Mind

Charlow,, Nate

doi: 10.1093/mind/fzz028pmid: N/A

Abstract This paper proposes a new model of graded modal judgement. It begins by problematizing the phenomenon: given plausible constraints on the logic of epistemic modality, it is impossible to model graded attitudes toward modal claims as judgements of probability targeting epistemically modal propositions. This paper considers two alternative models, on which modal operators are non-proposition-forming: (1) Moss (2015), in which graded attitudes toward modal claims are represented as judgements of probability targeting a ‘proxy’ proposition, belief in which would underwrite belief in the modal claim; (2) a model on which graded attitudes toward modal claims are represented as judgements of credence taking as their objects (non-propositional) modal representations (rather than proxy propositions). The second model, like Moss’s model, is shown to be semantically and mathematically tractable. The second model, however, can be straightforwardly integrated into a plausible model of the role of graded attitudes toward modal claims in cognition and normative epistemology. 1. Introduction Agents can bear graded attitudes (such as intermediate or high credence) towards epistemic modalities.1 Sentences expressing such graded attitudes are commonplace; consider the following triad (adapted from Moss 2015, p. 4). (1) It is probably the case that Trump might be impeached. (2) It is probably the case that Trump will be impeached. (3) Trump might be impeached. Moss remarks that ‘our judgements suggest that [(1)] is weaker than either [(2)] or [(3)]’. Believing (2) ‘is intuitively sufficient reason to bet at even odds’ that Trump will be impeached, ‘whereas merely believing [(1)] is not’ (Moss 2015, p. 4). Meanwhile, asserting (3) represents the speaker as believing that Trump might be impeached; (1) does not.2 The basic data point can be established in various ways. Consideration of (1)–(3) suggests the existence of sentences of natural language serving to express graded attitudes towards epistemic modalities. Graded attitudes towards epistemic modalities also appear to be presupposed by platitudes about the conversational role of epistemic modalities. Willer (2013), for instance, observes that assertions of epistemic modalities are understood as non-trivial proposals to add information to — that is, address a question within — a discourse. Assertion of a sentence like (3) addresses a question about whether Trump might be impeached: (4) Might Trump be impeached? But the notion of such a question seems to presuppose the possibility of a graded attitude (that is, a degree of confidence greater than 0 and less than 1) toward a sentence like (3). Such an attitude typically forms at least part of the cognitive basis for entertaining (or explicitly posing) such a question; the question is generally occasioned by the questioner’s bearing a graded attitude toward an epistemically modal representation. This paper proposes a new model of graded modal judgement. It begins (in §2) by problematizing the phenomenon for classical, truth-conditional accounts of the semantic content of epistemic modalities: given plausible constraints on the logic of epistemic modality, it is actually impossible to model graded attitudes toward modal claims as judgements of probability taking modalized propositions as their objects (Charlow 2016b, forthcoming b). In response to this problem, this paper considers two alternative models, on which modal operators are non-proposition-forming operators: In §3: The model of Moss (2015), on which a graded attitude toward a modal claim is represented as a degreed belief taking a ‘proxy’ proposition, belief in which would underwrite belief in the modal claim, as its object. In §4: A model on which a graded attitude toward a modal claim is represented as a degreed belief taking as its object a (non-propositional) modal representation (rather than a proxy proposition). The second model is shown to be theoretically tractable — a feature that does not ultimately distinguish it from Moss’s model. Since, however, Moss argues against accounts of the second type, such a model is worth developing, even if only as a proof of concept. In §5, I argue that such a model deserves attention, not only as a proof of concept, but also because it is straightforwardly integrated into a plausible understanding of the functional role of graded attitudes toward modal claims in both cognition and normative epistemology. A view of this shape has some claim to being regarded as the null hypothesis about the target phenomenon. 2. No uncertainty? Almost everyone would agree that the base semantic clause for the epistemic possibility modal ⋄ (and its dual operator □) is information-sensitive — that is, involves reference within the semantic metalanguage to a state of information — and that, relative to a ‘base’ state of information — for present purposes, this is modelled as a (possibly constant or partial) function from worlds of evaluation into sets of possible worlds — epistemic possibility modals quantify existentially over possibilities compatible with that state. Relative to a choice of information state σ and a choice of index of evaluation w, the appropriate semantic clause for ⋄ is as follows: A sentence of the form ⋄ϕ thus expresses a possible-worlds proposition, namely: Such a proposition is the sort of thing to which a probability function can, in principle, assign a probability, and is the sort of thing toward which agents can, in principle, bear graded attitudes (for example, being 10% confident in it). On the other hand, there is apparently strong evidence that sentences of the form ⋄ϕ cannot generally express possible-worlds propositions with these sorts of characteristics. First, assume that, for any w and σ, and . (5) # It must be raining, but it isn’t. (6) # It may be raining, but maybe it can’t be. In the present setting, this is equivalent to assuming that σ is governed by constraints of reflexivity and Euclideanness.3 Reflexivity: Euclideanness: These constraints imply that information states are epistemically transparent: Transparency: Given Transparency, epistemic modalities are ‘rigid’ relative to a choice of σ and w: if ϕ is a sentence of the form or and , then, for any . Hence, whenever : The basic difficulty this gives rise to is this: like an agent’s degree of belief in any proposition, an agent’s degree of belief in an epistemically modal proposition ought to be a probability. And yet it provably cannot be. (Conclusion: a graded modal judgement is not to be understood in terms of a degree of belief in an epistemically modal proposition.) To illustrate, consider a probabilistically coherent agent A bearing a graded attitude — confidence (0,1) — towards ⋄p (alternatively, □p).4 Although this may already be apparent to some readers, it is worth underlining that (as well as why) such an agent is impossible to represent within the present framework. Let represent A’s information at w. If p is possible (alternatively, necessary) for A at w, then (alternatively, ). But then, in view of Transparency, (alternatively, ). The difficulty is this: if A is probabilistically coherent, and A’s information entails ⋄p (alternatively, □p), then A must assign ⋄p (alternatively, □p) probability 1. It follows that A’s confidence in ⋄p or □p must be extremal (0 or 1).5 Proof Consider any probabilistically coherent agent A; let be A’s information at w and be a probability measure for A at w. Either or . If , then , in which case and . If , then , in which case and . Thus, either or . Within the ‘classical’ semantic setting presupposed in this section, it seems that we confront a hard choice, between a revisionary logic of epistemic modality and a revisionary understanding of the attitudes it is possible to bear toward sentences expressing subjective uncertainty. Given this sort of ‘classical’ setting, if we maintain the assumption that (5) and (6) are inconsistent, the phenomenon of degreed belief in epistemic modalities is mystified: such a degree of belief cannot be represented as a judgement of probability targeting an epistemically modal proposition. Nor, therefore, can we represent degrees of belief toward epistemic modalities using sets of probability measures taking epistemically modal propositions as their objects. According to the ‘Bayesian’ proposal for representing such attitudes, ‘[w]here an agent assigns a determinate probability to a proposition, every measure in their credal set [that is, the set of probability measures compatible with their information] assigns that probability to it. A probabilistic claim is true of a credal set just in case it is true on every probability measure in the set’ (Rothschild 2012, p. 110; see also Yalcin 2012). The difficulty is that, given the arguments of this section, a set of probability measures S is constrained so that if ϕ is epistemically modal: Attitudes of intermediate confidence (for instance, confidence n) toward a sentence ϕ are represented, according to the Bayesian proposal, with sets of probability measures, all of which assign probability to ϕ. No probability measure assigns intermediate confidence to ϕ if ϕ is epistemically modal. And so, given the Bayesian proposal, no set of probability measures can represent attitudes of intermediate confidence toward epistemic modalities. 3. Linguistic gradability and gradability of content Moss (2015) provides a semantics for epistemic operators, including a sentential operator expressing high confidence in its complement. On Moss’s semantics, a sentence of the form (read: it is probable that ϕ) expresses a constraint on probability measures, namely, the constraint that a probability measure satisfies just if it is in the following set: Epistemic operators are, in general, interpreted relative to contextually salient partitions (that is, contextually relevant questions); numerical indices (for example, subscripted ‘1’) are mapped to contextually salient partitions by a contextual variable assignment gc. Thus, expresses a constraint on probability measures that m satisfies if and only if m assigns this proposition a value exceeding 0.5: The object that receives a probability value (according to a probability measure m satisfying the constraint semantically expressed by the sentence) is the disjunction of those propositions in the salient partition that confirm ϕ — that is, the disjunction of those propositions p such that, if m were conditionalized on p, m would satisfy the constraint expressed by ϕ. Moss’s account handles iterated epistemic operators with ease. To illustrate, suppose . Then expresses a constraint on probability measures that m satisfies if and only if m assigns this proposition a value exceeding 0.5: This is, roughly, the constraint that m satisfies just if m regards a salient propositional disjunction as probable: in particular, the disjunction of those answers q to the contextually salient question such that conditioning m on q would make it the case that m regards p as necessary. In shorthand, it is the constraint that m satisfies just if m regards as likely some disjunction such that any way of m’s coming to accept that disjunction would amount to m’s regarding p as necessary.6 It is obvious why, on this model, it is sensible for a probabilistically coherent agent to think (as well as to express the thought) that p is probably necessary, without at the same time thinking (or being committed to expressing the thought) that p is necessary: encodes a stronger constraint on measures — in Moss’s system, it is the constraint that m satisfies just if, for any way r of answering gc(2), (Moss 2015, p. 27). Of course, this is a constraint that a measure satisfying the constraint encoded in will not generally satisfy. 3.1 Worldly representation A noteworthy feature of Moss’s account is that the semantic content of an epistemically modalized sentence — an epistemically modal representation — is not itself assessable within the semantic metalanguage as (for example) probable. For Moss, semantically rules out probability measures that do not regard as likely the disjunction of the propositions conditionalization on which is sufficient for believing ϕ. Strictly speaking, does not rule out measures according to which is not likely; when ϕ is epistemically modal, Moss denies that is the sort of thing over which a measure is defined (since it is not, on Moss’s semantics, a proposition). To be clear, a view like this certainly makes sense — indeed, given §2, would be forced — if probability measures (qua devices for representing agential degrees of belief) may be defined only over σ-algebras of sets of possible worlds. And — given that the theoretical purpose of invoking probability measures (for this application) is to model an agent’s degree of belief in some representation — assuming that probability measures are defined only over σ-algebras of sets of possible worlds makes sense, too, if the cognitive state of representation is to be understood in terms of representing some way the actual world could (not) be (as in, for example, Stalnaker 1984). 3.2 Generalizing representation This sub-section will try to motivate an alternative picture. It identifies some reasons in favour of adopting a ‘flexible’ or ‘generalized’ notion of representation, according to which (1) expresses, and (7) ascribes, a high degree of belief in the representation encoded in (3). (7) Marcy thinks it is probably the case that Trump might be impeached. This sub-section suggests, more generally, that it is theoretically natural to treat the semantic value of an epistemic modality — an epistemically modal representation — as a kind of content: an entity toward which we may represent agents as bearing representational attitudes (like degreed belief).7 Degreed attitudes towards representations of any type are rationally subject to the same laws of probability. Why is this? Why, for example, is it irrational to believe either of (8) or (9)? (8) It is probable that Trump might be impeached, and it’s probable that he can’t be. (9) Trump will probably be impeached, and he probably won’t be impeached. Generally, why is it that, for any ϕ whatsoever, it is irrational to believe while believing , that is, to think that both ϕ and its negation are probable? A natural, if incomplete, explanation: for any representation R, an agent’s degree of belief in R summed with her degree of belief in should never exceed a designated value representing full/outright acceptance. Notice that developing this sort of off-the-shelf explanation means countenancing degrees of belief as properties of representations of any type (in order to state, and ultimately account for, rational norms that govern the suite of graded attitudes agents can bear toward representations of any type).8 To similar effect, consider a decision situation in which the desirability of an action depends on an epistemically modal representation. Agents who are not risk-neutral confront such decision situations routinely. On a simple model of attitudes toward risk (see Buchak 2013), risk-inclined agents assign additional weight to low probability pay-offs, while risk-averse agents assign a negative weight to low probability pay-offs; rational agents who are not risk-neutral seek to maximize risk-weighted expected value. For any decision situation in which a risk-inclined agent believes to degree n that there is a low probability that p — for simplicity, imagine that the agent treats p as low probability just when the agent accepts — an agent who seeks to maximize risk-weighted expected value may compute (when her information allows her to do so9) an expected risk-weighted expected value.10 X X Open in new tab X X Open in new tab Suppose I strive to maximize risk-weighted expected value, and suppose I only care about the risk of D (=drinking from the lake) when it’s probable that A (=there’s an algae bloom). Suppose that my evidence is unspecific as to the ‘base’ probabilities in my decision situation — for example, it does not settle whether or not A is probable — although I do have evidence about the higher-order likelihoods, according to which it is 0.6 likely that A is not probable. Here is a decision table for this decision situation:11 D D Open in new tab D D Open in new tab If I strive to maximize risk-weighted expected value, and my decision situation is accurately represented in the above decision table, it seems to make rational sense for me to drink, if: Three (I hope modest) observations. First, decision situations like this should be regarded as fairly commonplace, once we decide to make room for higher-order uncertainty in language and thought — a shared aim of Moss (2015) and this paper. Second, a mathematical model of practical reasoning that could be extended to cover decision making under higher-order uncertainty would be desirable. As Joyce (2010, p. 314) observes, in frameworks admitting such uncertainty, ‘in many decisions there are no options that maximize … across [the probability functions compatible with one’s evidence], but one must still choose’. Third, a natural first step in developing such a model is to adopt a generalized notion of representation, according to which: (i) degrees of belief/confidence are treated as values that can attach to representations of any type, including epistemically modal representations; (ii) it is possible to formulate rational norms that govern the relations between the suite of degreed attitudes agents can bear toward representations of any type, and action. §§4–5 will propose just such an account. 3.3 Signpost As seen in §2, on ‘classical’ (that is, ordinary truth-conditional) accounts of the semantics of epistemic modalities, the content of epistemically modal ϕ, relative to a designated agent A’s information, is unfit for being the object of graded attitudes for A (for example, A being 10% confident that ϕ), according to standard ‘Bayesian’ techniques for modelling those attitudes (and the semantics of sentences ascribing those attitudes). Moss (2015) provides the first analysis of epistemic modality on which it makes sense for speakers to express graded attitudes towards epistemic modalities. But, on that analysis, the semantic value of an epistemic modality — what I’ve called an epistemically modal representation — is not itself the object of an attitude like degreed belief. This section has identified some tentative motivations for treating such entities as the objects of such attitudes. §4 will develop a semantics for epistemic modality that formalizes and implements this sort of intuitive cognitive model. §5 picks up where §3 leaves off, arguing that there is a sensible, generalized notion of representation undergirding this semantics. 4. Credences in representations 4.1 Introducing credence On the alternative I have in mind, we will define a new quantity, call it credence, for probability operators of natural language to uniformly express. To utter a sentence like (1) is simply to express high credence in the content of, that is, representation semantically encoded in, (3). Here is how I prefer to conceptualize this idea (at a very high level of abstraction). Some credences are probabilities: subjective estimates of objective chance of the truth of a worldly representation (alternatively, subjective estimates of actual-worldly truth-value). Some credences are not probabilities (when a subject’s credence cannot be understood as their estimate of objective chance of the truth of a worldly representation, or as a subjective estimate of actual-worldly truth-value). There is, nevertheless, no obstacle to defining credences so that they behave like probabilities, whether or not the object of credence is a worldly or non-worldly representation.12 Begin by assuming that a representation is a set of objects of any semantic type — a set of alternative possibilities that witness the truth of (‘satisfy’) sentences of our language.13 In general, sets of alternative possibilities represent sets of ‘candidates’ for different ways of representing; only some sets of alternative possibilities (that is, sets of possible worlds) represent candidates for actuality (that is, sets of alternative ways the world could be); other sets of alternative possibilities (for instance, sets of sets of possible worlds) represent candidates for ways of representing a set of candidates for actuality; and so on.14 Definition 1 Let be a representation. Then a set of representations partitionsif and only if, for all : Definition 2 An alternative set for is any set that partitions .15 Definition 3 If is an alternative set for ’s σ-closure Σ is ’s closure under . Consider any base representation , alternative set for , and ’s σ-closure Σ. We will say that a representation is based on  if and only if . We introduce the notion of a credence function that is based on , and Σ, by requiring that credence functions be normalized to the base representation , and that it be additive over disjoint elements of Σ. Definition 4 A credence function based on, and Σ is a function such that: Possibly, n = ∞, in which case Cr is constrained by normalization and countable additivity . Possibly, , in which case Cr is constrained by normalization and finite additivity . Definition 5 Given a credence function Cr based on , Σ, and the conditionalization of Cr on is a function such that: is based on If , then If , then Otherwise, Definition 6 Given Cr based on , and Σ, a conditional credence function based on , and Σ is a function such that . 4.2 A semantics of representations The driving semantic ideas are the following. Sentences of natural language semantically encode representations. Representations can be modelled as sets of possibilities. Possibilities come in many different semantic types: possible worlds, sets of possible worlds, sets of sets of possible worlds, credence functions, sets of credence functions, sets of sets of credence functions, and so on. Consider a language containing a stock of atomic sentences , Boolean compounds of sentences, the indicative conditional →, the ‘probably’ operator , and the epistemic modal ⋄. An interpretation function for this language maps sentences into representations. The obvious clauses would be as follows:16 These clauses require some elaboration. We have generalized the probability calculus to credence functions, by requiring that credence functions be specified relative to (i) a ‘base’ representation , (ii) an alternative set that partitions , and (iii) ’s σ-closure Σ. We will therefore say that the representation expressed by such sentences is determined relative to a base representation , alternative set for , and ’s σ-closure Σ. We will call a triple with these characteristics a space, and we will allow a context c to determine (via a contextually determined variable assignment gc) a space (of the requisite semantic type) for each space-sensitive expression of our language.17 I shall assume (for technical and empirical reasons) that the operators , and presuppose the ‘visibility’ (in the sense of Yalcin 2011, 2018) of their arguments in . (In general, a representation is visible in a partition if and only if for each or . Applying this notion of visibility, will be defined only if, for each or .) Taking these clauses in order: , when defined, is the property a credence function Cr based on gc(1) has if and only if Cr treats the representation expressed by as certain conditional on the representation expressed by . , when defined, is the property that a credence function Cr based on has if and only if it assigns the representation expressed by a value greater than 0.5. , when defined, is the property that a representation based on gc(1) has if and only if is compatible with the representation expressed by ϕ. Equivalently, where the semantic types demand it, such properties (modelled as λ-abstracts) may be understood as sets of possibilities that satisfy the corresponding λ-abstract. For instance, where the semantic types demand it — or, indeed, wherever a set-theoretic understanding is more convenient — we let . 4.3 Examples Example 1. Consider the case of scoping over a propositional atom. Let . Here, semantic types require that: be a set of worlds, for example, is a partition of in which is visible, for example, ; Σ is ’s σ-closure. A sentence of the form expresses the property a credence function (based on , and Σ) has when it assigns the worldly representation encoded in p a value greater than 0.5. Example 2 Next consider an example involving iterated over ⋄. Let and . Here the semantics requires that: As above, is a set of worlds; is a partition of in which is visible; is ’s σ-closure. is a set of sets of worlds, for example, is a partition of in which is visible, for example, As intended, expresses the property a credence function has when it assigns the (non-worldly) representation encoded in a value greater than 0.5. More precisely, it expresses the property a credence function defined over has when it assigns the set of possibilities such that is based on gc(2) (that is, such that ) and is compatible with — a value greater than 0.5. Example 3 Next consider the reverse iteration. (As the types increase, I’ll compress formal detail for the sake of readability.) As intended, expresses the property a set of credence functions has when it contains a credence function that assigns the worldly representation encoded in p a value greater than 0.5. Example 4 Finally, two examples involving iterated epistemics: As intended, expresses the property a set of sets of worlds has when it contains a set of worlds that is compatible with p. As intended, expresses the property a set of sets of worldly propositions has when it contains a set of worldly propositions that is compatible with ⋄p. 4.4 Compositionality and polymorphic types The interesting operators of our language (→, ) uniformly take set-type meanings (representations) as arguments. This gives our system the veneer of compositionality, but, for now, only the veneer. Set-type meanings are, strictly speaking, not typically the semantic values of these operators’ complements; the semantic values of the sentences of our language, in fact, constitute a manifold of functional types.18 Here is an illustration: ⋄ can semantically combine with a worldly representation ,19 an epistemically modal representation , an epistemically modal representation with epistemically modal content , and so on, ad infinitum. Epistemic operators therefore have the (perhaps surprising) property of being relatively unselective as to the semantic type of their complements — more precisely, of being type-polymorphic.20 The polymorphic type of epistemic operators can be given a recursive characterization: One way to think about polymorphic types is this. An expression like ⋄ has a semantic type, in two guises: qua expression-type (in which case its type is polymorphic) and qua expression-token (in which case its type, as tokened on an occasion of use, is a type drawn from the polymorphic type hierarchy). The semantic type of, for example, ⋄, as tokened on an occasion of use will ‘depend’ (very loosely speaking21) on the semantic type of its complement (but will always be drawn from the hierarchy of types introduced here).22 5. Two aspects of mental life The previous section showed that the notion of credences in epistemically modal and probabilistic representations, constrained by the probability axioms, is both mathematically and semantically tractable. But — and I intend this question seriously — does it make sense? We have introduced a semantic hierarchy of representations with no upper bound on the complexity of the semantic type of a representation. Is this cognitively realistic? (Here, I will argue: yes.) We have assumed that objects at any level of the type hierarchy can receive credences (where credences are constrained by assumptions of normalization and additivity). Is this normatively plausible — do standard justifications for normalization and additivity apply if credences are not assumed to be defined over worldly representations? (Here, I will argue: yes and no.) 5.1 Cognitive Moss offers an argument against a proposed extension of the Bayesian proposal pursued in Rothschild (2012) and Yalcin (2012) to graded modal judgements: [I]t is hard to imagine a reason for ruling that embeddings of epistemic vocabulary beyond a certain level of complexity are semantically uninterpretable. In the absence of such a reason, our theory should deliver semantic values for embeddings of arbitrary complexity. Hence in order to repair the [Bayesian] proposal, we would have to model subjects as having not just sets of sets of measures as mental states, but sets of sets of sets of measures, and so on. It is difficult to independently motivate such an arcane model of our mental life. (Moss 2015, p. 30) While our proposal isn’t quite Bayesian in the sense of Rothschild (2012) and Yalcin (2012), Moss’s critique clearly applies. The charge that this is an ‘arcane model’ does not, however, really bite. To illustrate, recall that, on the view defended here: On our view, expresses the property a set of sets of worlds (that is, a set of worldly propositions) has when it contains a set of worlds that is compatible with p. To call such a sentence probable is to express a property of credences in sets of sets of sets of worlds (that is, sets of sets of worldly propositions) — namely, the property of assigning a credence greater than 0.5 to . An agent can treat any set of objects as a set of alternatives for cognitive purpose P. An agent can represent sets of possible worlds for the purpose of representing different abstract alternatives (individual possibilia) for accurately representing the world. An agent can represent sets of sets of possible worlds (that is, sets of propositions) in order to represent different alternatives, not for the purpose of accurately representing the world, but for the purpose of representing alternative ways of representing the world (for example, alternatives that treat p as possible versus those that treat p as impossible). An agent can represent sets of sets of sets of possible worlds (that is, sets of sets of propositions) in order to represent different alternatives (sets of propositions) — not for the purpose of accurately representing the world, nor for the purpose of representing alternative ways of representing the world, but instead for the purpose of representing alternative ways of representing alternative ways of representing the world. Representations, as we understand them, have an iterative, or recursive, structure (but is that surprising?). But the cognitive state of representing for purpose P is not arcane: it is the attitude of representing the various alternatives of as candidates for fulfilling P. We have understood the attitude of representation more expansively than is traditional23 — in particular, we have relativized representations to cognitive purposes, and have declined to assume that the functional role of representation is uniformly about representing individual possibilia as candidates for actuality. Generalizing a familiar notion need not, however, render it arcane. Indeed, given this generalized understanding of representation, representing a set of alternatives as candidates for fulfilling P describes a sort of familiar cognitive activity in which agents plausibly can and do engage. 5.2 Normative Why represent agents as having credences in non-worldly representations? We cited two (related) motivations (§3.2). A theoretical motivation: to vindicate ascriptions of confidence in epistemically modal representations within our theoretical metalanguage. And a normative motivation: to describe and justify rational norms governing confidence in epistemically modal representations; and to describe and justify rational norms governing the relationship between confidence in epistemically modal representations, and action. The present account satisfies the theoretical motivation. What about the normative? This sub-section proposes that (i) the rational norms governing the relationship between credence and action are a generalization of the theory of expected value; (ii) generalizations of the decision-theoretic notions of a decision problem and of the expected value of an action relative to a decision problem can be stated in a basically standard form. 5.2.1 Generalizing expected value We begin by defining the notion of a generalized decision problem. Definition 7 A decision problem Π based on is a tuple where: is a partition of the possibilities relevant in Π. is a set of actions available in Π. Cr is a credence function based on . Val is a conditional value function, such that is a value representing the degree to which A is desired conditional on C (for each ). Decision problems can be presented in a standard tabular format, in which cells of the table correspond to ‘pay-offs’, here understood as degrees of desirability conditional on the corresponding representation.24 Π A1 Π A1 Open in new tab Π A1 Π A1 Open in new tab In contrast to standard presentations of decision tables, we do not assume that the contingencies relevant in a decision problem Π based on form a partition of W (or of a subset of W). Instead, , partitions a salient representation — picturesquely, the ‘base’ representation against which an agent’s deliberation occurs. This base representation is not, however, required to be of any specific semantic type. Given the notion of a generalized decision problem, a corresponding generalization of expected value is immediate. Definition 8 If is based on and , the expected value of A in Π is a weighted sum of credences multiplied by values: 5.2.2 Justifying credences Why should an agent in a generalized decision problem maximize generalized expected value? More specifically (and to bracket certain controversies about formulating a mathematical theory of rational action), why should an agent who wants to maximize expected value in a generalized decision problem compute expected value using a credence function (the properties of which are constrained by Definition 4)? There are two main ways of answering this type of question in the literature. First, Dutch Book arguments, on which, roughly, agents who have incoherent credences are irrational because subject to sure losses (for an overview, see Hájek 2009). Second, accuracy arguments, on which, roughly, agents who want to maximize expected epistemic value (roughly, the proximity of one’s credences to the truth), but who have incoherent credences, are irrational because coherent credences are always more proximal to the truth (originating with Joyce 1998). Let us investigate the prospects of extending these answers to the present account. Matters are, not surprisingly, less than straightforward with accuracy arguments. Accuracy is fundamentally a worldly notion: a representation is said to be accurate when it is satisfied (‘true’) as evaluated against a possibility taken to represent actuality (as also noted by Staffel 2019). Accuracy arguments purport to show that subjective estimates of objective chance that violate the axioms of probability are rationally defective, since for any such estimate, there is another way of estimating chances that (i) satisfies the axioms of probability, and (ii) is guaranteed to be overall more accurate in w, for any possible world w to which the agent assigns some credence (see especially the accuracy theorem of Joyce 1998). In order to adapt accuracy arguments to the framework proposed here, we would require a non-worldly proxy for the notion of accuracy (as well as a non-worldly proxy for the notion of actuality). The prospects here strike me as very dim — particularly given the conceptualization of our theory suggested in §4. Dutch Books, however, do appear to generalize to this application. The constraints on generalized credence functions we have introduced are therefore ultimately motivated by ‘pragmatic’ considerations (although in the case of probability measures over worldly propositions, they may still be motivated by considerations of accuracy). Nothing in the bare mathematics of the ‘Dutch Book theorem’ (see, for example, Hájek 2009) requires that decision-theoretic contingencies (‘states’) are worldly propositions. To illustrate, here is a Dutch Book for an ‘overconfident’ agent who commits herself to a sentence of the form (and thereby commits herself to regarding both and as probable). (I assume, just for the sake of illustration, that the operator expresses a credence greater than 0.6 in its complement representation.) Bet 1 -0.6 0.4 Bet 2 0.4 -0.6 Bet 1 -0.6 0.4 Bet 2 0.4 -0.6 Open in new tab Bet 1 -0.6 0.4 Bet 2 0.4 -0.6 Bet 1 -0.6 0.4 Bet 2 0.4 -0.6 Open in new tab For concreteness, suppose that ϕ is the proposition that there is an algae bloom in the lake. This table represents Bet 1 (for example, drinking lake water) as undesirable to degree 0.6 conditional on , and desirable to degree 0.4 conditional on ; it represents Bet 2 (for example, purchasing bottled water) as desirable to degree 0.4 conditional on , and undesirable to degree 0.6 conditional on . The agent of this decision problem regards as fair a series of ‘bets’ that, taken together, logically guarantee a ‘loss’ (from the vantage point of her own conditional degrees of desirability).25 This is, I claim, a defect of rationality. In general, an agent imposes a partition on a base representation , thereby generating an alternative set for , for the sake of representing alternatives whose adoption is relevant for a cognitive purpose P. Whichever alternative in such an agent should accept — however such an agent fulfils P — that agent will be subject to a loss (from the vantage point of their own conditional degrees of desirability) in a Dutch Book. Whether or not, that is to say, the agent concludes that there is probably an algae bloom, her incoherent credences make her such that she regards both drinking the lake water and purchasing bottled water as good bets in the Dutch Book presented above. Roughly speaking, in such a Dutch Book, such an agent regards it as acceptable to spend money on bottled water to avoid drinking lake water that she regards as acceptable to drink. That is irrational. 5.2.3 Nailing down Dutch Books Staffel (2019) develops both accuracy-style and Dutch Book-style arguments for coherent credences in non-worldly representations (while also registering doubts that such arguments actually meet the theoretical needs that prompt them). In Staffel’s expressivistic Dutch Book — which is in certain respects similar to the one advanced here — an ‘underconfident’ agent (say, one who assigns both and its negation credence 0.4) can avoid a sure loss by not becoming opinionated. The fact that the underconfident agent would lose money if she became opinionated does not point to any obvious rational defect. There are many things I might do that would put me at a great disadvantage in particular circumstances. But if I have no reason to think I’ll find myself in those circumstances, then I have little or no reason to avoid those actions. (Staffel 2019, p. 185; emphasis added) This difficulty certainly threatens the expressivistic Dutch Book developed in Staffel (2019). It might also seem to threaten the version I have pursued here. The ‘irrationality’ that, I claimed, characterizes an incoherent agent is as follows: relative to an alternative set that represents various candidate representations for fulfilling purpose P, the agent would be subject to a loss if she selected an alternative from to fulfil P. However, if she is not in a position to select an alternative from to fulfil P — if she is unable to settle on any particular way of resolving the relevant question (for instance, whether there is probably an algae bloom) — the negative conditional desirability (of, for example, Bets 1 and 2 conditional on ) is never ‘actualized’. The ‘loss’ to which the agent is subject in a Dutch Book is of a merely hypothetical character: the agent will be worse off if she becomes opinionated, but if she doesn’t, she won’t. What is irrational about that? In reply: we said that, in the above Dutch Book: An agent entertains a set of type (a set of credence functions). She partitions into: (i) a cell of credence functions according to which it is probable that there is an algae bloom; (ii) a cell of credence functions according to which it is not probable that there is algae bloom. The purpose the agent tries to achieve in so partitioning is, we said, to represent alternative ways of representing the world (for example, alternatives that treat an algae bloom as probable versus alternatives that do not). Conditional on either way of representing — that is, conditional on representing an algae bloom as probable and conditional on representing it as not probable — the agent is subject to a loss (from the vantage point of her own conditional degrees of desirability) in a Dutch Book. The irrationality here is, I submit, manifest: the agent is trying to achieve goal g (for instance, figuring out how to represent the likelihood of there being an algae bloom — as probable or not probable), but her credences are such that any way of achieving g presents her with a deficit in desirability in a Dutch Book. That is to say, her credences are structurally such that, conditional on any way of achieving what she is trying to achieve, she is subject to a deficit in desirability (from the vantage point of her own conditional degrees of desirability) in a Dutch Book. Claim: if your credences in context c are structurally such that they prevent you from doing what you’re trying to do in c, without being subject to a sure loss in a Dutch Book, your credences in c are irrational in c. 6. Conclusion This paper began by observing that standard models of the semantics of epistemic modals render the phenomenon of graded modal judgement, whether in thought or in language, unintelligible. In response, this paper developed a model of graded modal judgement, in both thought and language — one that represents graded modal judgement as a generalization of our cognitive capacity for reasoning with hypotheses about objective chance (that is, our cognitive capacity for probabilistic reasoning). The generalization was developed as a package of interrelated semantic, cognitive, and epistemological theses: Semantic: Modals compose with representations of arbitrary type. (§4.2) Cognitive: Agents entertain representations of arbitrary type for specific cognitive purposes; the state of bearing a graded attitude toward a representation of arbitrary type is a natural cognitive kind (instances of which are, broadly, governed by the purpose for which the agent is entertaining the relevant representation). (§5.1) Epistemological: Part of the functional role of credences in representations of arbitrary type (entertained for cognitive purpose P) is to determine fair ‘prices’ for bets against ways of representing that fulfil P. Agents whose credences violate normalization or additivity are thus subject to Dutch Books. (§5.2) On the model of graded modal judgement developed here, modal sentences are semantically evaluated against complex constructions out of possibilia. But various sentences of our language are not semantically evaluated relative to individual possibilia. And so our model exhibits the characteristic insensitivity of logics axiomatized by S5 to a choice of possible world taken to represent ‘indicative actuality’ (as in Kaplan 1989), or to a choice of possible world taken to represent a non-actual circumstance of evaluation. I take this to be one of the main virtues of the present theory: it can accommodate many of the intuitions that motivate axiomatizing the logic of epistemic modality with S5, without rendering the notion of graded modal judgement, whether in thought or in language, unintelligible (for more discussion, see Appendix B.1). A polemical note to conclude. Notice that ‘non-factual’ theories — theories that do not take modalities of the relevant type to be proposition-forming operators, a description satisfied by both our theory and Moss’s — offer the theorist at least two broadly workable models of the cognition, semantics, and epistemology of graded modal judgement. ‘Factual’ accounts of these modalities, so long as they are constrained by S5 — and, indeed, even a weaker logic like KD45 — are able to offer none of these attractions (see Appendix B.1). It is probably time to move past the philosophical preoccupation with the ability of non-factual theories of operators in natural language to account for environments embedding these operators. If anyone has such problems, it seems to be the theorists who have pushed such objections, rather than their targets.26 Footnotes 1 By ‘epistemic modality’, I mean a sentence (or sometimes the content of a sentence) of the form Oϕ, where O is an autocentrically interpreted epistemic operator and ϕ is its sentential prejacent. An autocentric interpretation of a sentence of the form Oϕ is an interpretation according to which the speaker is interpreted as making an epistemic claim, ‘based on’, or from the ‘vantage point’ of, their own information or evidence (see Lasersohn 2005). Epistemic operators are here understood to encompass genuinely modal operators (‘must’, ‘might’), epistemic or probabilistic adverbs (‘probably’, ‘certainly’, ‘possibly’), numerical probability operators (‘it is n-likely that’), and more. 2 Yalcin (2009) argues (in the service of a more general scepticism about the semantic productivity of iterating epistemic vocabulary) that a speaker who asserts that it might be the case that Trump might be impeached is committed to allowing that Trump might be impeached. I deny this (but also its relevance to the target phenomenon for this paper). Claim: the truth (or assertability) in a context c of a sentence of the form implies that ϕ is a relevant epistemic possibility in c (and vice versa). Therefore, a speaker who asserts this claim at c has not made a mistake, if Trump’s being impeached is not a relevant epistemic possibility at c (that is, it would be improper to assert (3) in c). This suggests the content of the speaker’s assertion is weaker than with (3). In any case, it is hard to see how to extend Yalcin’s argument — as it must be extended if the aim is to deny the semantic productivity of iterating epistemic vocabulary — to possibility modals scoping over ‘probably’. Someone who says that it’s possible that Trump will probably be impeached is not committed to allowing that Trump will probably be impeached (see Moss 2018, p. 46). Finally, even if successful, arguing that a speaker who asserts that it might be the case that Trump might be impeached is committed to allowing that Trump might be impeached is insufficient to establish that the content of the speaker’s assertion entails that Trump might be impeached — particularly given a plausible argument that the content of the speaker’s assertion is weaker than the content of (3). 3 On Euclideanness, see Appendix B.1 and N. Charlow (2016b). These are standard assumptions in the semantics and logic of epistemic modalities (see, for example, Holliday and Icard 2010; Gillies 2010; von Fintel and Gillies 2010, 2011, 2018). The phenomena of interest in this paper will also arise for modalities of belief (axiomatized by KD45, rather than S5). 4 A probabilistically coherent agent is one whose degrees of belief in some σ-algebra of some subset of W are representable with a probability function. 5 It is worth underlining that this is not an artefact of the use of sets of possibilia to model states of information. So long as the class of models for an epistemically modal language is required to satisfy object-language analogues of Reflexivity () and Euclideanness () — and logical consequence is closed under modus ponens, that is, implies that — the logic of epistemic modality will be constrained by the following entailments: (To preview, on the account defended here, these entailments will fail in the left-to-right direction, even though for any context c, and ; on this point, see Appendix B.1.) Let designate A’s information at w; we will not assume that is a set of possible worlds. Now either (if is compatible with p) or (otherwise). If , then , in which case is incompatible with (that is, entails ). Since A is probabilistically coherent, . If, on the other hand, , then , in which case is incompatible with ⋄p (that is, entails ). Since A is probabilistically coherent, . Thus, again, either or . 6 An informative comparison is with Gaifman (1988)’s classic approach to higher-order uncertainty. Gaifman represents higher-order uncertainty as uncertainty about the ‘true’ probabilities: the claim ‘there is a 70% chance that later this evening the chance of rain tomorrow will be 80%’ is represented as an assignment of probability to a proposition about some future event — in this case, the event in which the ‘true’ probability of rain tomorrow (according to some contextually salient estimate of the probability of rain tomorrow) is 80%. Like Moss, Gaifman treats higher-order uncertainty (to degree d) about epistemically modal ϕ as factual uncertainty (to degree d) about the truth of some proposition p such that, if the agent conditionalizes on p, the agent thinks that ϕ (here, see Gaifman’s discussion of his Axiom VI at pp. 281–2). Unlike Moss, Gaifman takes the content of ‘the chance of rain tomorrow will be 80%’ to be a possible-worlds proposition that is true at w if and only if the ‘true’ probability of rain tomorrow at w (according to the contextually salient estimate at w) is 80%. Apart from the metasemantic quandaries associated with this proposal — in virtue of what is the ‘true’ probability of some event at w determined relative to a context? — Gaifman’s account (unlike Moss’s) is caught up in the dialectic of §2: if ⋄ is a propositional operator, the logic of ⋄ is S5, which trivializes degreed belief in sentences of the form ⋄ϕ. (Thanks to a referee for this journal for drawing my attention to Gaifman’s paper.) 7 This is already part and parcel of the ‘classical’ propositional account of epistemic modalities discussed in §2. Moss relinquishes this bit of the ‘classical’ account to provide a treatment of graded modal judgement; this section offers some pushback. To be sure, the considerations raised in this section are tentative, and are not intended as a serious brief against Moss’s theory. The relevant context for this discussion is Moss’s remark that ‘It is difficult to independently motivate such an arcane model of our mental life’ as the sort of model that this paper goes on to propose (Moss 2015, p. 30). This section (and §5) argues that the model of mental life that this alternative picture evokes is relatively commonsensical, once we decide to make room for higher-order uncertainty in language and thought. 8 Schroeder (2011) and Staffel (2019) argue in a similar fashion. There is certainly work to do to build a ‘natural’ account of this phenomenon that is both functional and theoretically appealing — but that will be the work of this paper. To preview, on the account defended here (§5), an agent who believes makes a specific kind of rational mistake: they are subject to (generalized) Dutch Books. Interestingly, such an agent, unlike an agent who believes (9), will not be subject to having accuracy-dominated credences — something that should be seen as congenial to a generally ‘non-factualist’ or ‘Bayesian’ perspective on probabilistic thought and talk (see Yalcin 2011). 9 This qualification is necessary since I do not want to assume an agent is always in a position to compute such a value. That said, there are certainly decision situations in which an agent is in a position to compute such a value. Imagine α is entertaining two probability functions Cr1 and Cr2 such that: (i) Cr1(p) = 0.7 and Cr2(p) = 0.3; (ii) if α were to adopt either Cr1 or Cr2, would be defined for α. In such a decision situation, evidently, is just the value of REU(X) computed using a credence function Cr (=Cr2) , whereas is just the value of REU(X) computed using a credence function Cr (=Cr1) . This example provides a good opportunity to note that there is an epistemological debate (from which I will prescind in this paper) about how probabilistically unspecific evidence (evidence that fails to settle a precise probability for a relevant possibility) constrains an agent’s credences (and the choices an agent might base on her credences). (On certain models of ‘imprecise’ credence, entertaining a set of possibilities such as would be disallowed. See the discussion of the controversial ‘convexity’ requirement in Moss 2019, as well the overview in Joyce 2005.) This paper will also prescind from the debate over general decision rules for imprecise decision problems (for several criteria of adequacy for such a decision rule, see Joyce 2010, pp. 313 ff.). This paper does attempt to state a natural decision rule for one type of imprecise decision problem in which an agent seeks to maximize some value that is dependent on an epistemically modal representation. Although I believe the ability to formulate such a decision rule represents an advantage of this account over the relatively coarse-grained decision rules put forward in alternative frameworks for representing higher-order uncertainty (for example, Gärdenfors and Sahlin 1982), I defer comparison to another time. 10 Risk-sensitivity is just one illustration of how epistemically modal representations can impinge on practical reasoning: decision situations in which an agent confronts ‘higher-order uncertainty’ (uncertainty about the probabilities of the relevant states in their ‘base’ decision situation) serve the purpose too. In decision situations characterized by higher-order uncertainty, a maximizing agent — one who seeks to maximize any quantity that depends on an epistemically modal representation (including ordinary expected value) — will tend to prefer an action they regard as likely to maximize that quantity to actions they regard as unlikely to maximize it. Such preferences can be characterized straightforwardly in the framework this paper proposes. 11 An alternative picture: my decision in this situation depends on how likely I regard some ordinary proposition p (for example, that the pH of the lake water exceeds some value) such that updating on p implies thinking there is probably an algae bloom. But, by stipulation, my decision in this decision situation depends on how inclined I am to think that . We might additionally suppose that my inclination toward this claim is a prior, in the sense that there is no salient proposition p such that (i) I am 0.4 confident in p, and (ii) if I update on p, I’ll believe . Even if my inclination in this regard is a prior — if, for example, I regard as having low probability, without regarding any salient proposition as having low probability — it seems that I can confront the decision situation described here. 12 This general perspective draws inspiration from the ‘multi-dimensional’ approach towards the probabilities of indicative conditionals in Bradley (2012), as well as remarks in Sepielli (2012) and Staffel (2019) discussing how a non-cognitivist or expressivist might model the cognitive and normative characteristics of gradable attitudes towards non-factual semantic contents. Jonathan Weisberg (personal communication) alerts me to an earlier approach to higher-order probability (Hild 1998) that is similar in both spirit and certain modelling choices to the one developed here. 13 This paper ultimately argues that epistemic quantifiers over representations (domains of possibilities) are polymorphic in type (likewise for the representations they take as arguments). Type polymorphism, in the context of theoretical computer science, ‘refers to a range of language mechanisms that allow a single part of a program to be used with different types in different’ environments (Pierce 2002, p. 331). The polymorphic semantic types appealed to in this paper are constructed from the basic materials of possible worlds and credence functions (for details, see §4.4). Depending on the kind of language we are modelling, I am pragmatic about the basic materials: in my view, agents can represent sets of (and sets of sets of, and sets of sets of sets of) many kind of things — in addition to possible worlds and credence functions, preference orderings or decision rules for deontic modalities (Charlow 2015, 2016a, 2018), degree thresholds for gradable adjectives (Charlow forthcoming a), features of their subjective perceptual experience (Charlow 2019a), and more besides — as candidate possibilities for some purpose (on this generalized notion of representation, see §5.1). At the most general level, a representation is just a set of possibilities, and a possibility is just a semantic object against which a sentence can be evaluated for satisfaction, relative to a model. 14 This is only a rough first pass at stating a functional psychological role for the representation of a set of alternatives of arbitrary semantic type. For an elaboration, see §5. 15 Present purposes do not require using partitions here, but there are three advantages worth noting: invoking partitions (i) makes the extension of this architecture to decision theoretic applications very smooth (see §5.2), (ii) allows us to avoid certain apparent counterexamples to modus ponens (Charlow 2019b), and (iii) will eventually allow us to connect a way of imposing structure on a relevant set of possibilities to a more concrete fact about a discourse (for example, a set of relevant questions which generates the relevant partition, à la Roberts 1996 and Yalcin 2011, 2018). 16 Notation: designates a set of objects of semantic type , while is the universe of objects of semantic type . 17 This notation, and the implementation via contextually determined variable assignments, is from Moss (2015, 2018). Note, however, that indices in Moss’s semantics uniformly resolve to partitions of W. 18 Here I assume that semantic composition is always via function-argument application. 19 Notation: s is the type of worlds, t is the type of truth-values. A function of type is a function from objects of type τ into objects of type . 20 On type polymorphism in theoretical computer science, see especially Pierce (2002). For a recent application of polymorphic types to natural language semantics, see S. Charlow (forthcoming). 21 This is no violation of compositionality: the semantic type of ⋄, as tokened on an occasion of use, is not semantically determined by, or selected in virtue of, the semantic type of its complement. It is simply to say that if ⋄ occurs in a semantically well-formed expression, its semantic type must be drawn from the hierarchy of types defined above, and must be of the right type to compose, by function-argument application, with the semantic value of its sister. 22 To be compositional, our system requires a understanding of semantic coordination. We currently understand as expressing , but there are two reasons this will not work. First, the semantic values of ’s arguments are functions, not sets. (This is trivial to fix, and I will continue to talk as if the difference between a characteristic function and a set is no difference at all.) Second, the semantic values of ’s arguments are frequently sentences of different semantic type. This is less trivial to fix: we will require a generalized understanding of conjunction that allows it to coordinate constituents of different semantic type, as in Partee and Rooth (1983). To keep the main discussion maximally simple, I will ignore this sort of complication here (though I will address it in the Appendices). 23 The traditional view I am attempting to generalize here is, of course, that of Stalnaker (1984, Chapter 4). 24 Staffel (2019) remarks that, in an expressivistic system (alike in some, but not all, respects to the one proposed here), ‘wins and losses can’t be determined by checking what the world is actually like’ (if the relevant contingencies are not worldly propositions that can be ‘checked’ for truth against the actual world). But if, as seems correct, the conditional value is like the conditional probability — in that both track degrees of desire or belief, under the indicative supposition that C — there is no immediate need for worldly matters to ‘determine’ wins and losses in decision problems based on representations of arbitrary type. The degree to which an agent who indicatively supposes C desires to perform A will determine — nothing worldly required, so long as the degree to which an agent desires to perform A can depend on a non-worldly representation. (Such dependence appears to be commonplace; recall §3.2.) There may yet be a need for worldly matters to determine wins and losses, for a theorist who wants to use the notion of conditional desirability to run a Dutch Book argument. More on this just below. 25 If, for example, the agent is by assumption an REU maximizer, then the conditional degrees of desirability recorded in this decision table (when defined — recall fn9) may be taken to represent REUs computed conditional on the relevant representation. As noted above, once the notion of a decision problem is generalized, words like ‘bet’ and ‘loss’ lose their normal connotations: whether or not ϕ is probable does not ordinarily seem to be the sort of question on which an agent can bet (in the sense of making a wager that produces some payout if ϕ is probable, and another payout if it is not). (How, after all, would a winning bet be determined — particularly given the broader non-factualist setting of this paper?) In the present context, when a package of ‘fair’ ‘bets’ is said to guarantee a ‘sure loss’ for an agent, this means that (i) there is a set of actions such that for each , the expected value of doing A for the agent is at least as great as the expected value of not doing A for the agent (roughly, there is a set of actions all of which the agent regards as ‘fair’ or ‘worth doing’); (ii) the expected value of the complex action doing everything in  is less than the expected value of not doing everything in (roughly, the agent does not regard the complex action doing everything in  as ‘fair’, even though she regards all of the actions in as ‘fair’). 26 Special thanks to Simon Charlow for extensive feedback, and for introducing me to this paper's main formal tool. Thanks to this journal's referees, and to Simon Goldstein, Sarah Moss, Paolo Santorio, Julia Staffel, and Jonathan Weisberg, all of whom offered valuable feedback at important junctures for this project. 27 Our Probabilify rule is a close relative of the type-shifting rule introduced in Moss (2015, p. 34) (see also Moss 2018, pp. 234 ff.). Note, though, that the role that type-shifting plays in Moss’s system is quite different from the role it plays in ours. Moss’s account, unlike ours, requires that (understood as a set of worlds) be type-shifted into a probabilistic (‘ur’, for short) content — the set of probability spaces normalized to — in order to compose with an epistemic operator (and, more generally, to be the sort of object that can play the content role in Moss’s larger theory of probabilistic assertion and belief). This raises the following question: why does appear to lack a reading on which it means that p is not certain (that is, means the negation of p’s ur-content)? Moss addresses this question for a sentence of the form with the stipulation that ‘semantic types of sentences are shifted from sets of worlds to sets of probability spaces if and only if such type shifting is forced’ (Moss 2018, p. 238). But how will this work for the following sort of case? (10) A: It’s raining. B: That’s not true. Moss writes that ‘If a sentence contains no epistemic vocabulary at all, then assertion itself forces the semantic type of that sentence to be shifted from a proposition to a set of probability spaces’, that is, an ur-content (Moss 2018, p. 234; emphasis added). Presumably, B’s denial can be represented as targeting the content of A’s assertion with negation. But that represents B’s denial as possibly weaker than it could, in fact, be. Since our account utilizes type-shifting exclusively for inducing semantic coordination between syntactically coordinated sentences, this kind of worry does not arise for it. 28 γ is the type of credence functions (§4.4). So a function of type is of type . It is natural to assume that whatever credence, if any, someone assigns will determine (or perhaps rationally constrain — I do not have a good sense of what is at stake here) their credence in : if you think of p as i-likely, you are, or ought to be, i-certain of (though often there is no such that you are, or ought to be, i-certain of q). 29 Restrictable quantifiers are generalized quantifiers, in the sense of Barwise and Cooper (1981). 30 Except when no quantifier is provided, in which case a silent restrictable quantifier — which Kratzer (1986), for instance, took to be an epistemic necessity modal — is posited in logical form. 31 See Égré and Cozic (2011)’s adaption of the theorem of Hájek (1989) to an inexpressibility result for a language containing only unary probability operators. Sketch of the proof: consider a fair three-ticket lottery, with tickets numbered one, two, and three. According to any probability model for this situation, the conditional likelihood of one winning if three doesn’t is 1/2; but there are probability models for this situation in which no proposition is such that it has probability 1/2. 32 I am here thinking of consequence standardly, that is, in terms of set-theoretic inclusion. Notice that consequence, in this sense, can hold only between sentences of the same semantic type — for example, . This is not difficult to repair, however — should we decide that it is important to designate the relationship between, say, and ϕ as one of logical consequence. (I think this less important than predicting (as this section does) that sentences of the form express defective semantic contents (that is, ), but your mileage may vary.) Notice that is a set of p-entailing subsets of W; since each possibility consistent with is a p-entailing possibility, we say that p is a ‘bird’s-eye consequence’ of . More generally ψ is a bird’s-eye consequence of ϕ in c if . In general, ψ is a bird’s-eye consequence of ϕ0 if and only if there exists a sequence such that for each is a bird’s-eye consequence of , and ψ is a bird’s-eye consequence of . Thus, for instance, ϕ is a bird’s-eye consequence of , as desired. 33 Inducing semantic coordination via type-raising yields curious results under negation, as observed by Mandelkern (2019, §7.4) and Simon Charlow (personal communication). For instance, applying the coordination-via-type-raising strategy to and wrongly predicts that both are equivalent to . One can avoid this by stipulating that, when a logical operator takes scope over a syntactically coordinated (but semantically uncoordinated) sentence, the sentence is parsed in conjunctive/clausal normal form (see Abney and Keshet 2013). Conditional on this stipulation, is parsed as as ); on this parsing, both are (correctly) predicted equivalent to (proofs omitted). (Fans of type-shifting accounts of semantic coordination between epistemic and non-epistemic language should, however, be on the lookout for a less stipulative way of dealing with this kind of issue.) 34 I ignore the question of the possible order-sensitivity of the phenomenon (that is, whether swapping the restrictor clause for the nuclear scope affects the sentence’s acceptability). Since semantic coordinability is not order-sensitive, the account here predicts that the phenomenon is not order-sensitive — which is, Yalcin (2015) agrees, probably desirable. 35 I provide a syncategorematic semantics for quantification in lieu of a compositional version (which would make use of a polymorphic type for generalized quantifiers). In the general case, for any two-place quantifier Qx: Here, is the quantificational relationship between sets expressed by Q (as in Barwise and Cooper 1981). Thanks to Simon Charlow for raising the question of generalized quantification (and for suggesting the natural clauses used here). References Abney Steven , Keshet Ezra 2013 : ‘Using Conjunctive Normal Form for Natural Language Semantics’. Manuscript available at http://www.ezrakeshet.com/site/ezrakeshet/papers/Abney-Keshet-2012-Conjunctive-Normal-Form.pdf. Barwise Jon , Cooper Robin 1981 : ‘ Generalized Quantifiers and Natural Language’ . Linguistics and Philosophy , 4 , pp. 159 – 39 . Google Scholar Crossref Search ADS WorldCat Bradley Richard 2012 : ‘ Multidimensional Possible-World Semantics for Conditionals’ . Philosophical Review , 121 , pp. 539 – 71 . Google Scholar Crossref Search ADS WorldCat Buchak Lara 2013 : Risk and Rationality . New York : Oxford University Press . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Charlow Nate 2015 : ‘Prospects for an Expressivist Theory of Meaning’ . Philosophers’ Imprint , 15 , pp. 1 – 43 . WorldCat Charlow Nate 2016a : ‘Decision Theory: Yes! Truth Conditions: No!’ In Nate Charlow , Chrisman Matthew (eds.), Deontic Modality , pp. 47 – 81 . Oxford : Oxford University Press . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Charlow Nate 2016b : ‘ Triviality for Restrictor Conditionals’ . Noûs , 50 , pp. 533 – 64 . Google Scholar Crossref Search ADS WorldCat Charlow Nate 2018 : ‘ Decision-Theoretic Relativity in Deontic Modality’ . Linguistics and Philosophy , 41 , pp. 251 – 87 . Google Scholar Crossref Search ADS WorldCat Charlow Nate 2019a : ‘Gradable Predicates of Experience’. Unpublished manuscript. Charlow Nate 2019b : ‘Modus Ponens and the Logic of Dominance Reasoning’. Unpublished manuscript. Charlow Nate forthcoming a: In Dunaway Billy , Plunkett David (eds.), Meaning, Decision, and Norms: Themes from the Work of Allan Gibbard . Michigan : Maize Books . Google Preview WorldCat COPAC Charlow Nate forthcoming b: ‘The Spectre of Triviality’. To appear in Analysis. doi: 10.1093/analys/anz011. Charlow Simon forthcoming: ‘A Modular Theory of Pronouns and Binding’. To appear in Proceedings of LENLS 14. Égré Paul , Cozic Mikaël 2011 : ‘ If-Clauses and Probability Operators’ . Topoi , 30 , pp. 17 – 29 . Google Scholar Crossref Search ADS WorldCat von Fintel Kai , Gillies Anthony S. 2010 : ‘Must … Stay … Strong!’ Natural Language Semantics , 18 , pp. 351 – 83 . Google Scholar Crossref Search ADS WorldCat von Fintel Kai , Gillies Anthony S. 2011 : ‘Might Made Right’. In Egan Andy , Weatherson Brian (eds.), Epistemic Modality , pp. 108 – 30 . Oxford : Oxford University Press . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC von Fintel Kai , Gillies Anthony S. 2018 : ‘Still Going Strong’. Available online at https://semanticsarchive.net/Archive/GMzZTM0Y/. COPAC Gaifman Haim 1988 : ‘A Theory of Higher-Order Probabilities’. In Skyrms Brian , Harper William L. (eds.), Causation, Chance and Credence: Proceedings of the Irvine Conference on Probability and Causation, Volume 1 , pp. 191 – 219 . Dordrecht : Kluwer . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Gärdenfors Peter , Sahlin Nils-Eric 1982 : ‘ Unreliable Probabilities, Risk Taking, and Decision Making’ . Synthese , 53 , pp. 361 – 86 . Google Scholar Crossref Search ADS WorldCat Gillies Anthony S. 2010 : ‘Iffiness’ . Semantics and Pragmatics , 3 , pp. 1 – 42 . Google Scholar Crossref Search ADS PubMed WorldCat Hájek Alan 1989 : ‘Probabilities of Conditionals — Revisited’ . Journal of Philosophical Logic , 18 , pp. 423 – 8 . Google Scholar Crossref Search ADS WorldCat Hájek Alan 2009 : ‘Dutch Book Arguments’. In Anand Paul , Pattanaik Prasanta K. , Puppe Clemens (eds.), The Handbook of Rational and Social Choice: An Overview of New Foundations and Applications , pp. 173 – 95 . Oxford : Oxford University Press . Google Preview WorldCat COPAC Hild Matthias 1998 : ‘Auto-Epistemology and Updating’ . Philosophical Studies , 92 , pp. 321 – 61 . Google Scholar Crossref Search ADS WorldCat Holliday Wesley H. , Icard Thomas F. III 2010 : ‘Moorean Phenomena in Epistemic Logic’. In Beklemishev Lev , Goranko Valentin , Shehtman Valentin (eds.), Advances in Modal Logic , vol. 8, pp. 178 – 99 . London : College Books . Google Preview WorldCat COPAC Joyce James M. 1998 : ‘ A Nonpragmatic Vindication of Probabilism’ . Philosophy of Science , 65 , pp. 575 – 603 . Google Scholar Crossref Search ADS WorldCat Joyce James M. 2005 : ‘How Probabilities Reflect Evidence’ . Philosophical Perspectives , 19 , pp. 153 – 78 . Google Scholar Crossref Search ADS WorldCat Joyce James M. 2010 : ‘A Defense of Imprecise Credences in Inference and Decision Making’ . Philosophical Perspectives , 24 ( 1 ), pp. 281 – 323 . Google Scholar Crossref Search ADS WorldCat Kaplan David 1989 : ‘Demonstratives’. In Almog Joseph , Perry John , Wettstein Howard (eds.), Themes from Kaplan , pp. 481 – 563 . Oxford : Oxford University Press . Google Preview WorldCat COPAC Kratzer Angelika 1981 : ‘The Notional Category of Modality’. In Eikmeyer Hans-Jürgen , Rieser Hannes (eds.), Words, Worlds, and Contexts , pp. 38 – 74 . Berlin : Walter de Gruyter . Google Preview WorldCat COPAC Kratzer Angelika 1986 : ‘Conditionals’ . Chicago Linguistics Society , 22 , pp. 1 – 15 . WorldCat Lasersohn Peter 2005 : ‘ Context Dependence, Disagreement, and Predicates of Personal Taste’ . Linguistics and Philosophy , 28 , pp. 643 – 86 . Google Scholar Crossref Search ADS WorldCat Lewis David 1976 : ‘ Probabilities of Conditionals and Conditional Probabilities’ . Philosophical Review , 85 , pp. 297 – 315 . Google Scholar Crossref Search ADS WorldCat Mandelkern Matthew 2019 : ‘ Bounded Modality’ . Philosophical Review , 128 ( 1 ), pp. 1 – 61 . Google Scholar Crossref Search ADS WorldCat Moss Sarah 2015 : ‘On the Semantics and Pragmatics of Epistemic Modals’ . Semantics and Pragmatics , 8 , pp. 1 – 81 . Google Scholar Crossref Search ADS WorldCat Moss Sarah 2018 : Probabilistic Knowledge . Oxford : Oxford University Press . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Moss Sarah 2019 : ‘Global Constraints on Imprecise Credences’. Manuscript available online at http://www-personal.umich.edu/ ssmoss/Global Constraints.pdf. Partee Barbara H. , Rooth Mats 1983 : ‘Generalized Conjunction and Type Ambiguity’. In Bäuerle Rainer , Schwarze Christoph , von Stechow Arnim (eds.), Meaning, Use, and Interpretation of Language , pp. 361 – 83 . Berlin : Walter de Gruyter . Google Preview WorldCat COPAC Pierce Benjamin C. 2002 : Types and Programming Languages . Cambridge, MA : MIT Press . Google Preview WorldCat COPAC Roberts Craige 1996 : ‘Information Structure in Discourse: Towards an Integrated Formal Theory of Pragmatics’. In Toon Jae-Hak , Kathol Andreas (eds.), Ohio State University Working Papers in Linguistics, 49: Papers in Semantics . Columbus : Ohio State University Department of Linguistics . Available online at http://semanticsarchive.net/Archive/WYzOTRkO/. Google Preview WorldCat COPAC Rothschild Daniel 2012 : ‘ Expressing Credences’ . Proceedings of the Aristotelian Society , 112 ( 1 ), pp. 99 – 114 . Google Scholar Crossref Search ADS WorldCat Rothschild Daniel 2015 : ‘ Conditionals and Propositions in Semantics’ . Journal of Philosophical Logic , 44 , pp. 781 – 91 . Google Scholar Crossref Search ADS WorldCat Schroeder Mark 2011 : ‘ Two Roles for Propositions: Cause for Divorce?’ Noûs , 47 , pp. 409 – 30 . Google Scholar Crossref Search ADS WorldCat Sepielli Andrew 2012 : ‘ Normative Uncertainty for Non-Cognitivists’ . Philosophical Studies , 160 ( 2 ), pp. 191 – 207 . Google Scholar Crossref Search ADS WorldCat Staffel Julia 2019 : ‘Expressivism, Normative Uncertainty, and Arguments for Probabilism’. In Gendler Tamar Szabó , Hawthorne John (eds.), Oxford Studies in Epistemology , Vol. 6 , pp. 161 – 89 . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Stalnaker Robert C. 1970 : ‘ Probability and Conditionals’ . Philosophy of Science , 37 ( 1 ), pp. 64 – 80 . Google Scholar Crossref Search ADS WorldCat Stalnaker Robert C. 1984 : Inquiry . Cambridge : Cambridge University Press . Weatherson Brian 2004 : ‘Moore, Bradley, and Indicative Conditionals’. Unpublished manuscript. Willer Malte 2013 : ‘ Dynamics of Epistemic Modality’ . Philosophical Review , 122 , pp. 45 – 92 . Google Scholar Crossref Search ADS WorldCat Yalcin Seth 2007 : ‘ Epistemic Modals’ . Mind , 116 , pp. 983 – 1026 . Google Scholar Crossref Search ADS WorldCat Yalcin Seth 2009 : ‘ More on Epistemic Modals’ . Mind , 118 , pp. 785 – 93 . Google Scholar Crossref Search ADS WorldCat Yalcin Seth 2011 : ‘Nonfactualism about Epistemic Modality’. In Egan Andy , Weatherson Brian (eds.), Epistemic Modality , pp. 295 – 332 . Oxford : Oxford University Press . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Yalcin Seth 2012 : ‘Bayesian Expressivism’ . Proceedings of the Aristotelian Society , 112 ( 2 ), pp. 123 – 60 . Google Scholar Crossref Search ADS WorldCat Yalcin Seth 2015 : ‘Epistemic Modality De Re’ . Ergo , 2 , pp. 475 – 527 . WorldCat Yalcin Seth 2018 : ‘ Belief as Question-Sensitive’ . Philosophy and Phenomenological Research , 97 , pp. 23 – 47 . Google Scholar Crossref Search ADS WorldCat Appendices A Indicatives A.1 Scope-Taking and Type-Raising with Indicatives As intended, expresses the property a credence function has when it assigns the non-worldly representation encoded in a value greater than 0.5. Handling the narrow-scope representation is trickier. A first attempt: But this attempt fails, since is undefined in the present system, as and are of different semantic types. In common with Moss (2015, §2.4), and ultimately following Partee and Rooth (1983), we can address this by raising the type of :27 If , then . That is to say, raising the type of a worldly representation generates a set of worldly representations (equivalently, again, a characteristic function of worldly representations). In particular, it generates the set of worldly representations that involve representing as true. Therefore, if , then .28 Probabilifying a raised worldly representation (-ing) p yields the set of credence functions that assign probability 1 to some way of representing that p. Type-raising in hand, we have: As intended, expresses the property a credence function has when the ratio of the credence it assigns to the credence it assigns is 1. A.2 Domain restriction and triviality It bears noting that there is another, probably preferable, possibility for representing the scopal interactions of indicatives and modal operators, on which the latter are analysed as binary (that is, restrictable29) operators (Kratzer 1981, 1986): Kratzer denies that the indicative conditional contributes its own quantificational force; rather, indicative conditionals are syntactic devices for making explicit the restriction argument of a restrictable quantifier.30 There is no semantic distinction between the ‘wide scope’ and the ‘narrow scope’ : both are represented using the restricted modal . One motivation for adopting Kratzer’s analysis of indicative conditionals is explaining the sorts of judgements of equivalence that Stalnaker’s thesis (Stalnaker 1970) attempts to unify — for example, the judgement that (11) and (12) are equivalent. According to Stalnaker’s thesis, the probability that an indicative is true equals the conditional probability of C on A. Supposing that probability operators in natural language semantically express degrees of conditional probability, Stalnaker’s thesis predicts, correctly, that (11) and (12) are equivalent. (11) Rain is likely, conditional on atmospheric pressure being low. (12) It is likely that if atmospheric pressure is low, it will rain. In line with Stalnaker’s thesis, the Kratzerian story about probability operators under consideration here renders (11) and (12) equivalent — more precisely, is able to generate equivalent logical forms for these sentences. More generally, and regardless of whether Stalnaker’s thesis holds in its full generality, no version of the thesis — even massively restricted — can be accommodated without taking probability operators (and, by extension, modal operators) to be binary operators. That is because a language with only unary probability operators provably lacks the resources to express a sufficiently wide range of conditional probabilities.31 It is difficult to overstate the importance of this idea: it allows the theorist to accommodate (a perhaps appropriately restricted version of) Stalnaker’s thesis, without signing onto the thesis in the form in which it is usually presented: That is because the Kratzerian analysis does not represent as expressing a probability judgement whose object is a conditional proposition; rather, it expresses a restricted probability judgement: that C is likely (as assessed against the representation expressed by A). [I]n saying ‘there is one chance in two that if A then C’, the conditional ‘if A then C’ does not express any self-standing proposition … However, this remains compatible with the idea that if-clauses are devices of quantifier restriction. In the scope of an operator, if-clauses do make a systematic truth-conditional contribution to the whole sentence. (Égré and Cozic 2011, p. 22) This would seem to be just what is required to avoid challenges to Stalnaker’s thesis on grounds of triviality results in the mould of Lewis (1976) (see Rothschild 2015; Charlow 2016b). The dialectic in this neighbourhood of issues is, however, a great deal more vexed than this quick summary would suggest. Charlow (2016b) shows that triviality results in the mould of Lewis (1976) arise for restricted operators (and that such results do not depend on the (mis)understanding of logical form embodied in Stalnaker’s thesis). Indeed, as Charlow (2016b) argues, obstacles of triviality arise for any treatment of restricted quantification that takes , and to be elements of the same semantic algebra — that is, any treatment that takes and to be of the same base semantic type. This is precisely the assumption that the analysis in this paper discards. I take this to be another argument in favour of this paper’s analysis: unlike any competitor account of which I am aware, it allows the theorist to accommodate the intuitions of equivalence that underlie Stalnaker’s thesis, while also avoiding the spectre of triviality. B Epistemic contradiction B.1 (In)validating S5 If we introduce epistemic modal operators that do not raise the semantic type of their complements, we will observe that, for such operators, the axioms of S5 are validated. Either (if ) or (if ). Given a stipulation that , the logic of and its dual is the logic of S5: But no such consequences hold for the type-raising modal operators of our language:32 To illustrate, let . Then: is the property a set of worlds has when it contains a p-world; is the property a set of sets of worlds has when each element in this set is compatible with p. But if — as our account has it — expresses a property utterly distinct from — indeed, at a different level of the type hierarchy from — the property expressed by , why does a sentence like (13) of the form (and equivalent to ) sound so terrible? (13) #It may be raining, but maybe it can’t be. Perhaps because it is Moore-paradoxical (see Weatherson 2004)? Alas, it is not; note that Moore-paradoxicality dissolves in unasserted environments (see Yalcin 2007): (14) #Suppose it may be raining, but maybe it can’t be. (15) Suppose it is raining, but you don’t know it’s raining. Explanation: our account predicts that sentences of the form are semantically defective. (Moss’s semantics generates the same prediction, in basically the same fashion.) Notice that semantically coordinating and requires intersecting with: denotes the property a set of propositions has if and only if each q in that set is compatible with p. As noted above, denotes the property a set of propositions has if and only if some q in that set is incompatible with p. Obviously, no satisfies both properties: And so sentences of the form are predicted, on independent grounds, to be semantically anomalous (in spite of the fact that the left conjunct expresses a property utterly distinct from that expressed by the right conjunct).33 The data from natural language, therefore, do support a version of Euclideanness, namely, the version in (16). This represents an empirical edge over classical truth-conditional accounts of epistemic modality. Accounts of this type cannot, on the face of things, explain why sentences like (13) are semantically anomalous: in such frameworks, regarding sentences like (13) as inconsistent is equivalent to embracing a Euclideanness constraint on epistemic accessibility from the ‘actual’ world w (recall §2): Euclideanness: This makes vivid the dilemma confronting classical truth-conditional accounts of epistemic modals. Such accounts can at most do one of the following: Accommodate Euclideanness (while rendering graded modal judgement unintelligible). Accommodate graded modal judgement (while rendering (13) semantically impeccable). The account defended here skirts the dilemma: it accommodates both the clear semantic intuitions motivating Euclideanness, without sacrificing an intelligible model of graded modal judgement. B.2 Quantification Yalcin (2015) notes the following datum and observes that no standard theory of epistemic modality — including the theory of Yalcin (2007) — is able to account for it:34 (17) #Some/#Every person who is not infected might be infected. Here I will work through how this datum is accounted for, more or less automatically, on the present treatment, while also showing how to extend the theory of generalized quantification to the theory under consideration here. Assume a first-order version of the language defined in §4.2. Here is the natural clause for the two-place existential quantifier; the two-place universal quantifier is its dual.35 Roughly: expresses the constraint that satisfies if and only if some d of which represents ϕ to hold is such that represents ψ to hold of d. Picturesquely, it is the constraint of being such that there is some d such that d is represented as satisfying the quantifier’s restrictor and scope. expresses the constraint that satisfies if and only if every d of which represents ϕ to hold is such that represents ψ to hold of d. Picturesquely, it is the constraint of being such that any d such that d is represented as satisfying the quantifier’s restrictor is such that d is represented as satisfying the quantifier’s scope. With this understanding of generalized quantification in hand, we are in an immediate position to explain (17), using a strategy that is effectively the same as our strategy for (13). Notice that, in the case of a sentence of the form , semantic coordination of the quantifier's restrictor with its nuclear scope demands raising the quantifier’s restrictor: Consider any that satisfies . By assumption: In particular, for some , but . Clearly there is no such . Similarly, consider any that satisfies By assumption: In particular, for any , but . Clearly there is no such (on the standard assumption that restricted quantification in natural language requires non-vacuous satisfaction of the quantifier's restrictor clause). Thus, for any context c: © Charlow 2019 This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)