R E V IS E D P R O O F 12 3 Knowledge of objective modality 4 Margot Strohminger1 • Juhani Yli-Vakkuri2 5 6  Springer Science+Business Media B.V., part of Springer Nature 2018 7 Abstract The epistemology of modality has focused on metaphysical modality and, 8 more recently, counterfactual conditionals. Knowledge of kinds of modality that are 9 not metaphysical has so far gone largely unexplored. Yet other theoretically 10 interesting kinds of modality, such as nomic, practical, and 'easy' possibility, are no 11 less puzzling epistemologically. Could Clinton easily have won the 2016 presi12 dential election-was it an easy possibility? Given that she didn't in fact win the 13 election, how, if at all, can we know whether she easily could have? This paper 14 investigates the epistemology of the broad category of 'objective' modality, of 15 which metaphysical modality is a special, limiting case. It argues that the same 16 cognitive mechanisms that are capable of producing knowledge of metaphysical 17 modality are also capable of producing knowledge of all other objective modalities. 18 This conclusion can be used to explain the roles of counterfactual reasoning and the 19 imagination in the epistemology of objective modality. 20 21 Keywords Epistemology of modality  Modality  Metaphysical necessity  22 Conceivability and possibility  Counterfactuals  Imagination 23 A1 & Margot Strohminger A2 margotfs@gmail.com A3 Juhani Yli-Vakkuri A4 ylivakkuri@gmail.com A5 1 Humboldt University of Berlin, Berlin, Germany A6 2 Bielefeld University, Bielefeld, Germany 123 Philos Stud https://doi.org/10.1007/s11098-018-1052-4 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 24 The epistemology of modality has focused almost exclusively on knowledge of 25 metaphysical modality.1 However, other kinds of so-called objective modality (in 26 the sense of Williamson 2017a) such as nomic, practical, and 'easy' modality can 27 also appear epistemologically puzzling, and they are important topics in their own 28 right. Thus the neglect of the epistemology of other objective modalities may look 29 unmotivated or parochial. At worst, it may look similar to an approach to the 30 epistemology of mathematics that only deals with knowledge of some very weak 31 mathematical theory, such as Robinson arithmetic. 32 This paper makes a start on a more comprehensive approach to the epistemology 33 of modality, of which metaphysical modality is a special, limiting case. Knowledge 34 of other objective modalities and knowledge of metaphysical modality are puzzling 35 in many of the same ways. It will be argued that by and large the same cognitive 36 mechanisms that are capable of producing knowledge of metaphysical modality-in 37 particular, those we use for acquiring knowledge of counterfactuals-are also 38 capable of producing knowledge of all other objective modalities. This idea is 39 anticipated by Williamson in his classic discussion of the central role of 40 counterfactuals in the epistemology of metaphysical modality, where he says, in 41 passing, that 42 the connections [of metaphysical possibility] with restricted [objective] 43 possibility and with counterfactual conditionals are not mutually exclusive, for 44 they are not being interpreted as rival semantic analyses, but rather as different 45 cases in which the cognitive mechanisms needed for one already provide for 46 the other (2007: 178). 47 Given that all objective modalities are restrictions of metaphysical modality, it 48 should not be surprising that the relationship between the epistemologies of 49 objective modality and metaphysical modality turns out to be more like that 50 between the epistemologies of restricted quantification and unrestricted quantifica51 tion than that between the epistemologies of Robinson arithmetic and all arithmetic. 52 1 The epistemology of objective modality 53 Some modalities are restrictions of metaphysical modality-following Williamson, 54 we will call these the objective modalities.2 Metaphysical modality can be defined in 55 terms of restricted objective modality: a proposition is metaphysically necessary iff 56 it is necessary in every objective sense (Williamson 2017a: 3). (Objective 57 modalities can also be defined in terms of metaphysical modality, as we will see 58 in Sect. 2.) Thus whatever is metaphysically necessary is also necessary in any 59 objective sense, and whatever is possible in some objective sense is also 1 See Strohminger and Yli-Vakkuri (2017) for review. 2 Williamson is, of course, far from being the only philosopher to recognize a broad category of nonepistemic modalities that includes metaphysical modality: see, for example, Lange (2009), Hale (2013), Kment (2014) and Vetter (2015). Linguists have recognized a similar-perhaps the same-category of 'root', 'circumstantial', or 'dynamic' modality since the 1970s (Kratzer 1981, 2012; Portner 2009). M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 60 metaphysically possible. An objective modality that is restricted by a trivial 61 condition, such as a truth-functional tautology, is still an objective modality. 62 Metaphysical modality thus counts as a trivial restriction of itself, and so counts as 63 an objective modality. 64 Nomic (or nomological) modality is a paradigm case of a non-trivially restricted 65 objective modality. Many of the objective modalities we express using the modal 66 words ('possibly', 'necessarily', 'could', 'would', etc.) outside of theoretical 67 contexts are far more restricted than that paradigm. They include, inter alia, 68 'practical' and 'easy'3 varieties of modality. When we ask whether it is practically 69 possible that the Democrats won the 2016 U.S. presidential election, we are asking a 70 question equivalent to this: is it metaphysically possible given the practical 71 constraints that the Democrats won the 2016 U.S. presidential election? What 72 counts as a 'practical constraint' depends on the context. In this case they may 73 include, for example, the fact that Trump was the Republican nominee, and in every 74 case they will include the (actual) laws of nature. And to ask whether something is 75 easily possible-or, to use a more colloquial idiom, whether something could easily 76 have happened-is roughly equivalent to asking whether it is metaphysically 77 possible given that things are similar or close to how they actually are. What counts 78 as 'similar' or 'close' also depends on context, but in any actual context 'similarity' 79 to actuality requires sameness with respect to the laws of nature. 80 Not everything we express using the modal words is a restriction of metaphysical 81 modality. So-called epistemic modalities are a paradigm example. Both the 82 Generalized Continuum Hypothesis and its negation are epistemically possible, 83 since neither is known, but one of them is metaphysically impossible. Epistemic 84 'modalities' do not even seem to be modalities in that they are not properties of 85 propositions4: it seems that one and the same proposition can be both epistemically 86 possible when expressed by one sentence and epistemically impossible when 87 expressed by another. For example, it is highly plausible that the sentence 'It might 88 be that something is Greek and not Hellenic, but it might not be that something is 89 Greek and not Greek', where the 'might' is epistemic, is true in some contexts, even 90 though the proposition that something is Greek and not Hellenic is none other than 91 the proposition that something is Greek and not Greek. Logical 'modality' is even 92 more clearly not a modality (and so a fortiori not an objective modality): it is 3 See Sainsbury (1997), Peacocke (1999: 310–328) and Williamson (2000: 123–130). 4 Are all and only objective modalities properties of propositions? This seems to us more a matter to be decided than discovered. We think of the objective modalities as all and only those that can be characterized by a restricting condition in the sense of Sect. 2. This does not include all properties of propositions, since there are at least as many properties of propositions as there are functions from metaphysically possible words to propositions: such a function r characterizes the (or a) property P such that a proposition p has P at w iff p [ r(w). The alternative notion of an objective modality as a property of propositions is adequately captured by Scott-Montague 'neighborhood semantics', in which a modal operator is interpreted by an assignment of a set of sets of worlds, thought of as the set of relevantly necessary propositions, to each world. As we point out in Sect. 2, our own approach is equivalent to a relational (or 'Kripke') approach to the semantics of modal logic. Unlike neighborhood semantics, relational semantics cannot interpret a modal operator by an arbitrary property of propositions (see Bull and Segerberg 1984: §21). Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 93 logically necessary that p just in case the sentence 'p' is true under all 94 interpretations of its non-logical constants. Thus, for example, it is logically 95 possible that something is Greek and not Hellenic but not logically possible that 96 something is Greek and not Greek, because 'Greek' and 'Hellenic' are non-logical 97 constants. Deontic modality is an unclear case. Since deontic 'must' and 'may' 98 statements don't display any of the hallmarks of non-objectivity, it is tempting to 99 classify them as expressing restrictions of metaphysical modality, but there are also 100 reasons not to rush to judgment here: primarily, evidence suggestive of their non101 normality (in the logical sense) and hyperintensionality.5 (However that may be, it 102 will turn out, on our analysis, that there are objective modal operators that are 103 deontic in the sense of being restricted by the fulfilment of obligations.) 104 The epistemology of modality should study all knowledge of objective modality 105 and not only the limiting case of knowledge of metaphysical modality. The central 106 questions in the epistemology of metaphysical modality tend to generalize to all 107 non-trivial objective modalities. For example, the familiar question, 'How, if at all, 108 can we know whether it is metaphysically possible that p when it is not true that p?' 109 remains puzzling when we replace 'metaphysically' with 'nomically', 'practically', 110 'easily', 'technologically', etc. (cf. Williamson 2017a: 10). After all, knowledge of 111 such facts is just one valid inferential step away from knowledge of metaphysical 112 possibility: if one knows that it is in some objective sense possible that p, one can 113 come to know by deduction that it is metaphysically possible that p. 114 The importance of various restricted objective modalities to quotidian concerns, 115 engineering, policy planning, and planning and decision-making in general 116 contributes to an interest in their epistemology. In these contexts, we often use 117 modal words to express restricted objective modalities. In various theoretical 118 contexts, too, we are often interested in whether something is objectively possible in 119 a restricted sense. Epistemology itself is a salient example: in that field the 120 expressions 'reliable', 'knowable', 'risk', 'in a position to know', 'safe', and 121 'sensitive' are regularly used to express some kind of restricted objective modal 122 notion. For example, it is initially plausible that one is in a position to know only if 123 (and perhaps also if) one can know, in some objective sense of 'can'.6 Natural 124 science is concerned with nomic possibility, as well as with other objective 125 modalities whose importance to natural scientific inquiry has gone largely unnoticed 126 in the epistemology of modality until recently (Williamson 2017a, b). The 'can' or 127 'cannot' in standard formulations of Heisenberg's uncertainty principle expresses 128 some kind of objective modality. Even pure mathematics is rife with conjectures, 129 axioms, proofs, and theorems that manifest commitments to various objective modal 130 claims. Typically these commitments are implicit (Yli-Vakkuri and Hawthorne 131 MSb), but there are also some plausible examples of explicit objective modal 132 commitments in pure mathematics: for example, it seems plausible that Church's 5 See Fine (2014) for discussion. 6 See Yli-Vakkuri and Hawthorne (MSa) for discussion. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 133 thesis concerns computability, or what can be computed, in some objective sense of 134 'can'.7 135 2 Objective modalities as restrictions of metaphysical modality 136 We have already seen that metaphysical modality can be defined in terms of 137 objective modality. This section asks about the other direction: can objective 138 modality be defined in terms of metaphysical modality? (The answer is 'Yes.') Our 139 motivation here is epistemological. After offering a definition, we will discuss how 140 it enables us to extend certain familiar observations in the epistemology of 141 metaphysical modality to the less explored territory of the epistemology of objective 142 modality.8 143 Statements about what is possible or necessary in some (objective9) sense are 144 closely related to certain statements about what is metaphysically possible or 145 necessary. In particular, when evaluated in the same context, any statement of the 146 form (1) is necessarily equivalent to the corresponding statement of the form (10), 147 and any statement of the form (2) is necessarily equivalent to the corresponding 148 statement of the form (20), where 'R' expresses the property of being the conjunction 149 of all of the conditions that restrict 'possible' and 'necessary' in the context. 150 (1) It is possible that p. 151 (10) It is metaphysically compossible with the R-condition that p. 152 (2) It is necessary that p. 153 (20) It is a metaphysically necessary consequence of the R-condition that p. 154 For example, if (1) and (2) express nomic modality, 'R' in (10) and (20) will express 155 the property of being the conjunction of all laws of nature. The restricting conditions 156 corresponding to practical and easy varieties of modality are conjunctions of some 157 highly local conditions, and what 'R' expresses in these cases is highly sensitive to 158 the context of speech. In the limiting case, where (1) and (2) express metaphysical 159 modality, 'R' will express the property of being the conjunction of some necessary 160 truths (or of no conditions at all; see below). 161 It bears emphasis that our assumption about the necessary equivalence of (1) and 162 (10) and of (2) and (20) in any context does not commit us to the view that the 163 condition that restricts a restricted modal operator is something competent users of 164 the operator are able to articulate or express in any way other than by using that very 165 operator. (In this respect restricted modal operators resemble implicitly restricted 7 Thanks to Timothy Williamson for this example. See Yli-Vakkuri and Hawthorne (MSb: note 5) for discussion. 8 In doing so, we retread some ground covered by van Fraassen (1977), Humberstone (1981) and Hale and Leech (2017). None of these authors propose the analysis we give. Although there is a certain superficial similarity between our analysis and Hale and Leech's, there is also an important difference: see note 11. 9 Since we are only concerned with objective modalities in this paper, we will henceforth leave the 'objective' implicit. Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 166 quantifiers.) In some cases it is fairly easy to specify the restricting condition very 167 precisely using other words: the phrase 'the conjunction of all laws of nature' does 168 the job in the case of nomic modality. But in other cases it is not easy: 'the 169 conjunction of the practical conditions' is extremely vague and uninformative-it is 170 at best a stand-in for a fuller specification of the restricting condition, which we are 171 rarely able to supply. And in some cases it is difficult to come up with any words 172 that even gesture in the right direction (what condition restricts an 'easy' possibility 173 operator?). In cases of the latter kind, it is more natural to think of the restriction 174 associated with the operator as an accessibility relation: a binary relation R on 175 worlds, such that 'It is necessary that p' is true at a world w just in case 'p' is true at 176 every world v such that wRv. In the case of easy possibility, the accessibility 177 relation is some variety of closeness or similarity. But there is no need to have both 178 restricting conditions and accessibility relations, as long as we are working with 179 standard possible-worlds semantics. Conditions can be represented by accessibility 180 relations, and conversely: given an accessibility relation R, we can define the 181 condition that restricts the modal operators at a world w as {v| wRv}, and given an 182 assignment of restricting conditions to worlds, we can define the restricting 183 accessibility relation as the relation R such that wRv iff v [ r, where r is the 184 condition that restricts the modal operators at w, i.e., the proposition that is R in w.10 185 The necessary equivalence of statements of forms (1) and (2) in all contexts 186 suggests the following analysis of restricted necessity in terms of metaphysical 187 necessity, which we will assume in what follows.11 10 Here we are assuming a coarse-grained conception of propositions (conditions) as sets of worlds, but the inter-translatability of accessibility-relation talk with restricting-condition talk does not require that assumption. As long as, for each set of worlds W, there is a proposition f(W) that is true at exactly the worlds in W, f(W) can play the role of the restricting condition that holds at exactly the worlds in W. And it does seem plausible, even given a view on which propositions have arbitrarily fine-grained structure, that there is a function f that fits this description: f(W) might be, for example, the proposition that at least one of the worlds in W is actualized, where 'actualized' is understood in a non-rigid way, so that it is contingent which world is actualized. (Note that we are not assuming that, for each set of worlds W, there is a unique proposition that is true at exactly the worlds in W. That would be implausible on a structuredpropositions view. The axiom of choice guarantees the existence of a suitable function f even if there are sets of worlds that exactly verify more than one proposition. Nor are we assuming the consistency of views on which propositions are arbitrarily fine-grained-theories that posit extremely fine-grained propositional structure are inconsistent: see Dorr (2016a) and Goodman (2017). Given an inconsistent view, anything whatsoever is the case.) 11 Hale and Leech (2017: §6.3) propose an analysis of what they call 'alethic' modality that is superficially similar to but importantly different from (hR), which they illustrate with the case of nomic or (as they call it) 'physical' modality, as follows. H&L: It is physically necessary that p = def Aq(p(q) ^ h(q ? p)). Here 'p' is to be read as 'it is a law of physics that...' (Hale and Leech 2017: 13). This, however, is not an adequate analysis because it does not give the restricted modality being analyzed a normal modal logic: it validates neither necessitation (p ? hRp, where p is valid) nor the K axiom (hR(p ? q) ? (hRp ? hRq)). Hale and Leech try to solve this problem in a footnote (note 23). We don't think their solution works, but we'll save our criticisms of it for another occasion (Yli-Vakkuri plans to defend his own higher-order analysis elsewhere). M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F ðhRÞ hRp$ 9qðR qð Þ ^hðq! pÞÞ ð'It is R-necessary that p just in case there is a proposition that is R and p is a metaphysically necessary consequence of it':Þ 188189 While English does not have special modal words that express metaphysical 190 modality in every context, it is convenient to have ones that do, and for that reason 191 we will use is h and e as context-insensitive operators that express, respectively, 192 metaphysical necessity and possibility, and hR and eR as schematic necessity and 193 possibility operators restricted by the R-condition. So interpreted, it is plausible that, 194 and we will assume that, (hR) is logically valid. 195 In (hR), hR may be interpreted as expressing any restricted necessity. R 196 expresses a property of propositions: the property of being the condition that 197 restricts the necessity operator. The reader should think of R(q) as having the form 198 'q is the conjunction of propositions p such that ...'. For example, if hR expresses 199 nomic necessity, then R expresses the property of being the conjunction of the laws 200 of nature, and if hR expresses metaphysical modality, then R expresses some trivial 201 restriction, such as the property of being the conjunction of the empty set. (We 202 assume that every set of propositions has a conjunction. Consequently, the 203 conjunction of the empty set is a necessary truth: necessarily, all of its conjuncts are 204 true.) We also assume that, whatever property of propositions R may express, it is 205 necessary that, and it is a logical truth that it is necessary that, there is a unique 206 proposition that has it. That is, we assume that h9!qRðqÞ 208 and therefore hAqR(q), is valid. (Think of R(q) as having the standard form 209 q = ^{p|u(p)}, where ^ is an infinitary conjunction operator.12) 210 What about the restricted possibility operator eR? We will define it as the dual of 211 hR. Because h and e are duals, validity of ðeRdualÞ eRp$ :9qðRðqÞ ^hðq! :pÞÞ 213 follows immediately. Because (eR-dual) and hA!qR(q) are both valid, so is ðeRÞ eRp$ 9qðRðqÞ ^eðq ^ pÞÞ: 215 We will treat (eR) rather than (eR-dual) as the canonical equivalence relevant to eR: 216 it captures the intuitive idea that to be restrictedly possible is to be compossible with 217 the restriction. 218 It is important to read, as we have done, 'the restriction' as having wide scope 219 when it occurs in our informal glosses of restricted necessity and possibility: 'to be 220 restrictedly necessary is to be a necessary consequence of the restriction' and 'to be 221 restrictedly possible is to be compossible with the restriction'. Consider the narrow222 scope alternative to (hR): 12 This doesn't exactly get the logical form right, for several reasons, the least subtle one being that set theory isn't logic. If we want hA!qR(q) to come out as a logical truth, we'll have to resort to higher-order modal logic. The full higher-order analysis is beyond the scope of this paper. Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F ðhRnarÞ hRp$ h9qðRðqÞ ^ ðq! pÞÞ: 224 (hR-nar) is incorrect, as can be seen, for example, by interpreting hR as expressing 225 practical necessity. Necessarily, the conjunction of the practical conditions is true 226 (hAq(R(q) ^ q)), so by (hR-nar), something is practically necessary if and only if it 227 is metaphysically necessary (hRp $ hp). (hR-nar) collapses all modalities for 228 which hAq(R(q) ^ q) holds-which is to say, all of the factive or T-modalities- 229 into metaphysical modality. 230 Nor would it do to simulate wide scope for 'the restriction' by restricting 231 metaphysical modality by a sentence that expresses the same restricting condition 232 relative to every metaphysically possible world. To do this, we would have to say 233 that being restrictedly necessary is simply being a necessary consequence of r, 234 where r is the restricting condition, i.e., ðhRrigÞ hRp$ hðr ! pÞ; where r expresses the unique condition that satisfies R. 236 (hR-rig) inappropriately rigidifies restricted modality. Given that the logic of 237 metaphysical modality is S5 (in which all iterated modalities collapse) the validity 238 of (hR-rig) entails the validity of both the 4 (hRp ? hRhRp) and 5 (eRp ? 239 hReRp) axioms for all restricted modalities, resulting in the collapse of all iterated 240 restricted modalities. Given S5 for h, (hR-rig) also entails the metaphysical non241 contingency of restricted modality (heRp _ h:eRp and hhRp _ h:hRp). Both 242 consequences are unacceptable. The restricted modal claims we make outside of 243 philosophical contexts are typically metaphysically contingent, and iterations of the 244 restricted modal operators we typically use are not vacuous. (hR-rig) is an accept245 able analysis of actual restricted necessity, at least up to a standard of necessary 246 equivalence: when we replace hRp in (hR-rig) with @hRp ('It is actually restrict247 edly necessary that p'), the two sides of (hR-rig) become necessarily equivalent, 248 because it is a non-contingent matter which proposition is actually the restricting 249 condition. 250 (hR) gives restricted modalities a very weak logic: in particular it gives them the 251 weakest normal modal logic, K.13 This is the result we want, since we assume (see 252 note 4) that all objective modalities can be characterized by a relational semantics, 253 and K is the strongest logic obeyed by all modalities that can be characterized by a 254 relational semantics. 255 3 Knowledge of restricted modality 256 Let us now consider some epistemological consequences of the validity of each of 257 (eR) and (hR). First, if one knows one side of the biconditional as well as the 258 biconditional itself, one can come to know the other side by deducing it from these 13 With whatever restrictions to necessitation are mandated by the presence of the actuality operator @ and other indexicals in the language. By (hR), necessitation for hR inherits these restrictions from necessitation for h. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 259 items of knowledge. Second, whether or not one knows the biconditional, one can 260 come to know one side by deducing it from the other. Third, one can come to know 261 one side without performing any deduction, simply by evaluating it by whatever 262 method one could use to acquire knowledge of the other side. 263 The first generalization stands in little need of argument: knowledge can be 264 extended by deduction.14 The second and third are no less plausible upon 265 examination. 266 As regards the second generalization, coming to know p by deducing it from 267 q does not require knowledge of any conditionals connecting q and p. 268 As regards the third generalization, it is often the case that, when a deduction can 269 extend knowledge of p acquired by a certain method to knowledge of q, that same 270 method can be employed to produce knowledge of q directly.15 Recall the analogy 271 with quantification. Even if one typically comes to know that there are (in an 272 unrestricted sense) black squirrels in Canadian province x by first coming to know 273 that there are (in a sense restricted to x) black squirrels and then performing a 274 deduction, there is no obstacle to one's skipping the provincially restricted 275 knowledge and the deduction and coming to know that there are (unrestrictedly) 276 black squirrels by whatever method one typically comes to have the provincially 277 restricted knowledge. The converse is equally plausible. And the case of restricted 278 modality is not relevantly different. 279 The deep structural analogy between restricted modality and restricted quantifi280 cation bears emphasis here. Although the analogy is not controversial, it does not 281 wear its epistemological significance on its sleeve. 282 Like modal operators, quantifiers are normally implicitly restricted by (non283 trivial) conditions supplied by the context of speech. There is no beer-in a 284 contextually restricted sense-not because there is no beer in the universe, but 285 because there is no beer in your home (or whatever the relevant restricting property 286 is). Restricted quantifiers are analyzable in terms of unrestricted quantifiers as 287 follows, where (8x: R(x)) is a universal quantifier restricted by R. ð8x : R xð ÞÞF xð Þ $ 8xðR xð Þ ! F xð ÞÞ 289 By the duality of the restricted quantifiers, we also have: ð9x : R xð ÞÞF xð Þ $ 9xðR xð Þ ^ F xð ÞÞ 291 Now suppose we are after a story about how we can know some restrictedly 292 quantified claim to be true. Given the above equivalences, such a story will fall out 293 of an account of how we can know the equivalent unrestrictedly quantified claim to 294 be true, if we have such an account. One can come to know whether there is, in the 295 'in my home'-restricted sense, no beer by using any cognitive mechanisms by which 14 Of course it does not follow that every possible deduction extends knowledge. Knowledge-extending deductions must be 'competent' [see Williamson (2000: 117) and Hawthorne (2004: 34-35)]. 15 There may be exceptions. Consider a case in which one comes to know a highly non-trivial mathematical fact p by deducing it from some known axioms. We certainly don't want to claim that it is possible to come to know p simply by doing whatever one actually did to come to know the axioms and then judging that p on that basis. Thanks to Catharine Diehl for discussion here. Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 296 one can come to know whether there is, in the unrestricted sense, something in one's 297 home that is a beer. Of course, it is not inevitable that such a cognitive mechanism 298 will always be (at least easily) available. Even though we have cognitive capacities 299 that can deliver knowledge of some unrestrictedly quantified claims, they may in 300 some cases be fairly useless for deciding restrictedly quantified claims, among other 301 reasons because the restrictions in play are not transparent to us. For example, you 302 may know that, in a certain restricted sense, there is no beer while having very little 303 idea what property restricts your 'there is'. (The properties of being located in your 304 refrigerator, of being located in your home, and of being a thing you own may all be 305 equally plausible candidates.) In such a case you could not easily have come to 306 know that there is, in just that restricted sense, no beer by whatever method you 307 would evaluate the equivalent unrestricted 'there is' claim, because you are unable 308 to make the restriction explicit. 309 Just as the standard restricted quantifiers can be analyzed in terms of unrestricted 310 quantifiers, the restricting properties, and the truth-functional connectives, standard 311 restricted modal operators can be analyzed in terms of metaphysical modality, 312 restricting conditions (understood as having wide scope), and the truth-functional 313 connectives. If we are after a story about how we can know some restricted modal 314 claim to be true, then, given the equivalences provided by the analysis, such a story 315 will fall out of an account of how we can know the equivalent unrestricted modal 316 claim to be true, if we have such an account. Here, too, there is no guarantee that the 317 cognitive mechanisms that deliver knowledge of unrestricted modality will always 318 be (at least easily) available for deciding restricted modal claims. Here, too, the non319 transparency of the restrictions may sometimes get in the way, as we will see. 320 4 Extending knowledge of restricted modality by counterfactual 321 reasoning 322 In this section we will set aside the question 'How we can come to have any 323 knowledge of restricted modality at all?' Instead we will ask: 'How, given that we 324 do have some knowledge of restricted modality, can we extract further knowledge 325 of restricted modality from it?' 326 There are, of course, many such ways, but here we will explore ones that exploit 327 or are underwritten by logical relationships between restricted modal claims and 328 counterfactuals. The following principle concerning counterfactuals and unre329 stricted modality is widely thought to be logically valid,16 and we will assume for 330 now that it is (in Sect. 5 this assumption will face some complications). Possibility ðph! qÞ $ ðep! eqÞ: 332 POSSIBILITY is truth-functionally equivalent to this: if it is possible that p, and p 333 counterfactually implies q, then it is possible that q. In other words, POSSIBILITY says 334 that counterfactual modus ponens preserves possibility. Given that POSSIBILITY is 16 See Williamson (2007: 156), Lange (2009: 64) and Berto et al. (2017) for recent examples. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 335 valid, we can use our capacity for evaluating counterfactuals for extending our 336 knowledge of both possibility and necessity. POSSIBILITY underwrites a variety of 337 ways to extend knowledge of ep and ph? q to knowledge of eq, as well as (by 338 duality) to extend knowledge of hq and :ph? :q to knowledge of hp. As in the 339 case of the immediate epistemological applications of (hR) and (eR), for reasons 340 that are by now familiar, these ways can but need not involve knowing POSSIBILITY, 341 and they can but need not involve performing any deductions. (We will return to this 342 last theme in Sect. 5) 343 Similarly, we can use our capacity for evaluating counterfactuals for extending 344 our knowledge of any restricted modality for which Possibility  ðph! qÞ ! ðeRp! eRqÞ 346 is valid.17 But here we face a problem: it is not entirely clear for which restricted 347 modalities POSSIBILITY* is valid, and POSSIBILITY* is clearly not valid for some 348 restricted modalities. 349 The validity of POSSIBILITY* has been called into question even for one of our 350 paradigmatic restricted modalities: nomic modality. On David Lewis's (1973: 75, 351 1979) view, which we think cannot be lightly dismissed, just about any departure 352 from actuality would involve a violation of the laws of nature. The rough idea is 353 that, for example, if you had had one more cup of coffee this morning than you 354 actually did, then the history of the world up to your drinking that additional cup of 355 coffee would have been as it actually is, whereafter it would have diverged-a 356 'local miracle' would have occurred. (This claim could be justified in terms of a 357 Lewisian similarity-theoretic semantics for counterfactuals, but it need not be: 358 entirely independently of any semantic theory, certain natural anti-'backtracking' 359 judgments put a lot of pressure on one to draw Lewis's conclusion.18) But then, if 360 the laws of nature are deterministic, something actually nomically impossible would 361 have happened if you had had one more cup of coffee, etc.-a counterexample to 362 POSSIBILITY*, since it is nomically possible that you drink one more cup of coffee, 363 etc. Even independently of determinism, it is plausible that the kinds of awkward 364 transitions in world histories that the truth of various ordinary counterfactuals 365 requires on Lewis's picture will sometimes be actually nomically impossible. And if 366 there are such counterexamples to POSSIBILITY* for nomic modality, they will 367 invalidate POSSIBILITY* for any more restricted modalities-so for pretty much any 368 restricted modality we ordinarily express using the modal words. Consider the case 369 of practical possibility. It is practically possible for Clinton to have won. But on the 370 Lewisian picture, if Clinton had won, then she would have violated the actual laws 371 of nature, and it is not practically possible for Clinton to violate the actual laws of 372 nature. 373 Lewis's 'local miracle' view is controversial. But whether it is right or wrong, it 374 should be uncontroversial that POSSIBILITY* is not valid for every restricted modality. 17 Lange (2009: 64) endorses POSSIBILITY* for what he calls 'genuine' modalities. Genuine modalities are objective in the sense of this paper, but not vice versa (207-208, n. 5). According to Williamson (forthcoming), POSSIBILITY* is 'plausible for a wide range of restricted kinds of objective possibility'. 18 See Dorr (2016b) for discussion. Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 375 For note that the condition that restricts a modal operator need not be true, in the 376 sense that the proposition that satisfies R in (hR) may be false. So let eR be 377 restricted by a possibly true but actually false condition r, and let p be any truth378 functional tautology. Then ph? :r and eRp are both true, but eR:r is false, so 379 (ph? r) ? (eRp ? eR :r) is a false instance of POSSIBILITY*. 380 Of course, this non-constructive argument for the existence of restricted 381 modalities for which POSSIBILITY* is not valid does not immediately undermine its 382 epistemological applications (although it should make one worried). For all we have 383 said so far, we never have occasion to think or talk about the restricted modalities 384 for which POSSIBILITY* is not valid. But, in fact, we do think and talk about them 385 often. Legal possibility, where the restricting condition is (roughly) that the relevant 386 laws are obeyed,19 is a prominent example.20 Here is a direct counterexample to the 387 validity of POSSIBILITY* for that restricted modality: It was legally possible for Nixon 388 to win the 1968 Presidential election, and if Nixon had won the 1968 Presidential 389 election, then Nixon would have ordered his subordinates to commit burglary 390 (because he both won and ordered his subordinates to commit burglary), but it was 391 not legally possible for Nixon to order his subordinates to commit burglary. 392 Where does this observation leave our hope to be able to use counterfactual 393 thinking for extending our knowledge of restricted modalities? Happily, thanks to 394 (hR) and (eR), POSSIBILITY makes available, at least in principle, ways of extending 395 our knowledge of all restricted modalities by counterfactual reasoning. This is 396 because POSSIBILITY underwrites ways of projecting knowledge of what is actually 397 restrictedly necessary (or possible) from knowledge of what is restrictedly necessary 398 (or possible), and that something is actually restrictedly necessary (or possible) 399 logically entails that it is restrictedly necessary (or possible). 400 Here is an example of one such way. Suppose that you know that (1) it is 401 nomically possible that you drop a certain piano, P, from a fifth-floor balcony. And 402 suppose that you further know that, (2) if you dropped P from a fifth-floor balcony 403 and the actual laws of nature obtained, then P would shatter and the actual laws of 404 nature would obtain. Then you can deduce from (1) that it is (unrestrictedly) 405 possible that the actual laws of nature obtain and you drop P from a fifth-floor 406 balcony. And you can further deduce from this and (2), by POSSIBILITY, that it is 407 possible that P shatters and the actual laws of nature obtain. Finally, you can deduce 408 from this that the laws of nature are such that it is possible that they obtain and P 409 shatters-which is, by (eR), equivalent to the claim that it is nomically possible that 410 P shatters. If you competently deduce this, you will know it, and so you will have 411 used your capacity for evaluating counterfactuals for extending knowledge of nomic 19 Here we cannot, of course, simply think of the laws as what the law books explicitly dictate: what the law books explicitly dictate may be inconsistent, and therefore impossible to obey. The laws, rather, must be thought of as (in the typical case) a possible proposition determined by the explicit contents of legal texts and various features of the surrounding context, such as court decisions and perhaps the intentions of legislators. 20 In common uses of 'legally possible' (one can find many examples by searching Google News for 'legally possible' together with 'Trump') the restricting condition is not that some relevant laws are obeyed. A typical restriction seems to concern particular people obeying laws with respect to particular actions and also to require that certain practical conditions obtain. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 412 modality in a way underwriten by POSSIBILITY-and whether POSSIBILITY* is valid for 413 nomic modality is neither here nor there. 414 When fully spelled out using our preferred formalization, the deduction just 415 sketched has the following form. 1: eRp Assumption 2: 9rð@R rð Þ ^ r ^ pÞ h! 9rð@R rð Þ ^ r ^ qÞ Assumption 3: 9rðR rð Þ ^eðr ^ pÞÞ 1; ðeRÞ 4: 9rð@R rð Þ ^eðr ^ pÞÞ 3 5: e9rð@R rð Þ ^ r ^ pÞ 4 6: e9rð@R rð Þ ^ r ^ qÞ 2, 5, Possibility 7: 9rð@R rð Þ ^eðr ^ qÞÞ 6 8: 9rðR rð Þ ^eðr ^ qÞÞ 7 9: eRq 8; ðeRÞ 417 The transitions from 3 to 4 and from 7 to 8 are justified by the logic of actuality (u is 418 equivalent to @u), and the transitions from 4 to 5 and from 5 to 6 are justified by the 419 logic of necessity and actuality (u is equivalent to h@u).21 Similarly, one can use 420 (hR), the duality of e and h, and the equivalence of Aq(R(q) ^ h(q ? p)) with 421 Aq(@R(q) ^h(q ? p)) and of Aq(@R(q) ^h(q ? p)) with hAq(@R(q) ^ (q ? p)) 422 to extend one's knowledge of restricted necessity by an argument underwritten by 423 POSSIBILITY. 424 By our observation (in Sect. 2) that @eRp is necessarily equivalent to e(r ^ p), 425 where r is the condition that (actually) is R (for example, r is the conjunction of the 426 actual laws of nature), one might hope to cut some corners by using the known 427 counterfactual 2ð Þ ðr ^ pÞ h! ðr ^ qÞ 429 to extend one's knowledge of restricted possibility by POSSIBILITY (and similarly, 430 mutatis mutandis, for restricted necessity). In some cases this may be possible, but 431 in general it seems to require some rather impressive cognitive achievements. For 432 note first that, while @eRp and e(r ^ p) are necessarily equivalent given that r 433 expresses the condition that satisfies R, they are not logically equivalent; their 434 material equivalence follows by (eR) from !ð Þ 9!qR qð Þ 1⁄4 r ('r is the unique proposition with property R'Þ; 436 which in many cases seems quite difficult to know. In the case of nomic modality, 437 knowing (!) requires knowing of a particular proposition, r, that it is the conjunction 21 See the discussion of 'real-world validity' in Davies and Humberstone (1980) and Kaplan (1989: XVIII and 539: n. 65) on 'actually'. The 4-to-5 and 5-to-6 inferences also require the validity of the Barcan formulas for propositional quantifiers, which, in contrast with the first-order Barcan formulas, have tended not to be controversial. (In the recent debate on 'necessitism' sparked by Stalnaker (2012) and Williamson (2013), the validity of the propositionally quantified Barcan formulas has also been called into question (see Fritz 2016), but in the present dialectical context we take their validity to be sufficiently uncontroversial to assume without further commentary). Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 438 of the laws of nature, which would appear to be difficult. Second, suppose that, 439 contrary to appearances, it is not difficult to know that r is the conjunction of the 440 laws of nature. (Perhaps there are easy, stipulative ways: 'Let 'r' express the con441 junction of the laws of nature! Now I know that r is the conjunction of the laws of 442 nature.') Even if that is so, knowing something of the form (2*) seems quite 443 demanding. If one thinks r under a fairly uninformative guise (e.g., 'Things are this 444 way', where one somehow manages to refer to the conjunction of the laws of nature 445 by 'this'), it is difficult to know what follows counterfactually from r ^ p. If, on the 446 other hand, one has a robust enough conception of the laws of nature to be able to 447 know the relevant counterfactual, then that itself is a significant cognitive 448 achievement. In contrast, it takes very little to know facts about how things would 449 be if the actual laws of nature-whatever they may be-obtained and various other 450 matters were otherwise. One has the latter kind of knowledge when one knows the 451 premise 2ð Þ 9rð@R rð Þ ^ r ^ pÞh! 9rð@R rð Þ ^ r ^ qÞ: 452453 Of course, knowing (2) is still in general a more impressive cognitive 454 achievement than knowing ph? q. While it is fairly easy to know (2) when R is 455 the property of being the conjunction of the laws of nature, it may be much less easy 456 in the case of various more ordinary restrictions-a theme to which we will return in 457 the next section. 458 The key observation here is that, when it comes to using counterfactual reasoning 459 for extending our knowledge of restricted possibility and necessity, the difference 460 between restricted modality and actual restricted modality makes little difference. 461 When we are after knowledge of what is restrictedly possible or necessary we can 462 always use, mutatis mutandis, whatever means we have of coming to know that 463 something is actually restrictedly possible or necessary to come to know that it is 464 restrictedly possible or necessary, as long as no iterated modalities are involved. We 465 only cannot in general use those means to come to know what would have been 466 restrictedly possible or necessary had things been otherwise, or to come to know 467 what is restrictedly necessarily necessary, or restrictedly necessarily possible, etc. 468 5 Knowledge by imagination 469 We often evaluate restricted modal claims by exercises of the imagination, and at 470 least sometimes we acquire knowledge of restricted modality in this way: 471 Could we have hauled the piano upstairs, instead of taking it through the 472 window? One might answer the question-indeed, one might come to know 473 the answer-by imagining the piano being manipulated around the winding 474 staircase. (Byrne 2007: 136) 475 How might one come to know the answer in that way? Here, too, we can arrive at a 476 plausible answer by reflecting on the epistemological significance of (hR). M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 477 We assume, as is standard, that one of the cognitive mechanisms we use for 478 obtaining knowledge of metaphysical modality is a certain type of imaginative 479 exercise.22 To fix ideas, suppose that Williamson (2007: ch. 5) is right about the 480 nature of these imaginative exercises. (A broadly similar story could be told using 481 any of the competing accounts, but we use Williamson's as an illustration.) His 482 account relies on the validity of (hh?). ðhh!Þ hp$ ð:ph! ?Þ 484 As Williamson (2007: 155–158) observes, (h) is derivable from POSSIBILITY and the 485 principle that strict implication is at least as strong as counterfactual implication 486 (h(p ? q) ? (ph? q)) in K. (eh?) follows from (hh?) by duality. ðeh!Þ ep$ :ðph! ?Þ 488 On Williamson's view, the canonical way of evaluating a counterfactual is to 489 suppose (counterfactually) that the antecedent holds, to develop that supposition 490 using one's imagination-in effect, to imagine what else would be true if the 491 antecedent were true-and to see whether such development 'robustly' yields the 492 consequent or its negation. If it robustly yields the consequent, one accepts the 493 counterfactual, and if it robustly fails to yield the consequent, one accepts its 494 negation (2007: 152–155). If things go well, one thereby comes to know either the 495 counterfactual or its negation. Thanks to the validity of (hh?), we can come to 496 know claims of metaphysical necessity and possibility by the same process of 497 'counterfactual development', either by evaluating the logically equivalent coun498 terfactual or negated counterfactual and performing a deduction or by any of the 499 other ways we have discussed of exploiting logical equivalences for acquiring 500 knowledge. 501 By (hR), this epistemological story is immediately applicable to restricted 502 modalities. To evaluate hRp, one can evaluate the logically equivalent Aq(R(q) ^ 503 ((q ^ :p) h? \)) by the canonical method for evaluating counterfactuals, and, if 504 things go well, thereby arrive at knowledge of either hRp or :hRp. (And similarly, 505 mutatis mutandis, for eRp.) It is not easy, however, for things to go well in many 506 typical cases. For familiar reasons, it is not straightforward, even in the case of 507 nomic necessity, to know, concerning the conjunction r of the laws of nature, what 508 would be the case if r and something else were the case. In that case, again, one can 509 take a shortcut through the logic of actuality, asking instead what would be the case 510 if the actual laws of nature, whatever they may be, obtained. But in the case of 511 restrictions like those involved in various practical modal claims, this shortcut may 512 not help. It does not seem easy to know what would be the case if the actual 513 practical conditions, whatever they may be, obtained. How would one counterfac514 tually suppose that the actual practical conditions, whatever they may be, obtain? 515 The problem is that the second-order restricting condition R is not, in many cases, 516 any more transparent to thinkers of restricted modal contents than the condition 517 r that satisfies R. 22 Contemporary defenses include Yablo (1993), Chalmers (2002), Gregory (2004), Williamson (2007: ch. 5) and Hill (2014). Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 518 Luckily, there is another way to use counterfactual suppositional reasoning to 519 come to know restricted modal facts. It is even one that we commonly use for doing 520 so: we often evaluate restricted modal claims by the canonical method for 521 evaluating a counterfactual or a negated counterfactual that is restricted by the same 522 condition. Like ordinary English modal operators, ordinary English counterfactuals 523 are typically used, not for generalizing over absolutely all possibilities, but over the 524 possibilities that satisfy a certain restriction. In Lewis's semantics, this restriction is 525 represented, in effect, by an accessibility relation: an assignment 'to each world i of 526 ... a set Si of worlds, regarded as the set of worlds accessible from i' (Lewis 1973: 527 48)-call this set the sphere of accessibility (around i) associated with h? in the 528 context. (Since counterfactuals embedded within other counterfactuals or within the 529 scopes of modal operators are not at issue here, we will simply speak of 'the sphere 530 of accessibility associated with h?' and ignore its world-relativity.) Roughly 531 speaking, a counterfactual ph? q is true in a context just in case q is true at all of 532 the closest worlds within the sphere of accessibility supplied by the context at which 533 p is true.23 It follows that the equivalence ðhRh!Þ hRp$ ð:ph! ?Þ 535 and, by duality, ðeRh!Þ eRp$ :ðph! ?Þ 537 will hold in any context in which the sphere of accessibility associated with h? is 538 the set of worlds in which the restriction associated with hR is true. In such a 539 context, to discover whether it is restrictedly necessary or possible that p, one can 540 simply counterfactually suppose p or its negation, and proceed to develop that 541 supposition in imagination to see whether a contradiction follows. When one cor542 rectly evaluates a counterfactual by the canonical method, one's development in 543 imagination of the supposition of the antecedent is constrained by the restriction 544 associated with the counterfactual: one does not imagine possibilities that fall 545 outside of the sphere of accessibility. This is how ordinary counterfactual reasoning 546 proceeds when it proceeds correctly, and it requires no special cognitive achieve547 ments, such as supposing that a certain restriction holds, knowing what that 548 restriction is, or even being able to describe it in any informative terms. 549 Let us now return to Byrne's piano example. Here, in more detail, is how one 550 might come to know that the piano cannot be hauled upstairs through the stairway. 551 One visually imagines the piano being moved through the stairway. In doing so one 552 never visually imagines the piano beginning its journey with dimensions different 553 from its actual dimensions, and one never visually imagines the piano changing its 554 shape, or the stairway having dimensions different from its actual dimensions, and 555 so on. Never visually imagining these things does not require one to know what the 23 Less roughly, ph? q is true at a world w iff either (1) p is not true at any world in Sw or (2) p is true at some world v in Sw such that the material conditional p ? q is true at every world that is at least as close to w as v (Lewis 1973: 49). Because Lewis is not concerned with indexicality, there is no explicit context parameter in his semantics, but the point of the assignment of spheres of similarity to worlds is to represent a restriction supplied by context. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 556 piano's or the staircase's dimensions are, or even to counterfactually suppose that 557 the piano and the staircase have their actual dimensions, whatever they may be. 558 One's attempts to develop the supposition by visually imagining it robustly fail-in 559 effect, they lead to contradiction. (One need not explicitly derive a contradiction in 560 order to detect a failure. It is often sufficient that one detects that the development is 561 headed in an absurd direction: e.g., by being led to visualize parts of rigid bodies 562 being superimposed.) On this basis one judges, and one comes to know, that the 563 piano cannot be moved through the staircase. 564 In a case like the above, one typically does not come to know, because one does 565 not come to believe, the restricted counterfactual ph?\. Rather, one simply forms 566 the belief that :eRp by the same method by which one would canonically evaluate 567 ph? \, and that, together with the matching restriction being associated with h? 568 and eR, is sufficient for knowledge that :eRp. (In fact, there are good reasons not 569 to explicitly consider ph? \, since this tends to shift the context: see below.) 570 Similarly, the kinds of extensions of restricted modal knowledge underwritten by 571 POSSIBILITY and POSSIBILITY* discussed in Sect. 4 need not involve ever coming to 572 know, or even making a judgment on, a counterfactual. For example, a natural way 573 to get to know that one can A is to imagine oneself trying to A-with one's 574 imagining restricted by the condition associated with 'can'-finding that one then 575 imagines oneself succeeding, and judging on that basis that one can A (cf. 576 Williamson 2016: 116). Here one extends one's knowledge that one can try to A to 577 knowledge that one can A in a way underwritten by POSSIBILITY*. But one does so 578 without ever coming to know the relevant counterfactual. In a context in which 579 POSSIBILITY* holds, one can extend one's knowledge that eRp to knowledge that 580 eRq directly by the canonical method for evaluating the counterfactual ph? q. 581 And POSSIBILITY* does hold in any context in which the sphere of accessibility 582 associated with h? includes only possibilities in which the restricting condition 583 associated with eR is true. It is plausible that the contexts in which we find it natural 584 to attempt to extend our knowledge of restricted modality in ways underwritten by 585 POSSIBILITY* are also ones in which the restrictions of restricted modal operators and 586 of counterfactuals are coordinated in this way. 587 We have just touched upon a feature of POSSIBILITY* that we ignored in Sect. 4: 588 the context-sensitivity of its antecedent. One should not simply ask whether 589 POSSIBILITY* is valid for a particular restricted modality. Even for restricted 590 modalities for which it is not valid, POSSIBILITY* may hold in some contexts. And it 591 follows from what has been said that POSSIBILITY* does hold in every context, 592 regardless of the restricted modality, in which the sphere of accessibility associated 593 with h? includes only possibilities in which the restricting condition associated 594 with eR is true. 595 Does this mean that we can escape the strictures of Sect. 4's context-insensitive 596 discussion and use POSSIBILITY* for extending knowledge of any restricted modality 597 whatsoever simply by ensuring that we are in a context in which h? is suitably 598 restricted? 599 In fact, it does not. For a variety of restricted modalities, there are no 600 suitable contexts. This is so for the simple reason that the sphere of accessibility 601 associated with h? must always include the world of the context, whereas the Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 602 restricting conditions of many modal operators are not true in the world of the 603 context. One cannot, for example, restrict h? to worlds in which no violations of 604 the penal code occur. If one could, one would thereby produce counterexamples to 605 some of the most basic principles of counterfactual logic, including modus ponens.24 606 The difference between restricted modalities that obey the T axiom (hp ? p) 607 and others seems to be significant here. While non-T restricted modalities cannot 608 satisfy POSSIBILITY* no matter how we try to shift the context, there is no obvious 609 reason why every T-obeying restricted modality could not satisfy POSSIBILITY* in 610 some context. If so, POSSIBILITY* may have broader applications to the epistemology 611 of modality than the discussion of Sect. 4 suggests. 612 Finally, it's worth noting that the context-sensitivity of the right side of 613 Williamson's equivalence ðhh!Þ hp$ ð:ph! ?Þ 615 introduces a certain complication to his approach to the epistemology of meta616 physical modality. It is this: (hh?) holds only in contexts in which the sphere of 617 accessibility associated with h? includes all possibilities (Strohminger and Yli618 Vakkuri 2017: 833). Strictly speaking, then, we should not think of (hh?) as 619 logically valid. What is valid on Williamson's approach, rather, is ðhh!Þ hp$ ð:ph!kpp1⁄4> ?Þ; 621 where h? R is a counterfactual conditional connective restricted to the set of 622 possibilities in which the unique proposition r such that R(r) is true.25 Because the 623 tautology > is true in all possibilities, the counterfactual in (hh?*) generalizes 624 over all possibilities. When we are not idealizing away the context-sensitivity of 625 counterfactuals, the correct derivation of the Williamsonian equivalence (hh?*) 626 proceeds from NecessityR hðp! qÞ ! ðph!kpp1⁄4> qÞ 628 and PossibilityR ðph!kpp1⁄4> qÞ ! ðep! eqÞ 630 in K. While Williamson's derivation of (hh?) from NECESSITY and POSSIBILITY is 631 valid, POSSIBILITY holds only in some contexts-namely, those in which the coun632 terfactual conditional is unrestricted or, equivalently, restricted by a trivial condition 633 as in NECESSITYR and POSSIBILITYR. For if any possibility w falls outside its restriction, 634 there will be at least one proposition p-namely {w}-such that (ph? \) ? (ep 635 ? e\) is false. POSSIBILITY, and therefore (hh?), is only valid if we treat h? as a 636 logical constant that expresses what in ordinary English would be expressed by a 637 counterfactual with a trivial restriction. 24 Let r be the false but possible proposition that no violations of the (actual) penal code occur, and let p be any truth-functional tautology. Suppose that h? is restricted to worlds in which r is true. Then p and ph? r are true but r is false. 25 If that proposition is true; otherwise it is restricted to the empty set. M. Strohminger, J. Yli-Vakkuri 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 638 However, there is also another approach to the original Williamsonian analysis 639 (hh?), which is to treat the h? in it as an ordinary context-sensitive 640 counterfactual and to endorse it in all and only contexts in which its restriction 641 excludes no possibilities. This is easier than it might at first appear. As we have 642 previously argued (Strohminger and Yli-Vakkuri 2017: 833–834), having an 643 explicit contradiction as the consequent of a counterfactual tends to force a trivially 644 restricted reading of it. Again, the analogy with restricted quantifiers is illuminating. 645 It is not easy to get into a context in which 'Everyone is prepared for the exam' has a 646 trivially restricted reading-normally it expresses something like: Everyone 647 enrolled in the class is prepared for the exam. But adding an explicit trivial 648 restriction tends to get one into a context in which the resulting sentence does have a 649 trivially restricted reading: try interpreting 'Everyone in the universe is prepared for 650 the exam' in such a way that the 'everyone' is restricted to those enrolled in the 651 class. It isn't easy, even though in principle it should be possible: after all, computed 652 in the way semantics textbooks instruct us, the resulting restriction is to: xjx is in the universef g \ xjx is in the classf g 1⁄4 xjx is in the classf g: 654 The mechanism of semantic processing, however, does not deliver an intersective 655 reading when the explicit restriction is trivial (or, in general, less restrictive than the 656 attempted implicit restriction). A similar mechanism appears to be at work in the 657 processing of counterfactuals, where an explicitly trivially false consequent tends to 658 force the counterfactual to be evaluated with a trivial restriction. If so, getting into a 659 suitable context will not require much more than considering Williamson's (hh?). 660 The foregoing observation also introduces a complication to the epistemological 661 applications of both POSSIBILITY and POSSIBILITY*: unless we treat h? as a logical 662 constant, as described above, POSSIBILITY is not valid because there are contexts in 663 which at least one world is excluded by the restriction of h?, and POSSIBILITY* is 664 not valid for any restricted modality eR whose restriction fails to exclude at least 665 one world excluded by the restriction of h?. And even if we do treat h? as a 666 context-insensitive logical constant, neither POSSIBILITY nor POSSIBILITY* have any 667 immediate epistemological significance, because the counterfactual reasoning we 668 carry out in natural language and in thought is done using (possibly trivially) 669 restricted counterfactuals and not the envisaged context-insensitive logical constant 670 h?. 671 We suggest that this problem is not as serious as it may seem. We have already 672 noted that it is plausible that the contexts in which we find it natural to attempt to 673 extend our knowledge of restricted modality in ways underwritten by POSSIBILITY* 674 are ones in which the restrictions of restricted modal operators and of counterfac675 tuals are coordinated so that every possibility excluded by the restriction of the 676 counterfactual conditional is also excluded by the restriction of the modal operators. 677 It is also plausible that we find the same coordination in contexts on which we find it 678 natural to attempt to extend our knowledge of metaphysical modality in ways 679 underwritten by POSSIBILITY. Contexts of the latter kind are almost exclusively 680 philosophical ones in which metaphysical modality is at issue, in which our 681 counterfactuals are trivially restricted. They are contexts in which we naturally say 682 things like 'That would lead to a contradiction', and so the mechanism of semantic Knowledge of objective modality 123 Journal : Small-ext 11098 Dispatch : 27-2-2018 Pages : 21 Article No. : 1052 * LE * TYPESET MS Code : PHIL-D-17-00848 R CP R DISK R E V IS E D P R O O F 683 processing described in the previous paragraph is plausibly at work and will deliver 684 a trivial restriction for h?. 685 Acknowledgements This paper started out as Margot Strohminger's project. Juhani Yli-Vakkuri was 686 recruited as a coauthor in the final stages of preparation for publication. Credit (and blame!) should be 687 assigned accordingly. We would like thank Timothy Williamson for detailed comments on early drafts of 688 this paper, as well as Johannes Brandl, Jessica Brown, Catharine Diehl, Peter Fritz, Christopher Gauker, 689 Sören Häggqvist, John Hawthorne, Hannes Leitgeb, Julien Murzi, Christian Nimtz, Barbara Vetter, and 690 audiences at the Munich Center for Mathematical Philosophy (MCMP) at the Ludwig Maximilian 691 University of Munich, the Free University of Berlin, the University of Antwerp, the University of 692 Edinburgh, the University of Salzburg, and Bielefeld University for helpful comments and discussions. 693 This research was supported by the Alexander von Humboldt Foundation. 694 695 696 697 698 References 699 Berto, F., French, R., Priest, G., & Ripley, D. (2017). Williamson on counterpossibles. Journal of 700 Philosophical Logic. https://doi.org/10.1007/s10992-017-9446-x. 701 Bull, R., & Segerberg, K. (1984). Basic modal logic. In D. Gabbay & F. 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