervaluationist bets. For conditional supervaluationist bets, Field s suggested revisionism is not justifiable by way of Dutch-Book arguments. Specifically, one can give a Dutch- Book argument for the following constraint: For conditional supervaluationist bets regarding any set of hypotheses including sentences A, DA and DA D A, where p, q, and r are the betting quotients for A, DA, and DA D A respectively, if r > 0, p is to be equal to q r.

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