How to Count 2 1 2 Oranges Eric Snyder and Jefferson Barlew Draft of 7 March 2016 Please do not cite without permission. 1 The Counting Oranges Puzzle Here s a puzzle due to Nathan Salmon (1997). Suppose there are three oranges on the table. I take one of them, cut it in half, eat one of the halves, and set the remaining half back on the table. Now consider the Question: (Question) How many oranges are there on the table? (Answer) There are 2 1 2 oranges on the table. In the scenario described, the intuitively correct answer is the Answer: 2 1 2 oranges. Now, a half-orange is either an orange or isn t. If it is, then there are three oranges on table, and if it isn t then there are only two. In either case, we get an answer to the Question which is not the intuitively correct Answer. Call this the Counting Oranges Puzzle (COP). COP is first and foremost a puzzle about the meanings of cardinality expressions, i.e. expressions whose express purpose is to count. Its theoretical significance is that it purports to undermine traditional analyses of cardinality expressions. For example, according to Frege s (1884) familiar analysis, one, two, etc. are second-order concepts expressible in terms of first-order quantifiers and identity. Thus, examples like (1a) are analyzed as (1b), where F translates is an orange on the table. (1) a. There are two oranges on the table. b. x. y. F (x) F (y) x y z. F (z) [z = x z = y] So (1a) will be true just in case there are at least two oranges on the table, and those are the only oranges on the table. Put differently, (1a) will be true if the class of oranges on the table is two-membered. Now, classes are extensional: they have objects as members and are identified by which members belong to them. Moreover, cardinal numbers are classes of equinumerous concepts on Frege s view, where two concepts F and G are equinumerous just in case every F can be mapped to a unique 1
G and vice versa. As a result, cardinal numbers are necessarily whole. There cannot be a concept F such that more than two but less than three objects fall under F. But this is precisely what the Answer seems to require; it appears to be counting the number of oranges on the table in terms of a fractional number. Consequently, stating coherent Fregean truth-conditions for the Answer would appear impossible. We can put the point differently using Frege s cardinality-operator #x[φ]. This maps a concept to a cardinal number, namely the number of objects falling under that concept. For example, if there are two oranges on the table, then the cardinality-operator will map the concept expressed by orange on the table to the number 2. Suppose we analyze the Answer similarly. (2) a. The number of oranges on the table is 2 1 2. b. #x[orange-on-table(x)] = 2 1 2 Then the Answer would be true only if the number of objects falling under the concept orange-on-table is identical to the number 2 1 2. But since there is no such number, (2a), and thus the Answer, is predicted to be false. Why, then, is the Answer the intuitively correct answer to the Question? According to Salmon, the problem with the Fregean analysis is its extensionality. Since classes are extensional, cardinal numbers must be whole. Thus, the lesson of COP is that counting is inherently intensional. We don t count classes of objects, but rather pluralities or groups of objects, relative to some counting property. Depending on what that counting property is, we get different answers to how many -questions. Relative to the property of being a whole orange, for instance, the answer to the Question will be two oranges. Relative to the property of being an orange, however, the answer to the Question will be the Answer. That s because we assign different weights to members of the plurality in question; we assign weight 1 to the whole oranges, weight.5 to the half-orange, and adding these weights together, we get the Answer: 2.5 oranges. Some have suggested that a similar lesson applies to the traditional analyses of cardinality expressions within linguistic semantics, e.g. the analyses of Barwise and Cooper (1981), Partee (1986), or Landman (2004). On these accounts, count nouns denote individuated, and thus countable, objects. Cardinalities represent how many such objects constitute a collection, and so must again be whole. Thus, it might be thought that these analyses share the same problematic assumption Salmon attributes to Frege s analysis, namely that count nouns like orange are extensional, denoting sets of things which are in some intuitive sense indivisible wholes. But there is a different semantics for count nouns available, one which does not share this assumption. It is due to Krifka (1989). Basing his semantics on classifier languages, 1 Krifka analyzes all English nouns as basically 1 See Chierchia (1998) and Scontras (2014) for relevant discussions. 2
mass, so that all numerical modification amounts to measuring quantities of stuff quantities of water, quantities of rice, quantities of orange, etc. On Krifka s semantics, the primary difference between count nouns like orange and mass nouns like water is that whereas mass nouns denote quantities of stuff directly, count nouns do so indirectly thanks to a certain classifierlike element built into their meanings, which Krifka calls a natural-unit (NU). In essence, when I say that that there are two oranges on the table, I m saying that there is a quantity of orange on the table measuring two natural-units of orange. (3) a. orange = λn.λx. orange(x) NU(orange)(x) = n b. 2 1 2 oranges = λx. orange(x) NU(orange)(x) = 2.5 Here, NU is a function mapping individuals to real numbers with respect to some property. Intuitively, cardinality expressions measure how much of a property a certain thing has with respect to a natural unit of that property. As with Salmon s proposal, counting is thus property-relative, and different measurements will result depending on what the property is. In the context of COP, for instance, relative to the property of being a whole orange, NU will return the real number 2. But relative to the property of being an orange, nothing obviously prevents NU from returning a non-whole number as the correct answer to the Question, as suggested in (3b). Consequently, Krifka s semantics affords a solution to COP which is strikingly similar to Salmon s solution, and which Christopher Kennedy and Jason Stanley (2009) explicitly endorse. After mentioning Krifka s semantics, they say the following: This proposal is very similar to the one suggested by Nathan Salmon (1997, p. 10), who suggests that... numbers are not merely properties of pluralities simpliciter, but relativized properties. An advantage of this analysis is that it provides a semantics for nominals with fractional number terms... Assuming that the sorted NU function is not constrained to return whole numbers as values, a phrase like 2.5 oranges denotes a property that is true of orange-stuff whose measure equals 2.5 orange-units. However, this analysis involves a commitment to the position that count nouns are in some fundamental sense semantically the same as mass nouns, in that they denote properties of quantities of stuff, rather than properties of atomic objects. 2 The solution on offer is just this. In the context of COP, we are measuring quantities of orange relative to a natural unit of orange. Since measurements are given in terms of real numbers, and since real numbers needn t be whole, 2 Kennedy and Stanley (2009, p. 619, fn. 16). 3
it s little wonder that the intuitively correct Answer to the Question involves a non-whole number. Let s call this the Krifka-Inspired Solution [+ Salmon] (KISS). KISS is not the only available solution to COP. In fact, we believe that it is not really a solution at all. That s because while KISS correctly predicts one available interpretation of the Answer, it actually predicts the wrong interpretation in the context of Salmon s puzzle. We are going to argue that the Answer is ambiguous in a way resembling so-called container phrases like glass of water. Consider (4a). (4) a. There are four glasses of water in the soup. b. There s a plurality of glasses x s.t. x consists of four individuals, each of which is filled with water and is in the soup. c. There s a quantity of water x s.t. x measures four glasses worth and x is in the soup. This example is due to Rothstein (2009), who argues that it is ambiguous between what she calls an individuating interpretation (II) in (4b) and a measure interpretation (MI) in (4c). 3 For the II, suppose that Mary has a strange way of heating up water; she pours it into glasses and places those glasses into boiling soup. For the MI, suppose that Mary wants to make some soup, and that the recipe calls for a certain amount of water, namely four glasses worth. Therefore, she fills a certain glass four times with water, pouring the contents each time into the soup. One of our primary empirical contentions here is that I/M-ambiguities are not restricted to just container phrases. For example, orange is an ordinary count noun, yet four oranges is similarly ambiguous. (5) a. There are four oranges in the punch. b. There s a plurality of oranges x s.t. x consists of four individuals, each of which is in the punch. c. There s a quantity of orange x s.t. x measures four oranges worth and x is in the punch. For the II in (5b), suppose that Mary has already made the punch and that she wants to decorate it for the party. She thinks that adding some fruit would do the trick, and so she drops several apples, some pears, and four oranges into the punch. For the MI in (5c), suppose Mary wants to make some punch for the party and the recipe calls for four oranges worth of orange pulp. Accordingly, Mary takes four whole oranges, pulverizes them, and adds the resulting pulp to the rest of the punch. As preliminary evidence for there being a genuine ambiguity in both cases, consider the fact that there is an asymmetric entailment relation holding generally between IIs and MIs. For example, if Mary places four glasses 3 Cf. also Landman (2004), Partee and Borschev (2012), and Scontras (2014). 4
filled with water into the soup, then there must be four glasses worth of water in the soup. On the other hand, if she pours an amount of water measuring four glasses worth directly from the tap into the soup, it clearly does not follow that she has placed any glasses in it. Similarly, if Mary drops four whole oranges into the punch, then there must be four oranges worth of orange in the punch. But if she pours four oranges worth of prepackaged orange pulp directly into the punch, it clearly does not follow that she has placed four whole oranges in it. This indirectly suggests that four oranges, like four glasses of water, is I/M-ambiguous. We will develop a number of heuristics which demonstrate this more directly in 2. For example, Rothstein points out that while only the MI of four glasses of water is acceptably paraphrased in terms of -ful. Likewise, only the MI of four oranges is acceptably paraphrased in terms of worth. Here s the theoretical significance of I/M-ambiguities. IIs involve counting glasses qua individuated containers and oranges qua individuated bodies of fruit. MIs, on the other hand, involve measuring quantities of a substance with respect to some non-standardized unit of measurement, or what Partee and Borschev (2012) call an ad hoc measure. The traditional analyses of cardinality expressions naturally predict IIs, while KISS naturally predicts MIs. The trouble is that neither can obviously capture both interpretations. In the case where Mary pours four oranges worth of prepackaged orange pulp directly into the punch, for instance, the traditional analyses wrongly predict (5a) to be false since there needn t be individual oranges in the punch. The problem for KISS is interestingly different. Though it rightly predicts (5a) to be true in this scenario, it appears to wrongly predict that (5a) should only have a MI, or so we will argue later on. If so, then the problem with KISS is that it has no clear means of explaining the numerous semantic differences between IIs and MIs demonstrated in 2. We will argue that the Answer is also I/M-ambiguous, just like four oranges. It has an II paraphreasable as the Answer I, an MI paraphraseable as the Answer M, and these are truth-conditionally distinct. (Answer I ) There are two oranges on the table, and there is a half-orange on the table. (Answer M ) There are two and a half oranges worth of orange on the table. To show this, we will apply the same heuristics mentioned above to the Answer, revealing it to be I/M-ambiguous. The question thus becomes: How can the Answer come to have these distinct interpretations? To answer it, we will first need to explain how I/M-ambiguities arise more generally. This is done in 3. On our analysis, following Rothstein (2009) and Scontras (2014), IIs arise from combining one of the traditional analyses of cardinality expressions with countable meanings of the relevant expressions, e.g. glass of water or orange. To account for MIs, we propose 5
a certain type-shifting principle we call the Universal Measurer (UM). This takes a noun like glass or orange and converts it into an ad hoc measure on substances of the appropriate sort, e.g. water or orange. As a result, four oranges denotes quantities of orange measuring four oranges worth, thus effectively reproducing Krifka s semantics. However, because UM does not have an inverse, there is generally no way to sematnically recover the default denotation of orange the set of individual oranges from its measure-like denotation resulting from an application of UM. Consequently, there is no way to semantically recover the II from the MI, thus explaining the asymmetric entailment noted earlier. On our analysis, the Answer I entails the Answer M, but not the other way around, thanks to UM. However, the trouble is explaining how the Answer can have an II in the first place. After all, if IIs result from the traditional analyses of cardinality expressions, yet fractional cardinalities are incoherent on those analyses, then it is hard to see how an II could arise in the first place. We will argue that appearances are deceptive here: 2 1 2 in the Answer is not specifying a fractional cardinality of the oranges on the table. Rather, we are actually counting three things on the II of the Answer, namely two whole oranges and a half-orange. 4 is dedicated to defending this claim against certain objections raised by Salmon. We show how combining independently motivated analyses of the component expressions two, and, and a half in a natural way actually predicts that the Answer should have an interpretation paraphraseable as the Answer I, and more generally that fractions on IIs involve a form of counting. For instance, 2 2 3 oranges on the II counts four things two whole oranges and two more things, namely thirds of an orange. This in turn affords a neat solution to COP. Though there are strictlyspeaking only two oranges on the table in that context, the Answer remains the intuitively correct answer to the Question because we are individuating the two whole oranges from the half-orange and counting these three things separately. But since three is a whole number, COP presents no particular threat to traditional analyses of cardinality expressions. 2 Individuating and Measuring The purpose of this section is to develop four general heuristics for teasing apart individuating and measure interpretations. We will begin by applying those heuristics to uncontroversially I/M-ambiguous container phrases like four glasses of water, showing how they successfully disambiguate between IIs and MIs. We will then apply those same heuristics to the Answer, revealing it to be similarly ambiguous. Our first heuristic has to do with the acceptability of worth. Rothstein (2009) observes that the suffix -ful disambiguates for MIs in con- 6
tainer phrases, as revealed by the following contrast, where (6) is uttered in a measure context and (7) is uttered in an individuating context. M-Context: Mary wants to make some soup. The recipe calls for four glasses worth of water. Accordingly, Mary fills a certain glass four times with water, pouring the contents each time into the soup. John says As I was passing by the kitchen, I noticed there was some water in the soup. How many glasses of water did you put in the soup? (6) Mary: Four glasses / glassfuls. I-Context: Mary wants to heat up some water for coffee. Accordingly, she fills four glasses with water and places them into the boiling soup. John says As I was passing by the kitchen, I noticed some glasses in the soup. How many glasses of water did you put in the soup? (7) Mary: Four glasses /??glassfuls. As Rothstein explains, the acceptability of glassfuls in (6) but not (7) makes sense if measure contexts impose an MI, individuating contexts impose an II, and -ful disambiguates for MIs. Based on this observation, Rothstein proposes that the function of -ful is to convert container nouns like glass into ad hoc measures. 4 In other words, glassful denotes a measure of a substance like water in terms of a non-standardized glass-unit, e.g. the amount of water that would fill a certain glass used to pour water into the soup. One of our contentions here is that -ful is a special case of worth in this respect. In general, the function of worth is to convert a noun into an ad hoc measure. However, unlike -ful, worth is not limited to just container nouns. Rather it can be used with a variety of nouns, e.g. four oranges worth of orange, four grains worth of rice, or four ounces worth of salt. 5 Thus, our first heuristic is a natural extension of Rothstein s: worth is generally acceptable with MIs but not with with IIs, as seen by e.g. the unacceptability resulting from substituting glasses worth for -ful in (7). Now consider the following contrast. M-Context: John is on a special diet which requires him to have 2 1 2 oranges per day. At Smoothie World, smoothies are made with prepackaged orange pulp, and employees know how much orange pulp a typical orange 4 Cf. also Scontras (2014). 5 In four oranges worth of orange, for instance, worth converts orange into an ad hoc measure on quantities of orange. However, in a context in which various fruits are being washed, it is possible to use four oranges worth of water to measure amounts of water in terms of how much would be used to wash a single orange. Likewise with e.g. four glasses worth of (dish) water. 7
produces. In order to fulfill his dietary requirements, John orders an smoothie at Smoothie World. Mary makes John s smoothie. John says I just want to make sure that smoothie will meet my dietary requirements. How many oranges are in the smoothie? (8) Mary: Two and a half oranges / oranges worth. I-Context: There are three oranges on the table. Mary takes one of those oranges, cuts it in half, eats one of the halves, and places the remaining half back on the table. John says As I was passing by the kitchen, I noticed there were oranges on the table. How many oranges are on the table? (9) Mary: Two and a half oranges /??oranges worth. As before, the facts in (8) and (9) make sense if the M-Context imposes an MI of the Answer, the I-Context imposes an II of the Answer, and worth, like -ful, disambiguates for MIs. Note that the I-Context here is just Salmon s context for COP, thus suggesting that COP imposes an individuating interpretation of the Answer. Our additional diagnostics confirm this result. Our second diagnostic involves pluralization. As Rothstein points out, container phrases are acceptable with plural pronouns on IIs but not on MIs. M-Context: Same as (6). Pointing at the soup, Mary says: (10) a.?? Those are four glasses of water. b. That is four glasses of water. I-Context: Same as (7). Pointing at the soup, Mary says: (11) a. Those are four glasses of water. b. That is four glasses of water. The plural pronoun those refers to a plurality consisting of individuated objects, and we have a plurality of individuated glasses in the I-context but not in the M-context. Hence the difference in acceptability between (10a) and (11a). On the other hand, the singular pronoun that is acceptable in both contexts, plausibly for different reasons. On the II, we can think of the four glasses as constituting a single group, 6 thus making a referent for that available. On the MI, we have a quantity of water measuring four glassfuls, and this supplies an appropriate referent for that. We see a similar contrast with the Answer. M-Context: Same as (8). Pointing at the smoothie, Mary says: 6 See Landman (2004). 8
(12) a.?? Those are 2 1 2 oranges. b. That is 2 1 2 oranges. I-Context: Same as (9). Pointing at the table, Mary says: (13) a. Those are 2 1 2 oranges. b. That is 2 1 2 oranges. Again, this difference in acceptability makes sense if those refers to some salient plurality consisting of individuated objects. There is such a plurality in the I-Context but not in the M-Context, so there is something for those to refer to in (13) but not in (12). And both interpretations are acceptable with that since there is an appropriate referent available in both cases. Our third diagnostic involves the acceptability of modifiers like approximately and roughly, or what Lasersohn (1999) calls slack regulators. These are generally acceptable with MIs, but not always with IIs, especially when the cardinality of the plurality in question is small. 7 M-Context: Similar to (6), except that Mary pours water straight from the tap into the soup. She says: (14) There are approximately four glasses of water in the soup. I-Context: Same as (7). Mary says: (15)?? There are approximately four glasses of water in the soup. The difference in acceptability here is plausibly due to an implicature carried by the use of approximately. In general, approximately implicates that the speaker is unsure whether the amount specified is the exact amount which actually obtains. For instance, Mary s utterance of (14) implicates that she is unsure whether the amount of water poured into the soup measures exactly four glassfuls. 8 This sort of uncertainty is normal with measurement, at least to a certain degree of precision. However, if Mary just placed four glasses filled with water into the soup, then there would appear to be little room left for uncertainty as to how many such glasses there are. Once again, we see the same contrast in the Answer. 7 Slack regulators are generally acceptable with larger cardinalities where an exact measure is not so easily determined. For example, if Mary has a very large vat of soup and she has lost track of how many glasses she has placed in it, it is perfectly acceptable for her to say There are approximately twenty glasses of water in the soup. 8 This is plausibly a quantity implicature (Grice 1989). Since exactly four glasses of water entails approximately four glasses of water but not vice versa, the former is strictly speaking more informative. So if Mary knew there were exactly four glasses of water in the soup, she would have said so (assuming she s cooperative). Since she hasn t said that, she must not know whether there are exactly four glasses of water in the soup. 9
M-Context: Same as (8). Pointing at the smoothie, Mary says: (16) There are approximately 2 1 2 oranges in the smoothie. I-Context: John hates doing math problems but loves oranges. Mary offers him the following deal: for each math problem he solves, he will receive half an orange. So far, John has solved five math problems. Robin wants to make sure that John completes his homework, and so asks How many oranges has John earned so far?. Mary responds: (17)?? John has earned approximately 2 1 2 oranges. Again, the difference here makes sense if 2 1 2 is functioning as a measure on the MI, but is specifying a cardinality on the II. Whereas it is perfectly sensible for Mary to estimate when measuring the amount of orange going into a smoothie, if John has solved exactly five math problems, then there would appear to be little room for uncertainty as to how many oranges he has earned up to this point. 9 Our final heuristic involves the nouns number and amount. These can be used to tease apart IIs and MIs directly, as witnessed by (18) and (19). Context: John and Mary both want to heat some water for coffee, and both do it by placing glasses filled with water into boiling soup. However, John s glasses are exactly half the size of Mary s. Both place four of their glasses filled with water into their respective soups. (18) a. There are four glasses of water in Mary s soup, and there are the same number of glasses in John s soup. (true) b. There are four glasses of water in Mary s soup, and there is the same amount of water in John s soup. (false) Context: Same as for (18), only John places eight of his glasses into his soup. (19) a. There are four glasses of water in Mary s soup, and there are the same number of glasses in John s soup. (false) b. There are four glasses of water in Mary s soup, and there is the same amount of water in John s soup. (true) This makes sense if number relates pluralities specifically to their cardinalities, while amount relates substances to their measures more generally, 9 To be clear, Mary could acceptably utter There are approximately 2 1 oranges on the 2 table in the context of COP. After all, Mary may reasonably doubt that she has cut one of the oranges exactly in half. However, this is plausibly due to imprecision in the fraction word half (see Lasersohn (1999)), thus prompting our adjusted I-Context for (17). We will return to the meanings of fraction words in 4.2. 10
including but not limited to e.g. volume. 10 Thus, (18a) is true since Mary s glasses and John s glasses have the same cardinality, namely four. But since those glasses contain different volumes of water, (18b) is false. We see the reverse in (19). Since IIs involve counting the individuated members of a plurality but MIs involve measuring a substance according to some ad hoc measure, these examples directly reveal four glasses of water to be I/Mambiguous. We see the same pattern with the Answer. Context: Similar to (9), except that John has oranges which are exactly half the size of Mary s oranges. Like Mary, he has three oranges, cuts one in half, eats one of the halves, and leaves the other half on his table. (20) a. There are 2 1 2 oranges on Mary s table, and there are the same number of oranges on John s table. (true) b. There are 2 1 2 oranges on Mary s table, and there is the same amount of orange on John s table. (false) Context: Same as (20), except that John leaves five of his oranges on his table. (21) a. There are 2 1 2 oranges on Mary s table, and there are the same number of oranges on John s table. (false) b. There are 2 1 2 oranges on Mary s table, and there is the same amount of orange on John s table. (true) As before, these examples make sense if number here relates pluralities of oranges to their cardinalities, while amount relates quantities of orange to their volume. Like with four glasses of water, these examples directly reveal the Answer to be I/M-ambiguous. Altogether, these heuristics reveal two important facts. First, I/Mambiguities are not limited to just container phrases like glass of water. This is theoretically significant, as the literature on I/M-ambiguities has tended to focus exclusively on container phrases, suggesting perhaps that I/M-ambiguities are limited only to them. However, even ordinary count nouns like orange can give rise to IIs and MIs. 11 Secondly, the Answer is I/M-ambiguous, and COP imposes an individuating interpretation of the 10 See Scontras (2014) on amount. Note that amount is ambiguous in a way number is not. Suppose, for instance, that we are looking at four 1 lb. oranges. As Scontras suggests, that amount of oranges can then refer to two sorts of degrees, or representations of measurement, namely a weight (4 lbs.) or a cardinality (consisting of four singular oranges). On the other hand, that amount of orange can only refer to a weight, while that number of oranges can only refer to a cardinality. 11 An anonymous reviewer asks whether MIs are restricted to a limited set of count nouns, perhaps those with meanings that are easily or commonly construed as masses, such as foodstuffs (see e.g. Landman (2011)). Although at this point we do not believe MIs are restricted in this way, we leave this as a topic for future investigation. 11
Answer. The first observation raises the following empirical question: If I/M-ambiguities are not due specifically to the meanings of container nouns like glass, then how do I/M-ambiguities arise more generally? We will argue in the next section that these are made possible by a certain type-shifting principle we call the Universal Measurer. 3 The Universal Measurer We have seen that I/M-ambiguities are not limited to just numerically modified container phrases like four glasses of water. We see similar I/Mambiguities in four oranges, for instance. The purpose of this section is to provide a general account of I/M-ambiguities. More specifically, we extend extant analyses of I/M-ambiguities in a natural way to explain how both four glasses of water and four oranges can have IIs and MIs. Let s begin with I/M-ambiguous container phrases like that in (4a). (4a) There are four glasses of water in the soup. The question is how to derive both the II and the MI in a compositional manner. Following Scontras (2014), we assume that glass is lexically a monadic predicate true of individual glasses, or (22). (22) glass = λx. glass(x) However, glass takes on a distinctively relational character in container phrases: on the II, glass in glass of water roughly means glass containing water, for instance. We follow Rothstein (2009) in assuming that a certain type-shifting principle is responsible, one called the Construct State Shift (CSS). This is given in (23a), where P is a noun like glass, Q is a noun like water, and R is a contextually-supplied relation holding between members of P and Q. (23b) gives the result of applying CSS to the monadic meaning of glass in (22) and combining the result with of water. 12 (23) a. λp.λq.λx. y. P (x) Q(y) R(x, y) (CSS) b. CSS( glass of water ) = λx. y. glass(x) water(y) R(x, y) In essence, CSS transforms a monadic noun like glass into a relational noun denoting glasses which bear R to something else. In the case at hand, the relation determined is that of being filled with, and the something else in question is water. Ultimately, then, we get a predicate true of those glasses which are filled with quantities of water, or (23b). From here, it is straightforward to derive the II of (4a), repeated in (4b). 12 Following e.g. Rothstein (2009), we assume for expository convenience here that of contributes no semantic content. We drop this assumption in 4.2, however. 12
(4b) There s a plurality of glasses x s.t. x consists of four individuals, each of which is filled with water and is in the soup. We simply apply one of the traditional analyses of cardinality expressions. For concreteness, assume Scontras analysis given in (24), where CARD is a silent measure noun and µ # is a cardinality measure, or a function from pluralities x to numbers n such that there are n-many singular individuals constituting x. (24) a. four = 4 b. CARD = λn.λp.λx. µ # (x) = n P (x) c. four CARD = λp.λx. µ # (x) = 4 P (x) Applying (24c) to the denotation of glass of water in (23b) returns a predicate true of pluralities of four glasses, each of which are filled with water: (25) four CARD glasses of water = λx. µ # (x) = 4 glasses(x) water(y) R(x, y) Consequently, (4a) will be true just in case there s at least one such plurality of glasses in the soup. And this, of course, is the II of (4a). For the MI, Scontras proposes the type-shifting operation in (26a) he calls it SHIFT C M where k is a kind in the sense of Chierchia (1998) and is Chierchia s up -operator returning the instances of a kind. (26) a. λp.λk.λn.λx. k(x) µ P (x) = n (SHIFT C M ) b. SHIFT C M ( glass ) = λk.λn.λx. k(x) µ glass (x) = n c. four glasses of water = λx. WATER(x) µ glass (x) = 4 In effect, SHIFT C M transforms a monadic noun like glass into a measure term like ounce. 13 In (26b), µ glass is an ad hoc measure, and WATER in (26c) denotes quantities of water. Hence, according to (26c), four glasses of water denotes quantities of water measuring four ad hoc glass-units. This predicts correct truth-conditions for (4a) on the MI: it will be true if there s four glasses worth of water in the soup. But now consider (27). It too involves an I/M-ambiguous container phrase, namely four boxes of tires. Context: John and Mary work in a recycling plant specializing in recycling two items: aluminum cans and tires. Once the items are ground, they are sifted and the recycled material is packaged into boxes. John and Mary find some unmarked boxes, and there is some uncertainty as to what they contain. After dumping the contents of the boxes into two sorted piles, Mary points at one of the piles and says: 13 See Scontras (2014) for how these different kinds of expressions are related. 13
(27) That s four boxes of {tires/tire}. Obviously, a MI is intended here: Mary is talking about four boxes worth of tires, not four boxes containing tires. Moreover, (27) plausibly receives a grinding interpretation, in the sense of Pelletier (1975). For one thing, Mary is pointing at ground up bits of tire, and so she could have just as well used the massivized noun tire in the scenario described. Moreover, (27) isn t false just because Mary failed to point at piles of whole tires, contrary to what is predicted by SHIFT C M. 14 Happily, there is a fairly simple fix available. We propose generalizing Scontras analysis of MIs via the alternative type-shifting operation in (28a), which we dub the Universal Measurer (UM). 15 (28) a. λp.λq.λn.λx. γ(q(x)) µ P (x) = n (UM) b. γ(p ) = {x y. P (y) x y} c. four boxes of tires = λx. γ(tires(x)) µ box (x) = 4 The γ-operator in (28b) is a grinding-operation similar to ones proposed by Link (1983), Rothstein (2010), and Landman (2011). It takes the members of the extension of a predicate P and returns their parts, represented here as. Consider the set of pluralities of tires, for instance. These consist of individual, whole tires. But the latter also have parts all of the individual bits of rubber and metal constituting them, for instance. Now, let s assume that γ returns the set of all such parts. Then according to (28c), the result of applying UM to box and combining the result with the denotations of of tires and four (in that order) is a predicate true of those parts of pluralities of tires which measure four boxes worth. Accordingly, Mary s utterance of (27) would be true if she were pointing at a pile of whole tires or a pile of assorted bits of tire, so long as the tire-matter she is pointing at collectively measures four boxes worth. But what about I/M-ambiguous non-container phrases, e.g. (5a)? (5a) There are four oranges in the punch. Can the II and MI for these be derived in a similar manner? We believe that the answer is Yes, but with a couple important caveats. Let s begin with the II. Like box, orange is inherently monadic. Thus, we may plausibly assume that orange denotes the set of oranges, or (29). 14 According to Chierchia (1998) s analysis of mass nouns, applying the -operator to the kind TIRE returns the set consisting of all singular tires plus the pluralities formed from them. Crucially, however, this denotation does not include bits of tire-matter, and so (26a) wrongly predicts that (27) should be false in the context provided. Many thanks to an anonymous reviewer for helping us see this point. 15 In homage to Pelletier (1975) s famous Universal Grinder and Universal Packager. UM is plausibly the meaning of worth, though suitably restricted in various ways. 14
(29) orange = λx. orange(x) Unlike box, however, orange is not relational on IIs. In other words, we count oranges directly qua bodies of individuated fruit, not qua containers of orange. Since this is precisely the meaning of orange assumed in (29), we can derive IIs for four oranges by combining this denotation directly with one of the traditional analyses of cardinality expressions. For example, combining it with the meaning assumed in (24) correctly predicts the following truth-conditions: (5a) will be true just in case there is a plurality of four oranges, each of which is in the punch. (5b) There s a plurality of oranges x s.t. x consists of four individuals, each of which is in the punch. And this is precisely the II repeated in (5b). In order to derive the MI in (5c), we propose reflexivizing UM. (5c) There s a quantity of orange x s.t. x measures four oranges worth and x is in the punch. As the paraphrase suggests, it seems that orange needs to supply both the ad hoc measure and the substance being measured on MIs: we are measuring four oranges worth of orange. Intuitively, what s needed then is a way of guaranteeing that the first two arguments of UM are identical. We see something similar with relational verbs like bathe when occurring without overt objects. Witness (30b), for instance. (30) a. John bathed the baby. b. John bathed. Unlike (30a), (30b) can only be understood reflexively, i.e. as claiming that John bathed himself. Similarly, the MI of four oranges doesn t measure four oranges worth of just anything, but only of orange. According to Reinhart and Siloni (2005), the reflexivization of bathe can be seen as the consequence of applying a certain type-shifting function call it VREF to the basic relational denotation of bathe in (31a). (31) a. bathe = λx.λy. bathe(x, y) b. λr.λx. R(x)(x) (VREF) c. VREF( bathe ) = λx. bathe(x, x) In effect, VREF resets the arguments of bathe to be identical, thus reflexivizing the verb. We assume that VREF is one instance of a generalized reflexivization principle, one which takes relations of various types and reflexivizes them. Thus, we assume that another particular instance of this generalized relflexivization principle is REFL in (32a), where Q is an expression having the same type as UM. 15
(32) a. λq.λp.λn.λx. Q(P )(P )(n)(x) (REFL) b. REFL(UM) = λp.λn.λx. γ(p (x)) µ P (x) = n (RUM) Similar to (31c), applying REFL to UM effectively resets its first two arguments to be identical, thus resulting in (32b), or what we call the Reflexivized Universal Measurer (RUM). RUM effectively guarantees that the substance being measured, i.e. γ(p ), is of the same kind as the ad hoc measure being employed, i.e. µ P. Thus, it does precisely what we need it to do: it allows us to measure quantities of orange in terms of an ad hoc orange-unit. Applying RUM to the monadic meaning of orange from above results in (33a), thus generating the meaning of four oranges in (33b). (33) a. RUM( orange ) = λn.λx. γ(orange(x)) µ orange (x) = n b. four oranges = λx. γ(orange(x)) µ orange (x) = 4 Applying the grinding-operation γ to the extension of orange returns all parts of all singular oranges, including the oranges themselves. Thus, according to the resulting denotation in (33b), four oranges will denote those orange-parts measuring four ad hoc orange-units, 16 and so (5a) will be true just in case there is an amount of orange in the punch measuring four oranges worth. And this, of course, is just the MI paraphrased in (5c). The resulting analysis reveals the flaws in both the traditional analyses of cardinality expressions and KISS. The traditional analyses wrongly predict that the MI of (5a) should be false in the smoothie scenario since there needn t be four individual oranges in the smoothie. KISS rightly predicts that (5a) should be true in that same scenario, but because it appears to predict that (5a) should only have a MI (see 5), KISS has no obvious way of semantically explaining the various semantic contrasts between MIs and IIs noted in 2. In contrast, on the present view four oranges does not uniformly count oranges or measure quantities of orange in terms of ad hoc units of orange. Rather, it can do both, and which semantic function it performs depends wholly on context. In sum, both the traditional analyses and KISS capture a genuinely available interpretation of e.g. four oranges, but a completely adequate semantics would relate those different interpretations in a natural way. This is exactly what UM does. 16 An anonymous reviewer observes that ad hoc orange-units are different from ad hoc glass-units in that the latter but not the former are plausibly based on containment relations, e.g. how much liquid would fill a certain glass, thus leading to the question of how ad hoc measures are generally determined. Though we are tempted to think that they are constrained to at least what Vikner and Jensen (2002) call natural relations, we do not yet have a fully satisfactory answer to this question, nor do we know of any in the literature. This is something we hope to pursue in future research, however. 16
4 How to Count 2 1 2 Oranges We saw that the Answer is I/M-ambiguous in 2. The primary question here is whether the individuating and measure interpretations of the Answer are derivable in the same manner suggested previously for four oranges. It should be fairly evident how the MI arises. (Answer M ) There are two and a half oranges worth of orange on the table. Just as with the MI of four oranges, we apply the Reflexivized Universal Measurer to orange, thus shifting it into a reflexivized measure term. As a result, we predict truth-conditions for the Answer witnessed in the Answer M : it will be true just in case there is a quantity of orange on the table measuring two and a half oranges worth. This would be true if, for instance, there happened to be 2 1 2 oranges worth of orange slices, each from possibly different oranges, on the table. However, this is not the interpretation of the Answer naturally suggested by Salmon s puzzle. The diagnostics in 2 show that Salmon s context for COP is an individuating context, and therefore imposes an II of the Answer. IIs involve individuating members of pluralities, e.g. glasses containing water or boxes containing tires. In the context of COP, the relevant plurality is the one denoted by the 2 1 2 oranges on the table. This plurality consists of three individuals, we submit: two whole oranges and one half-orange. Put differently, half in two and a half oranges is functioning as a modifier of orange, so that two and a half oranges is really being interpreted as two oranges and a half-orange. This leads to the II of the Answer, or the Answer I. (Answer I ) There are two oranges on the table, and there is a half-orange on the table. If so, then 2 1 2 isn t specifying a fractional cardinality on the II of the Answer after all. Rather, it s functioning as a complex modifier, similar to large and small apples in (34a). (34) a. There are large and small apples on the table. b. There are large apples on the table, and there are small apples on the table. Like the Answer on the II, (34a) can be paraphrased as a conjunction of modified noun phrases. And just as one would not reasonably conclude from (34b) that there are apples on the table which are both large and small, one would not reasonably conclude from the Answer I that there are oranges on the table numbering both two and also one half. 17
If this informal characterization is correct, then COP presents no special threat to the traditional analyses of cardinality expressions. It would be threatening only if 2 1 2 in the Answer specified a fractional cardinality. But our claim is that it fails to do so on either interpretation. Anticipating this sort of response, perhaps, Salmon (1997) argues the Answer I is not a genuine interpretation of the Answer. More specifically, he challenges our claim that two and a half can function as a complex modifier. To quote him: The numeral 2 occurring in [the Answer] has been separated from its accompanying fraction, and now performs as a solo quantifier. The fraction itself has been severely mutilated. The numeral 1, which appears as the fraction s numerator in [the Answer], has ascended to the status of anonymous quantifier, functioning independently both of its former denominator and of the quantifier in the first conjunct. At the same time, the word half appearing in [the Answer I ] has been reassigned, from quantifier position to predicate position. In effect, the mixed number expression 2 1 2, occurring as a unit in [the Answer], has been blown to smithereens, its whole integer now over here, the fraction s numerator now over there, the fraction s denominator someplace else... Even the schoolboy knows that the phrase and a half in [the Answer] goes with the two and not with the orange. 17 Colorful rhetoric aside, we take Salmon to be making a fairly simple but substantial point here, namely that it is far from obvious how 2 1 2 functions as a complex modifier, as is required if the Answer I is genuinely available. If it does not, i.e. if the phrase and a half in [the Answer] goes with the two and not with the orange, then it is hard to see how the Answer I could arise compositionally from two and a half oranges. Thus, we interpret Salmon s remarks here as posing a serious challenge: compositionally derive the Answer I from independently plausible meanings of the relevant component expressions. We will address Salmon s challenge head on in what follows. We will show that it is possible to compositionally derive the Answer I from independently motivated analyses of the component expressions, namely two, and, and a half. We address the semantic contribution of and via Krifka s (1990) analysis of generalized cumulative conjunction in 4.1. The contribution of half is addressed via Ionin, Matushansky, and Ruys (2006) analysis of fraction words in 4.2. We derive the Answer I by combining these analyses with one of the traditional theories of cardinality expressions in 4.3. 17 Salmon (1997, p. 6). 18
We end this section in 4.4 by responding to two further objections raised by Salmon. 4.1 Generalized Cumulative Conjunction Let s begin with the semantic contribution of and in two and a half oranges. As with large and small apples, we assume that and in the Answer is an instance of cumulative conjunction. Consider the following examples from Krifka (1990). (35) a. John and Mary met at the mall. b. This is beer and lemonade. c. That flag is green and white. (35a) cannot mean that John met at the mall and Mary met at the mall, just as (35b) cannot mean that this liquid concoction is beer and it is lemonade. That s because the various occurrences of and here are witnesses to cumulative conjunction, as opposed to propositional conjunction. 18 What s more, because the different instances of and coordinate expressions of different semantic types, they must themselves take on different types in those different syntactic environments, thus mirroring generalized propositional conjunction on the treatment of Partee and Rooth (1983). Thus, what s needed is a theory of generalized cumulative conjunction. Krifka supplies one such theory. Adapting his analysis of coordinated modifiers like green and white in (35c) and simplifying somewhat, Krifka s semantics predicts that (36a) should have the logical form in (36b) or (36c), depending on whether and is understood propositionally or cumulatively. (36) a. There are large and small apples on the table. b. x. large(apples(x)) small(apples(x)) on-table(x) c. x. y. z. x = y z large(apples(y)) small(apples(z)) on-table(x) Here, is a join-operation on individuals; in general, if a and b are individuals, then a b denotes the plurality consisting of a and b. The incoherent truth-conditions given in (36b) result if and is understood as propositional conjunction, in which case (36a) is true if there are apples on the table which are both large and small. The coherent truth-conditions given in (36c) result if instead and is understood cumulatively, in which case (36a) is true if there are apples on the table, parts of which are large and parts of which are small. Our primary contention here is that and in two and a half oranges in the Answer, like and in (36a), is only plausibly interpreted cumulatively. 18 Also known as boolean and non-boolean conjunction. See Partee and Rooth (1983) and Lasersohn (1995) for related discussion. 19
4.2 Fractions To account for the meanings of fraction words such as half, quarter, third, etc., we adopt the analysis of Ionin, Matushansky, and Ruys (2006; henceforth, IM&R). IM&R model numerators as cardinality expressions, while denominators package parts of things into individuals which numerators then count: halves, quarters, thirds, etc. For example, two thirds of an orange counts certain proper parts of an orange, namely the thirds. Similarly, two thirds of the oranges counts certain proper parts of a plurality of oranges, namely the thirds. In either case, fractions like one half, two thirds, etc. essentially involve counting parts of a whole. According to IM&R, third has a fairly complicated meaning consisting of three components. We separate those components here for convenience. (37) third = λp. λx. y. i. P (y) z[p (z) z y] ii. S [Π(S)(y) S = 3 x S iii. µ [µ M s 1, s 2 [(s 1 S s 2 S) (µ(s 1 ) = µ(s 2 ))]]] Ultimately, third is a predicate-modifier: it takes a predicate like of an orange or of the oranges and returns a property true of the relevant parts. Suppose for concreteness that third modifies of the oranges, and assume with Ladusaw (1982) that of denotes a mereological relation, alá (38a). (38) a. of = λx.λy. y x b. of the oranges = λy. y ιx[oranges(x)] Then of the oranges plausibly denotes a predicate true of the parts of some uniquely salient plurality of oranges, as suggested in (38b). According to (37), third of the oranges expresses a property of these parts, and clauses (i)-(iii) determine which property that is. More specifically, clause (i) guarantees that there is a maximal part of the oranges, namely the entire plurality. Clause (ii) guarantees that there is a partitioning Π of that plurality into three countable, non-overlapping parts. 19 The non-overlap condition ensures that we do not count the same part twice. Finally, clause (iii) guarantees that the partitioning of the oranges is quantity uniform: the three parts partitioned by Π all measure the same amount, according to some measure µ in the domain of contextuallydetermined measures M. In the present case, the contextually-relevant measure is most likely cardinality. Hence, third of the oranges is a predicate true of parts of the oranges which, taken together, divide them into three non-overlapping parts having the same cardinality. For example, suppose we 19 Π(S)(y) makes S the set of elements of the partition of y. The way that Π and S are defined ensures that S includes all of y and that elements of S do not overlap. See IM&R (2006, p. 318) for details. 20