TY - JOUR
AU - Burgess, Alexis
AB - Abstract Cognitive partitions are useful. The notion of numerical identity helps us induce them. Consider, for instance, the role of identity in representing an equivalence relation like taking the same train. This expressive function of identity has been largely overlooked. Other possible functions of the concept have been over-emphasized. It is not clear that we use identity to represent individual objects or quantify over collections of them (numerically or otherwise). Understanding what the concept is good for looks especially urgent in light of the fact that numerical identities themselves are tangibly trifling. There is something odd about numerical identity. On the one hand, we appear to use the concept of identity all the time in ordinary thought and talk (not to mention metaphysics or mathematics). ‘Is that the same shirt you wore yesterday?’1 On the other hand, the distribution of the identity relation seems completely trivial and uninteresting: everything is identical to itself and nothing else. So why do we bother representing identity? Why do we even have the concept? What exactly is it good for? The present paper probes the possible functions of our thought and talk about identity, in the hope of solving this riddle and illuminating the concept’s emergence.2 §1 distinguishes our motivating riddle from Frege’s Puzzle, and argues that we cannot solve the former simply by recycling materials from the literature on the latter. §2 considers the possibility that we somehow leverage the concept of identity to represent facts about cardinality. There I offer reasons to resist a line of argument for this view derived from recent work of Susan Carey’s. §3 debunks the popular, Quinean thought that we need the notion of numerical identity just to comprehend basic, non-numerical quantification. Finally, §4 develops and defends my positive proposal that the primary function of identity is to help us represent equivalence relations and thereby partition domains of interest. 1. Introduction: qualitative functions Part of our riddle is the intuition that identity facts or states of affairs ‘out there in the world’ seem trivial and uninteresting. Call this the triviality intuition. To sharpen it a bit, we might say that de re knowledge of numerical identities seems to lack practical value. Something like this intuition is almost universally respected in the literature on Frege’s Puzzle, and I plan to take it for granted.3 The Puzzle, very roughly, is to explain how de dicto knowledge of numerical identities can be significant, given that de re knowledge is not. Almost everyone accepts this ‘given’ and tries to solve the Puzzle by explaining how an identity statement featuring distinct yet coreferring terms can be used to ‘convey’ substantive qualitative information about the terms’ common bearer.4 That is: information about the non-identity-involving properties of that object. For example, Catwoman might assert that (0) Batman is Bruce Wayne, in order to let Penguin know things like: (1) Batman has hair. (2) Bruce Wayne has scars. (3) Someone has hair and scars. (4) Every property of Batman is a property of Bruce Wayne, and vice versa. Things like these can be worth knowing. So (0) can certainly be worth saying. More generally, it seems clear enough that one useful thing we do with the notion of numerical identity is communicate qualitative claims.5 These include claims about property instantiation (1 & 2), coinstantiation (3), and indiscernibility (4). Call expressive functions like these ‘qualitative’ functions of identity. The fact that our concept of identity has qualitative functions does mitigate the sense of mystery I tried to cultivate in opening. It is not really problematic that the distribution of the identity relation is trivial, one might think, if we use the concept to keep track of non-trivial facts like (1)-(4). To this extent, the literature on Frege’s Puzzle makes helpful contact with the line of questioning I am interested in. Arguably, however, the qualitative functions of identity do not adequately explain why we have the concept to begin with. After all, Catwoman does not need to use the concept of identity to convey things like (1)-(4). She could just say (1)-(4) themselves. Or, to be more succinct, she could just say (4), and rely on Penguin’s background knowledge to deliver (1)-(3). Nor does (0) provide a more direct inferential route to (1)-(3). Any valid reasoning from (0) to (1)-(3) will have to invoke something like Leibniz’ Law, effectively passing through (4) along the way.6 So saying (4) is actually a bit more efficient than saying (0), in this respect at least. Of course, (4) contains more words, so saying it would be less efficient in another respect. But ordinary English has a convenient shorthand for (4), namely: Batman and Bruce Wayne are identical (in the qualitative sense). Why do we need the numerical sense of identity too? Is there something else we use it to do, beyond conveying qualitative information?7 An objector might reply that it would not be significantly easier to convey (1)-(4) using the concept of indiscernibility in place of numerical identity. It might be more accurate to say that the two concepts are roughly on a par as far as such qualitative functions are concerned. If so, we could presumably begin to explain the emergence of our concept of identity by pointing out that it is an adequate, undominated tool for these expressive purposes. Even if there are many ways to skin a cat, one can still account for the development of tools involved in whatever method we end up selecting by citing our cat-skinning ambitions. The same goes for cognitive tools and representational aims. As Buridan’s ass illustrates, we sometimes simply have to choose, however arbitrarily, between comparably good solutions to our practical problems. This position is a decent fall-back. But we should not resign ourselves to it before exploring whether we use the concept of identity for anything that could not easily be done without it. If we find such a function, we will have a better explanation of the concept’s emergence. §4 accordingly pursues my positive proposal that our concept of identity developed to help us partition domains of interest via concepts for various equivalence relations. But first I want to argue against two other suggestions, from cardinality and quantification respectively. 2. Cardinality Russell may have been the first to observe that we can use the notion of numerical identity to formulate claims about uniqueness. To say that there is a unique F (or exactly one F) is at least truth-conditionally equivalent to saying: (8) ∃x [Fx ∧ ∀y (Fy → y=x)] Similarly, the claim that there are exactly two Fs is equivalent to: (9) ∃x ∃y [Fx ∧ Fy ∧ x≠y ∧ ∀z (Fz → z=x ∨ z=y)] And so on, for any finite cardinality. Facts about uniqueness and cardinality more generally are often quite significant. ‘Are there two bears circling our camp, or just one?’ Insofar as these are logically complex identity facts, they limit the scope of our triviality intuition. (On the other hand, one can imagine a Tractarian metaphysics on which the truthmakers for cardinality claims are just collections of atomic, identity-free facts.) But however that may be, the manifest significance of statements like (8) and (9) suggests an answer to our animating question. In short: we have the concept of identity to help represent facts about quantity; regardless of whether the identity relation ultimately enters into the metaphysics of such facts. But this cannot be quite right. After all, ordinary English speakers do not actually use the ‘=’ sign (or the ‘is’ of identity) to represent cardinality facts in the way illustrated by (8) and (9). We use adjectival numerals instead. The problem here is not that there is another way we could have gotten the job done (as with the qualitative functions of identity). The problem is rather that we do not seem to do the present job using identity at all. One could try to reply that statements featuring adjectival numerals are properly ‘analysed’ as quantified identity claims. Thomas Hofweber (2005: pp. 182-3) reports that something like this view is implicitly or explicitly at play in much of the philosophy of arithmetic. Now, when it comes to artificial dialects like the ‘English’ of mathematics or philosophy, we can simply stipulate that adjectival numerals shall be used to abbreviate cumbersome quantificational phrases. But of course the relevant version of the view for our purposes here is an empirical hypothesis about natural language. And as Hofweber points out (pp. 185-6), this hypothesis implausibly fragments the semantics of determiners.8 Generalized quantifier theory (GQT) provides a more unified semantics, but its account of numeral determiners does not involve the notion of numerical identity in any obvious way. Unlike the Russellian hypothesis, GQT does not have a place in the syntax for the identity sign. And its semantic values are just mathematical functions (from the values of nouns to the values of complete NPs). If there is a non-obvious role for identity to play here, it is probably to be found in the psychology of language acquisition. Maybe we need the concept of identity in order to grasp the semantic values of adjectival numerals. The standard way to confirm semantic competence with adjectival numerals is just to test whether the subject can reliably provide n objects upon request. By this criterion, children master the first few number words in discrete, laborious stages. But once they have learned ‘three’ or ‘four’, the remaining entries in their counting sequences quickly and collectively acquire content. Children somehow figure out that they can satisfy a request for n objects by intoning the antecedently meaningless lists of numerals they have memorized, setting one object aside for each entry, and halting when they hear themselves say ‘n’. We should therefore try to see whether psycholinguistics offers compelling evidence that we ever leverage the notion of numerical identity during this developmental process. The final stages of catching on to the general counting routine I just described do not seem to involve identity. But there is some reason to think we may use the concept to acquire our initial number words. Susan Carey (2009) argues that the main cognitive mechanism exploited in these early developmental stages is what is called (enriched) parallel individuation.9 This mechanism basically provides mental models of very small sets of objects that we can then map onto perceptually-encountered arrays of objects to check for 1-1 correspondence. And indeed, as the term ‘individuation’ suggests, the default view in this area seems to be that the basic task of representing objects (and sets of them) requires the concept of identity. Simply put, the claim is that we cannot think about individual objects (or mentally quantify over collections of them) until we are capable of entertaining thoughts like: this is the same one again. One way to challenge this case for the ‘cardinality’ function of identity would be to raise doubts about the role of parallel individuation in this developmental process. Carey herself considers the rival hypothesis that children use the system of analogue magnitude representation (AMR) instead of parallel individuation to acquire cardinal number concepts.10 She presents empirical evidence against (and tentatively rejects) this rival en route to her positive view. But if the analogue magnitude hypothesis could be salvaged, then the foregoing argument for countenancing the cardinality function would collapse. For there is no reason to think we use the notion of numerical identity to generate the unitless representations of approximate size provided by AMR. I will not pursue this challenge here, however, as I do not have any special insight about which (if either) hypothesis is right. Psychological opinion seems to be divided, and I am not aware of any empirical or a priori considerations that could currently settle the question. But I do have something to say about the conventional wisdom that representing objects presupposes identity. The standard citations in this area include Spelke and Kestenbaum (1986), Baillargeon and Graber (1987), and Spelke (1988). In a paper intriguingly titled, ‘Infants’ Metaphysics: The Case of Numerical Identity’, Carey and Xu (1996: pp. 113-4) offer the following summary of a classic object-tracking study due to Spelke and Kestenbaum: Four-month-old infants were introduced to two screens, separated in space [figure omitted]. They were shown an object emerging from the left hand edge of the left screen and then reemerging behind it. No object appeared in the space between the two screens. Then a physically identical [i.e., intrinsically indiscernible] object emerged from the right hand edge of the right screen, and then reemerged behind it. This sequence of events was repeated until the baby reached a habituation criterion, at which point the screens were removed revealing either two objects (expected outcome) or one object (unexpected outcome). Babies looked longer at the unexpected outcome, overcoming a baseline preference to look longer at two objects. Adults viewing the display also express surprise at the unexpected outcome of one object. Spelke and Kestenbaum conclude that babies (and adults) analyze the possible paths connecting the appearances of the objects, and infer from the spatiotemporal discontinuity that there must be two numerically distinct objects involved in the event. (Emphasis added) After all, if there were just one object involved in the event, it would presumably have to cross the (visible) gap between the two screens before its next appearance; but throughout habituation, nothing was seen in the gap. According to Carey and Xu (pp. 112-3), experiments like this one therefore provide ‘considerable evidence’ that children have the general sortal concept object from a surprisingly young age, and that they use ‘spatiotemporal information to establish representations of the individual objects in their immediate environments, and to trace identity through time’ (emphasis added).11 Let us grant all this except for the last, italicized conjunct. If visual perception is ever a fancifully fanciless medium of unvarnished news, our brains organize the news into stories about individual objects after just a few months. Or so I will suppose for the sake of argument. What we want to know is whether these concessions somehow implicate the concept of numerical identity specifically. Do the relevant experiments in developmental psychology really go to show that babies represent identity? I do not think so. Take the study described above. In the unexpected event, the screens are removed to reveal a single object. Say it is behind the left screen. Thus the identity-involving hypothesis would be that subjects are surprised to learn something like: the object I saw on the right is actually identical to the object now on the left. This seems like an odd thought to have under the circumstances. Consider an adult subject first, or imagine yourself in the experimental scenario. I submit it would be much more natural to think: the object I saw on the right is now gone. After all, there was ample opportunity during habituation to catch a glimpse of the object on the left passing through the gap between the screens to make an appearance on the right; yet that never seemed to happen. And the idea that one of the objects vanished is still surprising (despite being true). Attributing the second italicized thought to adult subjects would therefore explain the experimental results at least as well if not better than attributing the first. Nor can I see any reason why these considerations would not apply equally to infant subjects. So, since the second italicized thought does not involve the concept of identity in any obvious way, we have no basis to infer from this study that infants are using the concept (or even have it). One might concede that the data do not disqualify this rival hypothesis, but insist that test subjects still need the notion of numerical identity to form their initial judgment that there are ‘two numerically distinct objects’ on the scene during habituation. Only with this distinctness assumption in place, one could argue, will subjects evince surprise when the screens are finally lifted. If for some odd reason you had originally assumed there was just one object on the scene, you would never come to think ‘The object I saw on the right is now gone’. For you can see the object on the left at the end of the experiment, and by hypothesis, you had not expected to see another. So, putting these pieces together: even if my argument from the previous paragraph succeeds, we cannot escape the conclusion that subjects deploy the concept of identity earlier in the experiment. This rejoinder is unconvincing. My opponent’s interjection of the phrase ‘numerically distinct’ into her description of the subjects’ initial judgment seems superfluous and ad hoc. It would be simpler and more natural to suppose that subjects just think there are two objects on the scene (eliding any mention of identity). Accordingly, we could represent the final revelation as: there is only one object on the scene.12 One might be tempted to reply that we need the concept of identity to grasp one and two. Carey certainly thinks so. But her reason for holding this view is that the first few cardinal number concepts are furnished by the core cognition system of parallel individuation, whose operation requires the rudimentary ability to represent objects, which she takes to involve the concept of identity. So it would be question-begging to invoke the view that we need the concept of identity to grasp one and two in the current context, where we are trying to determine whether object representation does indeed involve identity. The present reply therefore awaits some independent reason to believe this view about one and two. To make matters worse for my opponent, I would argue further that we can actually explain the experimental data without assuming subjects ever deploy the concepts one and two. I only introduced the hypothesis that they initially take there to be two objects on the scene as a simple, salient alternative to Carey and Xu’s more loaded, distinctness-involving description of the study. I did not mean to suggest it was the only alternative. Here is a particularly austere, identity- and cardinality-free account of what happens in the minds of infant subjects, elaborating my original proposal that they are surprised by the absence of the object on the right: An object appears on the left. Baby notices. Some antecedently uninterpreted mental symbol is recruited to help baby think about the object. Maybe the symbol is a name in mentalese, which we can translate here as ‘Lefty’. Lefty disappears behind a screen. Then an object that looks exactly like Lefty appears on the right, in a location where (baby thinks) Lefty cannot possibly be. So an undedicated name, ‘Righty’, is recruited to represent it. The habituation routine transpires, alternately activating baby’s representations of Lefty and Righty. The screens are removed. And baby is surprised to find that Righty is missing. My opponent will predictably insist that the infant subject recruits a new name precisely because she realizes the object on the right is numerically distinct from Lefty. But other things being equal, it seems no less likely that the subject just uses the spatiotemporal information at her disposal to infer that Lefty is still behind the screen on the left, and thus that she should not use ‘Lefty’ to think about the object now visible on the right (without bothering to draw any intermediate, identity-involving conclusions). More plausibly still: the infant’s perception of spatiotemporal discontinuity could immediately and sub-personally trigger the recruitment of a new name when Righty first appears (obviating inference altogether). Either way, there is no need to say that the subject exploits the concept of numerical identity, distinctness, one, or two at any point in the experiment. Granted, she might. But the data do not establish that she does. I am therefore inclined to conclude that the standard empirical reasons for believing object representation trades on the concept of numerical identity are inconclusive, at best. And since this premise was a lynchpin in the argument adumbrated earlier for the cardinality function of identity, it is starting to look as though we may need to turn elsewhere for an account of the concept’s utility.13 3. Quantification Quine’s dictum, ‘No entity without identity’, will have occurred to readers by this point. As I understand it, this slogan was meant to summarize a normative view in the methodology of professional ontology, orthogonal to any descriptive theory of ordinary thought and talk. The view is that we should not countenance objects of a given kind unless we are prepared to articulate individuation criteria for them. This is not an empirical hypothesis in cognitive psychology about the nature of object representation. If anything in Quine makes direct contact with the topics of present interest, it is rather his discussion of the relationship between identity and quantification. In his review of Geach, Quine (1964) seems to argue that we need the notion of numerical identity just to understand basic, non-numerical quantification.14 As John Hawthorne (2003: p. 100) puts the point: ‘Without mastery of the concept of identity it is not clear how we would understand the significance of the recurrence of a variable within the scope of a quantifier’. Colin McGinn (2000: p. 11) agrees: ‘The apparatus of variable-binding, or pronominal anaphora, invokes the notion of identity’. To be a bit more precise, the suggestion seems to be that you cannot fully understand sentences of first-order logic that feature multiple occurrences of a single variable (bound by the same quantifier) unless you grasp the notion of numerical identity. If this were true, one could try to explain the emergence of identity by citing the manifest significance of quantification. Unfortunately, I have yet to find a philosophical argument for this view that does not just reduce to wishful thinking about the deliverances of cognitive psychology. Here is a reconstruction of Hawthorne’s argument, which is probably the best on offer.15 Let us grant him the preliminary assumption that understanding basic, non-numerical quantification minimally involves the ability to appreciate differences in meaning between claims like: (10) ∃x (Fx ∧ Gx) (11) ∃x ∃y (Fx ∧ Gy) Both claims require for their truth that something in the domain of quantification is F, and something is G; but only the former requires that something in the domain is F and G. In other words, while (10) and (11) both entail (12), only (10) entails (13): (12) ∃x Fx ∧ ∃y Gy (13) ∃x ∃y (Fx ∧ Gy ∧ x=y) Now for the Crucial Premise: understanding the difference in meaning between (10) and (11) inevitably involves recognizing this asymmetry in their entailments. But of course, to recognize this asymmetry, you have to comprehend (13), which overtly displays the identity sign. So, putting this all together, we secure the desired result that you cannot understand quantification unless you have the concept of identity. But Hawthorne’s Crucial Premise is false. There are other ways to register the semantic difference between (10) and (11). For instance, the latter but not the former follows from: (14) Fa ∧ Gb And since (14) does not contain the ‘=’ sign, Hawthorne’s reason for thinking we need the notion of numerical identity to understand (13) will not carry over.16 In light of this oversight, Hawthorne might switch tactics and try to argue that we need the notion of identity to recognize that (14) does not entail (10). Presumably the thought would have to be that we only see the invalidity of this inference insofar as we are aware that a and b might be distinct (that is, that some models of (14) assign different values to ‘a’ and ‘b’). Conversely, we only see the validity of the inference from (15) to (10) because we know that b is identical to b (i.e., that every model of (15) assigns both occurrences of ‘b’ the same value): (15) Fb ∧ Gb But this rejoinder proves too much. If we need the notion of identity for these purposes, then presumably we will need it even just to understand the difference in meaning between sentences like (14) and (15), which do not involve quantification. Hence our original Quinean thesis expands under pressure into the blanket view that representing an object using the same symbol more than once somehow requires a grasp of numerical identity. This broader view is not indefensible, but it is hostage to a broader range of empirical considerations.17 Four-month-old infants may not have mental representations akin to (10)-(13), but they can certainly think about objects in their immediate environments using mental symbols. The present view therefore entails that our infant subject has to understand identity just to redeploy ‘Lefty’ whenever Lefty reappears. Given our discussion in the previous section, it is entirely unclear whether this claim is true. Which is why I say that arguments like Hawthorne’s ultimately reduce to wishful thinking about the deliverances of cognitive psychology.18 Let us take stock before moving on. In recognition of the triviality intuition from §1, we have been trying to figure out why the concept of numerical identity might have developed in cognitive systems like ours. One thing we evidently use the concept to do is capture ‘qualitative’ information about property instantiation, coinstantiation, and indiscernibility. As I argued, however, this family of expressive functions does not provide a very satisfying teleological explanation of the concept’s emergence. If one could show that we also use it to acquire cardinal number concepts, represent objects, and/or understand quantification, then our motivating puzzle would be solved. But §2 and §3 challenged some of the strongest reasons to think that identity actually has such functions. Of course, the concept may serve these purposes yet, despite my sceptical worries. And I will happily admit that its qualitative functions might partially explain why we have it. But there is something else we do with the notion of numerical identity that arguably offers a better explanation. 4. The partition function A partition of a given domain is a set of subsets such that each element of the domain belongs to exactly one of the subsets. We can picture partitions as systems of cubby-holes, or grids overlying domains. As anyone with a calendar or chest of drawers can attest, partitions are extremely useful organizational tools. Instead of grappling with a sprawling, unstructured array of entities—whether seconds in the year, or clothing on the floor—we can use a partition to divide the array into a manageable number of clusters, dealing with individual entities qua elements of their respective cubbies: days or weeks; socks, shirts, sweaters, etc. Every equivalence relation induces a unique partition on any domain it is defined over: the set of maximal subsets of equivalent elements. Take geometric congruence. Our ability to think and talk about the congruence relation allows us to organize figures by shape. When variations in orientation are irrelevant for the purposes at hand, we can lighten our cognitive loads by representing individual figures en masse, as undifferentiated members of their respective equivalence classes. Some of these groupings are so salient in daily life that we have individual concepts for them, lexicalized in ordinary English as ‘circle’, ‘square’, etc. Compare Frege’s account of directions as equivalence classes of lines under the relation of being parallel. Natural languages tend to have fewer words for specific directions than they do for common shapes, perhaps because the latter are visible and tangible. But we can represent any direction at all (within some margin of error) using bodily rotation and an indexical like ‘forward’ or ‘left’. And even when we cannot find North, there is practical value in partitioning the space of possible paths. Lost in the woods, we want to keep walking in the same direction, whatever it may be. And if a bear attacks, we will do well to head off in different directions. Philosophical interest in examples like these is typically limited to the ontological (and epistemic) implications of ‘abstraction principles’ like: (16) The shape of A=the shape of B just in case A and B are congruent.19 (17) The direction of L=the direction of M just in case L and M are parallel. So I should emphasize that our own interests are quite different. In the philosophy of mathematics, biconditionals like (16) and (17) are usually invoked to argue for the view that (it is knowable a priori that) shapes or directions exist. But the present project does not actually rest on any such commitments. All we really need to take for granted are domains of objects themselves: geometric figures like A and B, linear paths through space like L and M, or whatever else. For the moral I want to draw from these examples is merely that we can benefit from clustering such things in cognition, using networks of mutually exclusive and jointly exhaustive categories. Strictly speaking, we do not have to countenance abstractions or equivalence classes ‘out there in the world’ corresponding to the cells of our mental and linguistic partitions. The point is just that value attaches to the practice of organizing objects in thought and talk via various notions of equivalence: congruent, parallel, etc. What exactly do these examples have to do with the notion of numerical identity? Obviously the ‘=’ sign appears on the left-hand sides of (16) and (17). But I just said that we are only interested in the concepts featured on the right. And even if we availed ourselves of the controversial view that these abstraction principles are knowable a priori, it would still be quite a leap to the conclusion that ‘congruent’ means having the same shape, or that ‘parallel’ means having the same direction. In fact, neo-Fregeans typically define these terms independently, to forestall the objection that they are begging the ontological question. Euclidean lines are parallel iff they never intersect. Planar figures are congruent iff they coincide up to rigid transformation: translation, rotation, and reflection. One can reconcile these definitions with the metaphysical view that being congruent (parallel) just is having the same shape (direction). But they seem to be genuine rivals to our identity-involving definitions. Nevertheless, it is a plausible empirical hypothesis that many of our words and concepts for equivalence relations are properly defined/analysed in terms of identity. However ‘congruent’ is officially introduced in geometry textbooks or essays on neologicism, it seems safe to assume that we coined the word using something like (16).20 Our concepts of shape and identity are much more basic than rigid transformation. Now, the situation with ‘parallel’ is admittedly less clear. And it would be excessive to insist that we always represent equivalence relations by invoking identity. But ordinary English provides plenty of other examples where identity quite plausibly shows up: ‘synonymous’, ‘simultaneous’, ‘siblings’, ‘cellmates’, ‘homosexual’, etc. It is hard to resist the idea that ‘synonymous’, for instance, is synonymous with ‘having the same meaning’. Indeed, we can directly observe this expressive function of identity in a range of common constructions, obviating the need to guess just how often we have actually leveraged the concept to coin a word. Instead of using ‘simultaneous’, we will often say explicitly that two events occurred at the same time. And when an equivalence relation of interest has not been given its own lexical entry in English, speakers predictably use this numerical sense of ‘same’ to pick it out. Consider for example: (18) They have the same last name. (19) Were we all born in the same decade? (20) The coffee should go on the same shelf as the creamers. Having the same last name is an equivalence relation, uniting the Hatfields against the McCoys. In each of these three examples, a different equivalence is expressed by a complex verb phrase of the same basic form: VP the same NP.21 In cases like (20), ‘as’ introduces another noun phrase to generate a larger construction corresponding to one specific cell of the partition at issue: everything that should go on the same shelf as the creamers. And we can easily pick out the membership of any equivalence class we like by simply modulating NP* in a phrase of the form: VP the same NP as NP*.22 My positive proposal is therefore that one particularly important reason we have the notion of numerical identity in natural language is to help represent equivalence relations and the partitions they induce on domains. To hone and buttress this proposal, let me now address a series of five, interrelated objections. Objection 1. Imagine a circular dinner table with five settings. Audrey, Ben, Camille, Dan, and Eloise are seated in alphabetical order, clockwise around the table, so that Audrey and Eloise are sitting next to each other. Both of the following claims sound true: (21) Audrey and Camille are sitting next to the same person. (22) Camille and Eloise are sitting next to the same person. If sitting next to the same person were an equivalence relation, it would follow that: (23) Audrey and Eloise are sitting next to the same person. But they are not. Audrey and Eloise are sitting next to each other, flanked on either side by Ben and Dan respectively. So sitting next to the same person is not transitive. We should therefore reject the proposal that phrases of the form ‘VP the same NP’ always express equivalence relations.23 I agree. Some such phrases are not even intelligible: sneeze the same Obama. The verb head has to admit of a transitive reading, NP cannot be a singular term, and we must somehow rule out semantically anomalous pairings of VP and NP. But the interesting thing about the present example is that all three of these conditions are met, and yet the phrase at issue does not express an equivalence relation. Nor is this a fringe case. There are plenty of other mundane examples. ‘Ben and Dan have the same tattoo’. That is generally true even if they each have a dozen (but only one in common). Sometimes ‘the same’ seems to function semantically like ‘a common’.24 It would be nice to know exactly when (and we will have a bit more to say about this below). But more importantly for present purposes, the mere fact that some phrases of the form ‘VP the same NP’ do not express equivalence relations is no reason to doubt that others do. We have already seen several illustrations of the equivalence function of identity, and it is easy enough to generate more. Ben and Dan have the same astrological sign, live in the same building, work on the same block, and so on.25 Objection 2. Examples like these just pick out atomic identity facts, which we have already dismissed as insignificant (in the sense that registering them lacks practical value). For example: the fact that (i) Ben and Dan are the same height, just is the fact that (ii) Ben’s height=Dan’s height, which just is the fact that (iii) H=H; for whatever height H they happen to share. The partition function of identity therefore runs afoul of our triviality intuition. Granted, one can always say ‘Ben and Dan are the same height’ in order to convey additional, qualitative information about Ben, Dan, and/or H itself (given some conversational common ground). If we have antecedently determined H to be instantiated by Ben, then the quoted claim could be used to communicate that it is instantiated by Dan as well. But the equivalence function of identity was advertised to provide a better explanation of the concept’s emergence than the qualitative functions we canvassed in §1. The letter of this second objection is accurate enough. But it is based on a misunderstanding of the value that I have claimed for our ability to represent equivalence relations. My proposal is not that equivalence statements pick out significant states of affairs (identity-involving or otherwise). It is rather that concepts of equivalence allow us to partition domains of interest. Again, the partition function of identity consists in our ability to quickly isolate the elements of any equivalence class we care to discuss by simply varying NP* in a given phrase of the form: VP the same NP as NP*. And the value of this talent lies in the common but considerable utility of organizing objects, whatever we may go on to say about them. Compare some physical partition. You do not build a bookcase just to sit around admiring the fact that Benjamin and Dante are now shelf-mates. The point of buying a new calendar is not to confirm the triviality that May 2nd and 4th once again fall in the same month. Different partitions have their own specialized uses. But at minimum, they all help us grapple with otherwise unwieldy domains by clumping individual elements together. Mental and linguistic partitions let us think and talk about medium-sized ensembles of objects, for whatever intellectual or practical purposes we may have, and then just as easily set those objects aside to focus on a mutually exclusive fragment of the same domain. Objection 3. The bulk of §2 was devoted to debunking the cardinality function of identity, but the foregoing discussion of abstraction principles like (16) and (17) would seem to reinstate it. After all, the single most prominent abstraction principle in the philosophical literature concerns cardinal numbers: (HP) The number of Fs=the number of Gs just in case the Fs and the Gs are equinumerous’ where the Fs and the Gs are any two collections of entities, and equinumerosity is the equivalence that relates two collections when their members stand in 1-1 correspondence (that is, when there is a bijection between them). Presumably it is at least as useful to organize groups of objects by size as it is to organize figures by shape, or lines by direction. So it would be disingenuous to champion the partition function of identity without acknowledging that the concept plays an important role in the representation of cardinality. Indeed, considering the enormous practical value of information about number, this particular illustration of the partition function is arguably among the most compelling we have discussed. This is too quick. To begin with, I never meant to deny that we use the notion of numerical identity to help represent cardinality. My arguments against Carey and Xu in §2 were merely supposed to shift the burden of proof in order to motivate taking a further look at other possible functions of identity. In fact, I would welcome the result that its cardinality function is actually a special case of the partition function we have been developing. But I do not think this special case should overshadow all the other important partitions we use identity to represent. Nor is it clear what (HP) really adds to the observation that we often use the phrase ‘same number’ to assimilate groups of objects. After all, we rarely use the term ‘equinumerous’ in ordinary discourse. And even if we originally acquire cardinal number concepts by representing equinumerosity (in parallel individuation), it is a further question whether our grip on that relation hinges on having the concept of identity. The idea inspired by (HP) would presumably be that infants combine the concepts of identity and number to grasp individual cardinalities like two or three. But that is a pretty bold empirical hypothesis. Carey, for one, does not think children have the concept of cardinal number (full stop) until they have extrapolated from two and three to master the meanings of number words further down their counting sequences.26 Objection 4. All this rests on the unarticulated assumption that ‘same’ expresses numerical identity when it is used in phrases of the form: VP the same NP. But the underlying structure of a sentence like ‘Ben and Dan are the same height’ does not display a sign for identity. It is really just an existential quantification over heights: (24) ∃x (Rxb ∧ Rxd)27 That is: the properties of ‘bearing the height-of relation to Ben’ and ‘bearing the height-of relation to Dan’ are coinstantiated (by some unspecified value). Addressing the first objection above, we already acknowledged that ‘the same’ sometimes seems to function semantically like ‘a common’. The present objection effectively suggests that it always works this way. Of course, coinstantiation is non-transitive. But when R in (24) determines a function of its second argument, the analysis simulates transitivity. For instance: since no one has multiple heights, the height-of relation determines a function. So we can reason successfully from ‘Audrey and Camille are the same height’ and ‘Camille and Eloise are the same height’ to ‘Audrey and Eloise are the same height’. Analysed using (24), this inference is not logically valid. Yet it is truth-preserving given the background fact that we only have one height apiece. And the same goes for examples involving shape, direction, number, etc. There is actually quite a lot to say here, but let me just offer two initial reasons to resist this coinstantiation analysis, semantic and syntactic respectively. First, I do not think the analysis always gives the right truth-conditions for the sorts of sentences we are interested in. Say P and Q are philosophers. P specializes in both metaphysics and language, while Q specializes in both language and mind. So they have an AOS in common. But it does not generally sound true to say: (25) P and Q have the same area of specialization. Or consider a couple of half-siblings. You would not say they have ‘the same parent’. Yet (24) predicts no infelicity. In examples like these, the definite article seems to impose some sort of uniqueness constraint. One might therefore be tempted to reply by simply amending (24) with a pair of clauses ensuring that each conjunct is only satisfied by a single member of the domain. Russellian uniqueness clauses would of course reintroduce identity, which our objector is trying to avoid. But GQT offers identity-free numerical quantifiers: (26) ∃x (Rxb ∧ Rxd) ∧ ∃!y Ryb ∧ ∃!z Rzd The main problem with this response is that the coinstantiation analysis was originally motivated by examples where ‘the same’ does seem to function semantically like ‘a common’. Recall our circular seating arrangement, or Ben and Dan’s shared tattoo. Insofar as (24) still looks plausible in those cases, we should resist the amendment offered by (26). Second, even when the coinstantiation analysis delivers the right truth-conditions, I doubt that it limns logical form. Here is a simple reason for scepticism. In any of the constructions we have considered, one can always replace ‘the same’ with ‘similar’ or ‘very different’, pluralize the relevant NP, and recover an equally meaningful, well-formed sentence of English. For example: Audrey and Camille are sitting next to similar people; but they have very different tastes in music. Presumably such transformations should have a negligible impact at the level of logical form. Yet it is not immediately clear how we might tweak (24) to accommodate these new sentences. Notice that (27) would pose no comparable problem, since it contains a symbol we can toggle to produce our qualitative variants: (27) ∃x ∃y (Rxb ∧ Ryd ∧ x=y)28 Again, however, our opponent wants to avoid invoking identity. Perhaps she could say that pluralizing the relevant NP introduces a second quantifier in logical form. But I think it is more plausible that there were two quantifiers all along. Consider the following exchange: (28) You and I have the same last name. (29) No we don’t. They have similar spellings, but very different pronunciations. This disagreement scans fairly smoothly. (Much more smoothly, I submit, than the discourse resulting from replacing (28) with something like ‘You and I have a last name in common’.) The pronoun ‘they’ appears to be used anaphorically in (29) to pick out the surnames of both speakers.29 If (28) only featured a single quantifier, this anaphora should not be possible. Which is not to say that (27) ultimately captures the logical form we want. For one thing, it has the same problem with (25) that frustrated (24). But I do think these syntactic considerations suggest that (27) trumps (24). Moreover, the left-hand sides of abstraction principles like (HP) provide an identity-involving alternative to (24) that reflects our syntactic and semantic observations. The idea would be to replace the existential quantifiers we have been using with a pair of definite articles. Using iota notation, we could represent the underlying structure of a sentence like ‘Ben and Dan have the same last name’ with something as simple as: (30) ιx Rxb=ιy Ryd30 In English, ‘The last name of Ben is the last name of Dan’. An obvious mark in favour of (30) is that phrases of the form ‘VP the same NP’ already feature the definite article. And with two NPs built into the syntax, we can easily explain the possibility of plural anaphora. Unlike (27), however, (30) allows us to assimilate our semantic puzzle about uniqueness to a pre-existing issue in the literature on definite descriptions. A familiar fact about definites is that they often but not always appear to impose some sort of uniqueness constraint.31 Whatever the best account of this phenomenon ultimately turns out to be, (30) at least gives us reason to hope it will help explain why ‘the same’ sometimes but not always seems to function semantically like ‘a common’. Objection 5. Even if we do in fact use the notion of numerical identity to represent equivalence relations and partition domains, we could just as easily have done these things without it. The coinstantiation paraphrase from the previous objection provides one viable alternative. Instead of saying that Ben and Dan VP the same NP, we can just say there is an NP they both VP (taking for granted that they each VP exactly one). Or to recycle the strategy from our introductory discussion of Frege’s Puzzle, we could say that Ben and Dan VP indiscernible NP(s). So the putative ‘partition’ function of identity does not seem to offer a very satisfying explanation of the concept’s emergence, for much the same reason we tabled the qualitative functions of identity at the outset. In general, the fact that we use A to do B helps explain why we have A only to the extent that it would be harder to do B without using A. The main problem with this rejoinder arises equally for both paraphrase proposals, so I will just focus on coinstantiation. Given my reply to Objection 2, a worthy surrogate for identity should provide the means to isolate arbitrary cells of a given partition, comparable to phrases of the form: VP the same NP as NP*. Notice, however, that the result of replacing ‘the same’ with ‘a common’ in this construction is ill-formed. You cannot say that Ben has a common last name as Dan, or that Audrey works in a common office as Camille.32 (Mutatis mutandis for ‘indiscernible’.) One could try to circumvent this syntactic obstacle by lambda-abstracting on ‘b’ in (24).33 In other words, one could paraphrase ‘VP the same NP as NP*’ as: __ has the property of there being an NP such that __ VP NP and NP* VP NP For instance, ‘lives in the same house as Dan’ would become something like: __ has the property of there being a house that __ lives in and Dan lives in. This predicate has the right extension. And toggling ‘Dan’ does enable us to pick out other equivalence classes in this same partition (that is, other sets of housemates). But our original, identity-involving construction obviously does the job much more efficiently. I am therefore inclined to conclude that the partition function of identity accounts for the concept’s emergence better than its qualitative functions; and indeed, better than any other expressive function we have considered in the pages above.34 5. Conclusion Teleological explanations in philosophy can sound suspiciously similar to ‘just so’ stories. And even readers open, in principle, to a teleological explanation of identity representation might think I have given certain functions of the concept short shrift. But however this may be, I hope you will at least agree that one important thing we do with identity is organize objects by partitioning domains. Or more modestly yet, given the likelihood that my list of possible objections is incomplete, and that my replies will inspire their own rejoinders: I hope the present essay has at minimum persuaded you that the (putative) partition function of identity deserves more attention than it has received to date.35 Footnotes 1 Or if ‘same’ seems pleonastic here: ‘Is that the same police car following us, or a new one?’ Any given example can of course be challenged. The mere fact that a sentence features the word ‘same’ (or ‘identical’, or ‘is’) is no guarantee that its semantic analysis actually involves numerical identity. But there seem to be enough examples to justify the claim that we use the concept of identity fairly frequently. 2 Let me clarify that the riddle is not about the monadic concept/property of self-identity. Granted, the distribution of this property also seems completely trivial and uninteresting: everything instantiates it. But we do not appear to use the concept of self-identity very often in ordinary thought or talk. Moreover, as readers will be able to verify, appealing to self-identity will not help us solve the puzzle of present interest. 3 As, for example, Salmon (1987) points out: when A=B, the fact that A=B just is the fact that A=A. I say ‘almost universally respected’ because Gallois (2005) tries to safeguard the possibility that identity facts are nevertheless non-trivial. His strategy is to explain the felt triviality of identity statements featuring the same name twice by appeal to a pragmatic ‘identity convention’ effectively coordinating repeated uses of a name within a discourse. But Gallois does not actually provide an argument against the triviality intuition. 4 Descriptivists and Millians typically disagree about whether the mechanisms of conveyance are semantic or pragmatic; but this choice point will not matter for our purposes. 5 Clear enough, but not entirely beyond dispute. It has occasionally been denied that examples like (0) actually feature the ‘is’ of identity. But we can afford to ignore this issue. If identity does not have any qualitative functions, our motivating riddle is that much more acute. 6 This claim holds up even under the supposition of descriptivism about proper names and Russellianism about descriptions. Any two names used in a single sentence or discourse will be replaced by distinct variables on this analysis. So we still need some version of Leibniz’ Law to ensure that the value of one variable satisfies the descriptive material semantically associated with the other. To wit: if x and y are assigned the same object, then anything true of the value of x is true of the value of y, and vice versa. 7 A variant of the explanation from qualitative functions would be to say that identity statements serve as instructions to combine ‘mental files’. Given (0), it is cognitively inefficient to keep separate books for Batman and Bruce Wayne (assuming the books do not contain information about other people’s beliefs). But the same holds given (4). So the question remains: why do we not just use the notion of qualitative identity to issue such instructions? 8 On the Russellian picture, ‘some’ is categorematic while ‘one’ is syncategorematic. And ‘most’ evades analysis altogether, inasmuch as it cannot be expressed in first-order quantificational logic. A related objection is that the quantificational regimentations of claims concerning larger cardinalities (e.g., Cruella wants 101 Dalmatians) have implausibly high degrees of syntactic complexity. 9 Chapter 8 presents her positive picture in much more detail, alongside critical discussion of the AMR proposal I mention below. Chapter 4 introduces both of these core cognition systems, and Chapter 7 explains how parallel individuation gets ‘enriched’ by set-based quantification. See LeCorre and Carey (2007) as well. 10 Carey (2009), Chapter 8, ‘proposal 1’. She cites a number of psychologists associated with this hypothesis. See for example Gallistel and Gelman (2000). 11 See also Carey (2009), Chapter 3. 12 Notice how awkward and artificial it would be to interject identity here. There is only one numerically identical object on the scene? Nothing is numerically identical simpliciter. And everything is numerically identical to itself. 13 Let me emphasize that Carey can jettison the idea that representing objects requires identity without compromising any of the larger plot points in her book; or even the thesis of her paper with Xu (i.e., that we have the concept object before we develop more discriminating sortal concepts). 14 Kai Wehmeier (2017) suggests that this traditional interpretation of Quine is actually mistaken. 15 Humberstone and Townsend (1994), which predates Hawthorne’s paper, contains persuasive responses to a few other possible arguments that quantification somehow involves or presupposes identity. See especially Section II. 16 Nor is the availability of this response just a fortuitous artefact of his chosen example. The recipe that generated (11) was to take a sentence of first-order logic without identity like (10), which contains a quantifier binding more than one occurrence of a single variable, and replace each subsequent occurrence with a unique variable to be bound by a new quantifier (with the same scope as the original). Because this recipe never introduces an identity sign, it is easy to verify that its input is always stronger than its output in first-order logic without identity. So we can always find identity-free sentences to illustrate the semantic difference between them. See Wehmeier (2012) for more on this score. 17 Humberstone and Townsend (1994: pp. 258-9) mount a more philosophical argument against the broader view. Assuming we do not need identity to understand sentences like (14), they appeal to Evans’ Generality Constraint to argue that we do not need identity to understand (15) either. In the present setting, however, our opponent will not accept their first premise. She will insist that fully understanding (14) requires recognizing that ‘a’ and ‘b’ need not be assigned the same value. 18 To make matters worse, notice that the present view is actually much stronger than the mere empirical hypothesis that infants in fact use identity for these purposes. Writers like Hawthorne and McGinn seem to think mastery of identity is the only thing that could conceivably explain our proficiency with quantification; and thus, taking up the rejoinder I have offered, the only thing that could explain how we understand repeated uses of a name. 19 This should really be ‘similar’ rather than ‘congruent’, since figures of different sizes can have the same shape. But I will continue to use ‘congruent’ in order to avoid evoking the generic notion of qualitative similarity. 20 So our ordinary talk of congruence may well commit us to the existence of shapes. But this does not undermine my earlier point that we, the theorists, need not countenance shapes in order to articulate the equivalence function of identity. For we can catalogue the ontological commitments of ordinary language without embracing them ourselves. 21 Geach (1962) was preoccupied with cases where the verb is ‘to be’, but for reasons that are not relevant here. 22 We can sometimes make do without identity, using a possessive transformation instead: VP NP*s NP. For example, ‘lives in the same building as Sam’ would become ‘lives in Sam’s building’. But possessives are notoriously ambiguous. Sam’s building might not be the building he lives in. Depending on the larger context, it could be the building he owns, the building where he works, the one he would most like to climb, etc. So the phrase ‘lives in Sam’s building’ does not stably correspond to the relevant equivalence class. Using the notion of identity coerces the intended reading. 23 One might be tempted to reply on my behalf that there is no single context in which (21) and (22) are both true, and hence no pressure to accept (23). Perhaps an utterance of (21) establishes Ben as the denotation of ‘the person sitting next to Camille’ in context, so that a subsequent utterance of (22) would either be infelicitous or require shifting to a context where ‘the person sitting next to Camille’ now denotes Dan. But neither option seems particularly plausible to me. 24 Interestingly enough, however, we cannot replace ‘the’ with ‘a’ in these constructions. Other determiners are equally bad. We cannot say ‘two same tattoos’, though we can say ‘the same two’ or ‘two of the same’. See Humberstone and Townsend (1994: p. 255) for further examples and discussion. 25 We sometimes use ‘exactly’ to ensure that the relation we are picking out is an equivalence. Ben and Dan like exactly the same bands, commute on exactly the same freeways, etc. 26 One might reply by setting (HP) aside and defining equinumerosity as neologicists do, without recourse to the notion of number. Interestingly enough, the standard definition does involve several occurrences of identity: in the definition of a function, total-definition, injection, and surjection. But it is far from clear that infants have these concepts. And in any case, we can define them all in ZFC without using identity, by simply construing the axiom of extensionality as an eliminative definition of set-identity. 27 Someone like Hawthorne would argue that you cannot understand (24) without recognizing that it is equivalent to the identity-involving: ∃x ∃y (Rxb ∧ Ryd ∧ x=y). So he would insist that we still need the notion of numerical identity to comprehend the original English sentence, even if its syntax is identity-free. But we cannot avail ourselves of this response without waiving our concerns about the Quinean view of variable binding from §3. 28 This syntax came up in the previous note. But the way Hawthorne would use (27) is importantly different from how I am using it here. For our purposes, the relevant hypothesis is that (27) rather than (24) gives the logical form of the sentence at issue, not that we need to appreciate their equivalence in order to understand (24). 29 One can of course construct a version of the case where ‘they’ is used demonstratively instead. (Perhaps the speakers are wearing name tags.) But the point remains that the anaphoric reading seems available in other versions. 30 For plural NP, we will want to use plural quantification: ιxx (Rxxb)=ιyy (Ryyd). For example, the pastries Ben loves are (identical to) the pastries Dan loves. Whereas proponents of (24) can swap their existential quantifier for a universal: ∀x (Rxb iff Rxd). Every pastry is such that Ben loves it just in case Dan loves it. It is unclear to me whether this observation cuts either way. 31 See, for example, Abbott (2006), and additional references therein. 32 You can say that Ben and Dan have a common last name, but only at the risk of misinterpretation. You can say that Ben has a last name in common with Dan; but you cannot say that Audrey works in an office in common with Camille. You can say they share an office, but that is not quite the same thing as working in the same office. These issues arise across a wide range of cases. For any given case, one can usually find some serviceable coinstantiation paraphrase. But there does not seem to be a single paraphrase strategy that works well across the board. 33 To see the parallel point for indiscernibility, use (27) instead; reading ‘=’ as qualitative identity. 34 A fuller account would also explain why (a) English syntax did not develop to permit phrases of the form ‘VP a common NP as NP*’, and (b) English semantics did not develop in line with the coinstantiation analysis from the previous objection. Are these just linguistic quirks, or is there some deeper reason identity emerged to serve the partition function? Cross-linguistic data would presumably be helpful here. But now we are really running up against the limits of my competency. I hope readers better equipped will be interested to pursue these further questions. 35 Thanks first and foremost to Susan Carey and Kai Wehmeier, without whom not. To audiences at Stanford, Irvine, and Dartmouth. And to JC Beall, Liz Camp, Herman Cappelen, Nate Charlow, Mala Chatterjee, Mark Crimmins, Sam Cumming, Kenny Easwaran, Mark Greenberg, David Hills, Ben Holguin, Justin Khoo, Boris Kment, Krista Lawlor, Sarah Lawsky, Karen Lewis, Penelope Maddy, Eliot Michaelson, David Plunkett, Mark Richard, Gideon Rosen, Nathan Salmon, Sam Shpall, Ted Sider, Brian Skyrms, Rachel Sterken, Tim Sundell, and especially Christie Thomas, who gave me some very detailed comments on an earlier draft. This paper has been in progress for a long time, so I apologize for the likelihood that I have left people out. I would also like to acknowledge the two referees, for their uncommonly helpful feedback. References Abbott Barbara 2006, ‘Definite and Indefinite’, in Brown ed., Encyclopedia of Language and Linguistics , Elsevier. Baillargeon Renee, Graber Marcia 1987, ‘Where’s the Rabbit? 5.5-Month-Old Infants’ Representation of the Height of a Hidden Object’, Cognitive Development  ( 2): 375– 92. Carey Susan 2009, The Origin of Concepts , Oxford. Google Scholar CrossRef Search ADS   Carey Susan, Xu Fei 1996, ‘Infants’ Metaphysics: The Case of Numerical Identity’, Cognitive Psychology  ( 30): 111– 53. Gallistel Charles, Gelman Rochel 2000, ‘Non-Verbal Numerical Cognition: From Reals to Integers’, Trends in Cognitive Sciences  ( 4): 59– 65. Gallois André 2005, ‘The Simplicity of Identity’, The Journal of Philosophy  ( 102): 273– 302. Geach Peter 1962, Reference and Generality: An Examination of Some Medieval and Modern Theories , Cornell. Hawthorne John 2003, ‘Identity’, in Loux, Zimmerman eds., The Oxford Handbook of Metaphysics , Oxford. Hofweber Thomas 2005, ‘Number Determiners, Numbers, and Arithmetic’, The Philosophical Review  ( 114): 179– 225. Humberstone Lloyd, Townsend Aubrey 1994, ‘Co-Instantiation and Identity’, Philosophical Studies  ( 74): 243– 72. LeCorre Mathieu, Carey Susan 2007, ‘One, Two, Three, Four, Nothing More: An Investigation of the Conceptual Sources of the Verbal Counting Principles’, Cognition  ( 105): 395– 438. McGinn Colin 2000, Logical Properties: Identity, Existence, Predication, Necessity, Truth , Oxford. Quine Willard 1964, ‘Review of P. T. Geach, Reference and Generality’, The Philosophical Review  ( 73): 100– 4. Salmon Nathan 1987, ‘The Fact that x = y’, Philosophia  ( 17): 517– 8. Spelke Elizabeth 1988, ‘The Origins of Physical Knowledge’, in Weiskrantz ed., Thought Without Language , Clarendon. Spelke Elizabeth, Kestenbaum Roberta 1986, ‘Les Origines du Concept d’Objet’, Psychologie Francaise  ( 31): 67– 72. Wehmeier Kai 2017, ‘Identity and Quantification’, Philosophical Studies  ( 174): 759– 70. Wehmeier Kai 2012, ‘How to Live Without Identity—And Why’, Australasian Journal of Philosophy  ( 90): 761– 77. © Burgess 2017
TI - The Things We Do with Identity
JO - Mind
DO - 10.1093/mind/fzw031
DA - 2018-01-01
UR - https://www.deepdyve.com/lp/oxford-university-press/the-things-we-do-with-identity-ah1jXT5J0l
SP - 105
EP - 128
VL - 127
IS - 505
DP - DeepDyve
ER -