Rigidity and predicates (Arthur Sullivan, draft May 19/03)

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Rigidity and predicates (Arthur Sullivan, draft May 19/03)

1. the problem

‘Rigid designation’ is Kripke’s name for a concept that has been in the air at least since the development of quantified modal logics (although, in a sense, the concept has been around as long as the notion of a de re modal property1). The concept of rigidity might have been baptized by Smullyan (1948), as it affords a neat way to state his refutation of Quine’s (1947) argument against the intelligibility of quantified modal logics.2 Rigidity, unbaptized, is very much there in Kripke’s (1959, 1963) semantics for modal logic; and it lies just beneath the surface of much other semantic and logical work in the 1950s and 1960s. Kaplan (1968: 379) perhaps comes closest to explicitly defining the concept, in his search for a kind of designator whose “reference is freed from empirical vicissitudes”. However, like Smullyan, Kaplan’s interests are more narrow, his goals less comprehensive, than Kripke’s. Kripke first explicitly introduces the notion in the course of developing a wide range of arguments about reference and content.

Two seminal conclusions for which Kripke (1971, 1972) argues are that proper names are rigid designators, and that there are some deep semantic affinities between proper names and various sorts of predicates or kind terms. However, even though he does, at places, explicitly attribute rigidity to certain kind terms,3 Kripke nowhere gives a definition of rigidity that applies to kind terms. This presents a challenge: Precisely which predicates or kind terms ought to be classified as rigid designators? More fundamentally: What should we take the criterion for rigidity to be, for predicates or kind terms? As the notion of rigidity is at the core of the new theory of reference, and of closely associated, deep criticisms of traditional ideas about language, these questions have been thought to have relevance to various sorts of points in the philosophy of language and mind (and, more broadly, in metaphysics and epistemology).4 There exists a considerable sub-literature, stretching back over 30 years, addressed to this issue of which predicates or kind terms should be counted as rigid designators.5

One relevant question concerns what Kripke himself had in mind, in extending the notion of rigidity to certain predicates or kind terms. On this question, Soames (2002: 263) says:

… in Naming and Necessity Kripke simply did not have a clear conception of what it would mean to characterize anything other than a singular term as rigid. It is clear that he was impressed by the semantic similarities between proper names and natural kind terms of many different grammatical and semantic categories, including natural kind predicates. He appears to have sometimes used the term ‘rigid’ in a rather loose and imprecise way, to indicate, without precisely specifying, the similarities these predicates bear to proper names.

Soames reports that, in informal discussion, Kripke has assented to this diagnosis. Given this, I consider that exegetical question closed. I focus on conceptual questions, about the substance and theoretical worth of the concept of rigidity, as it pertains to predicates or kind terms.

Two immediately evident options for extending the notion of rigidity to predicates or kind terms are, in rough outline: (a) take the designata of these terms to be their extension, and so classify a term as rigid if it designates the same extension in every possible world, and (b) take the designata of these terms to be their intension, and so classify a term as rigid if it designates the same intension in every possible world. The problem with (a) is that almost all predicates and kind terms (including especially all natural kind terms) turn out nonrigid. That is, there might have been more (or less, or different) tigers than there actually are, and so ‘tiger’ does not designate the same extension across counterfactual situations. Surely, the same is true of ‘gold’, ‘water’, and ‘pain’, of virtually any term that designates contingently existing concrete stuff and things.6 There is also a very quick objection to (b), which many find compelling—i.e., on this approach, rigidity seems to become trivial, as almost all predicates or kind terms (including the likes of ‘bachelor’, ‘hunter’, and ‘pencil’) come out rigid. The reason for this is expressed by Kripke (1972: 77) as follows: “One doesn’t say that ‘2+2=4’ is contingent because people might have spoken a language in which ‘2+2=4’ meant that seven is even”. The link between an expression and its intension must be held constant across counterfactual situations, if we are to study the modal properties of the content of our thought and talk.

So, although views roughly along the lines of (b) have been defended,7 many take this triviality objection to be at least a serious obstacle that no one has yet shown the way round, and perhaps even a conclusive refutation of the (b)-type approach.8 Within this sub-literature, a prevalent desideratum is a general definition of rigidity that counts natural kind terms in and various other sorts of terms out.9 More generally, the aim is a partitioning of the set of predicates and kind terms into rigid vs. nonrigid, somewhere in between the too conservative (a) and the too liberal (b), that marks off a principled proper subset as the rigid designators; invariably, the target set are terms that are in some ways tied up with other aspects of the arguments against descriptivism and for the new theory of reference.10

Against this line of thought, I will defend a version of option (b). I will present two sorts of argument for it. The first is positive evidence: in Section 2 I argue that the original substance and intent the concept of rigidity best fits with this particular way of extending the notion of predicates or kind terms. The second is negative evidence: in Section 3 I argue that many of the things that get brought up in developing the triviality objection involve mistakes, involve confusing rigidity with other notions with which it is historically connected but conceptually quite distinct. (Some objections are addressed in the Appendices.) There is a very clear sense in which most ordinary kind terms (including the likes of ‘bachelor’, ‘hunter’, and ‘pencil’) are rigid designators—i.e., from a modal point of view, their semantics is relevantly similar to ‘Benjamin Franklin’ and relevantly different from ‘the inventor of bifocals’. I will argue that it is a mistake to think that this should render the notion of rigidity suspect or worthless.

As a preliminary, I should sharpen (at least a bit) the version of option (b) I’ll defend. First I’ll simplify the disjunction ‘predicate or kind term’ to ‘kind term’. A kind term can be characterized (à la Frege) as a function from arguments to truth-values—i.e., kind terms are true of, or satisfied by, some number of instances (and, typically, true of, or satisfied by, different instances in different contexts). All monadic predicates include at least one kind term; (in a related, nominal, form) kind terms can also occur in the subject position.11 (Intensionally speaking) kind terms are semantically associated with a feature or characteristic that stays constant throughout changes of the term’s extension. (Extensionally speaking) this feature or characteristic provides a means by which a domain of individuals is (more or less roughly) divided into sets that could be designated as ‘that kind of thing’, ‘not that kind of thing’, and (where appropriate) ‘neither clearly that kind of thing nor clearly not that kind of thing’. In short, I’ll say that kind terms designate kinds12; I’ll defend the view that if an expression designates the same kind in all possible worlds, then it should be classified as a rigid designator.

Note that, thus far, this use of ‘kind’ is metaphysically neutral. This is as it should be, as the question of what kinds really are (or whether such entities as the meanings of kind terms exist, somewhere out there) is orthogonal to the question of rigidity. Rigidity is a semantic claim about a designator, not a metaphysical claim about the essence of what it designates.13 Just as the rigidity of proper names is compatible with a wide variety of views on individual essence, the rigidity of kind terms is compatible with a wide variety of views on the metaphysics of kinds. (That is, in the case of names: Haecceitists and anti-haecceitists can agree that names are rigid designators, they’d just have rather different views on precisely what names track from world to world—i.e., non-qualitative essences, or (something like) qualitative roles. Similarly, the view that names are nonrigid designators is not closed off by any stance on the metaphysics of essence—for instance, one could espouse a robust account of non-qualitative individual essences, but still favor a nonrigid descriptivism when it comes to the descriptive semantic question of how speakers use proper names.14 Just as there is nothing incoherent about an anti-haecceitist who thinks that names are rigid designators, or a haecceitist who thinks that names are nonrigid, in answer to the descriptive semantic question of how speakers use kind terms, a nominalist might favor rigid designation, and a Platonist might hold that kind terms are (sometimes, or usually, or always) used nonrigidly. Platonism, conceptualism, and nominalism about kinds are one and all consistent with the view that kind terms are rigid designators (and with the view that kind terms are nonrigid designators); that dispute concerns the mind- and language-independent metaphysical status of what speakers use the terms to designate.15) So, the existence, or nature, of essences or kinds is not the issue here. Rather, rigidity is a question about what is relevant to truth-conditions across different contexts of evaluation. Therefore, for our purposes here, we should leave open a wide range of options about the metaphysics of kinds.

2. the positive case

My thesis is that all semantically unstructured kind terms are rigid designators, and that any intuitions to the contrary are accommodated by a proper understanding of Russell’s (1905) referring/denoting distinction, as it pertains to kind terms. That is, just as in the case of individuals, kinds can be designated in two ways: they can be denoted (e.g., ‘Dan’s favorite color’) and they can be referred to (e.g., ‘yellow’). Denoting expressions can either be rigid (e.g., ‘the element with atomic number 79’) or nonrigid (e.g., ‘the element most highly prized by local jewelers’). However, any expression that refers to a kind designates it rigidly, and all semantically unstructured kind terms refer to (as opposed to denote) kinds.

According to Russell (1905), as distinct from reference (which is a conventional or stipulative relation between an expression and what it is used to designate—examples might include ‘I’, ‘this’, ‘nine’, ‘Jones’, ‘gold’), denoting is the connection between certain semantically structured expressions (‘a squirrel I saw yesterday’, ‘my least favorite Quaker’, and so on) and that, if anything, which uniquely satisfies the compositionally determined condition they express.16 These two semantic relations correspond to different sorts of propositions. Call a proposition ‘object-dependent’ if its subject-term refers to a specific individual, and ‘object-independent’ if its subject-term denotes whatever might happen to satisfy its linguistic meaning. Consider:

  1. Nine is odd.

  2. The number of planets is odd.

The subject-term in (1) is a semantically unstructured referring expression; (1) is object-dependent. The subject-term in (2) is a semantically structured denoting expression; (2) is object-independent.17

An expression is a nonrigid designator only if (holding linguistic conventions fixed) what it designates will vary relative to accidental changes throughout contexts of evaluation. No semantically unstructured term—be it ‘gold’, ‘bachelor’, ‘tiger’, or ‘hunter’—could meet this condition. Examples of nonrigid designators include ‘the number of planets’, ‘our most precious resource’, ‘Dan’s favorite color’. It is no accident that these are all semantically structured denoting expressions, for that is a necessary condition for nonrigidity. The question of rigidity—i.e., take a given token of an expression, survey a number of contexts of evaluation, and identify what that token designates, at each stop—does not get a foothold in the case of (semantic) reference, because the (stipulative, conventional) relation of (semantic) reference is indifferent to the sort of accidental changes that distinguish different contexts of evaluation. A given token of an expression could designate different things in different possible worlds only if what it designates at a world is determined by contingent, qualitative facts; this condition can only be met by denoting expressions, which express a (compositionally determined) condition that could be satisfied by different individuals or kinds in different worlds.

So, regardless of whether it is individuals or kinds under discussion, the question of nonrigidity only really comes up in the case of a denoting expression. Further, the connection between ‘bachelor’ and bachelor, akin to the connection between ‘gold’ and gold, is stipulative, conventional (semantic) reference, not satisfactional denoting. There is no mechanism that goes from context to context, in the case of our ‘bachelor’ thoughts and utterances, and seeks out which kind in that context is relevant to the truth-conditions. The semantic glue that holds ‘bachelor’ to bachelor stays constant throughout contexts of evaluation. So, as compared with the different kinds of semantic link between, say, the rigid and nonrigid designators ‘Benjamin Franklin’ and ‘the inventor of bifocals’ and their common designatum, from a modal point of view, the semantic links between the expressions ‘gold’ and ‘bachelor’ and that which they designate is on a par.18

It should prove instructive to draw a distinction between kind-dependent and kind-independent propositions. Call a proposition ‘kind-dependent’ if and only if the subject-term refers to a specific kind, and ‘kind-independent’ if and only if the subject-term denotes whatever kind might happen to satisfy its linguistic meaning. Thus, compare the following:

(3) Honeybees might have been birds.

(4) The species typically farmed for honey might have been birds.

The designator in the subject-position in (3) specifies a particular kind, whereas in (4) the designator expresses a complex condition, satisfied by different kinds in different contexts. Clearly, these differ significantly in possible-world truth-condition.19 Consider further:

  1. Bachelors should have been our target market.

  2. The set of men who are of the same marital status as Prince William in 2002 should have been our target market.

Exactly the same contrast holds between the possible-world truth-conditions of (5) and (6), for exactly the same reason: i.e., (5) is kind-dependent, while (6) is kind-independent.

So, denoting expressions can be nonrigid, as the relevant compositionally determined condition might be satisfied by different individuals or kinds in different contexts. However, not so for expressions whose job is to refer to (as opposed to denote) a kind (including terms that refer to manifest descriptive features, to contingent accidental characteristics, or to artificially-imposed functions). You cannot rigidly designate an individual or a kind via its contingent accidental properties, but you can rigidly refer to manifest descriptive (contingent, accidental, artificial, …) features or characteristics, and that’s what unstructured kind terms typically do.

In the case of singular terms, whose role is to single out an individual as the subject of discourse, it is the extension that matters for truth-conditions (and hence for the question of rigidity). This is the grounds for why it is plausible—although contentious—to identify a singular term’s contribution to propositional content with its referent. The role of a kind term, in contrast, is to categorize or classify, not simply to identify or single out. Notoriously, extension and truth-conditions are separate questions, when it comes to categorizing or classifying.20 Given the semantic role of a kind term, its contribution to propositional content must remain constant throughout changes of extension (over time and throughout possible worlds). (Otherwise it is not a kind term, but rather a proper name for a set.) It follows that, for kind terms, it is the intension that matters for truth-conditions (and hence for the question of rigidity). A singular term is rigid iff one and the same individual is relevant to the possible-worlds truth-condition of propositions expressed by sentences containing it; a kind term is rigid iff one and the same kind is relevant to the possible-worlds truth-condition of propositions expressed by sentences containing it.

Thus, in accord with Kaplan (1977) and Recanati (1993), I hold that all referring expressions are rigid designators, that the modal phenomenon of rigidity falls right out of the semantic phenomenon of referring to and expressing object-dependent information about specific individuals. To this I add that, with respect to the origin and intent of the distinction between rigid vs. nonrigid designators, semantically unstructured kind terms are relevantly similar to referring expressions, and relevantly different from denoting expressions. Hence, the modal phenomenon of rigidity falls right out of the semantic phenomenon of referring to and expressing kind-dependent information about specific kinds.

What I am calling the positive case for this conception of rigidity boils down to this: semantic structure is a necessary (but not sufficient) condition for nonrigidity; (semantic) reference (as opposed to denoting) is a sufficient (but not necessary) condition for rigidity. Therefore, a term’s lacking semantic structure implies that it cannot satisfy the necessary condition for nonrigidity; and a term’s being a referring (as opposed to denoting) expression implies that it is a rigid designator. Both lines of thought hold true of ‘bachelor’, ‘philosopher’, ‘hunter’, and ‘pencil’, just as surely as of ‘gold’ and ‘tiger’. For these reasons, all semantically unstructured kind terms should be classified as rigid designators.

3. the negative case

The negative case for this conception of rigidity concerns irrelevant and troublesome turns taken by attempts to satisfy what I called the prevalent desideratum—i.e., a definition of rigidity that applies exclusively to some favored subset of unstructured kind terms. Here my contention is that many moves in the debates over how the notion of rigidity should be extended to kind terms have embodied mistakes about the concept, failures to distinguish it from other things to which it is historically, but not intrinsically or conceptually, connected. The concept of rigid designation grew out of work in quantified modal logic. Subsequently, Kripke uses the notion to motivate a number of theses about reference and content; subsequent to that, philosophers influenced by Kripke have continued to try to put rigidity to further work. In some cases, to suit these subsequent ends, things have become associated with the concept or rigidity that are, at least, distinct from, and, at worst, in tension with, the original conception. I will cite two specific instances of this general trend. The first involves bringing potentially counter-productive metaphysical concerns to bear on the question of rigidity, the second involves clouding the issue with epistemological distractions.

First, under the influence of the idea that, on pain of triviality, a general definition of rigidity should count natural kind terms in and the likes of ‘bachelor’ or ‘philosopher’ out, Soames (2002: Chapter 9) is tempted toward essentialism in characterizing the desired sense of ‘rigid’. (Cf. p.251ff, where Soames formulates the hypothesis that “a predicate is rigid if and only if it is an essentialist predicate”, and investigates some more precise characterizations of what it means to be an essentialist predicate.) Soames does come to reject this line of thought,21 but nonetheless I think that his very temptation in this direction is illustrative of a widespread and potentially grave mistake. Kripke—along with Smullyan, Marcus, Kaplan, and others—had to do much hard work to prove that rigidity is itself not a metaphysical thesis at all, let alone an objectionable one, before it and its ilk (i.e., the terms of quantified modal logic) were commonly acceptable philosophical parlance. The point that these terms are, as Kaplan (1986: 265) puts it, “prior to the acceptance (or rejection) of essentialism, not tantamount to it”, was crucial and hard won. To link rigidity with essentialism—indeed, with any particular metaphysical doctrine or stance—risks losing some of this hard won ground, risks adding currency to vague and mistaken objections that rigid designation is an obscure essentialist doctrine. Rigidity is a claim about the semantics of a designator, not about the essence of what is designated.

Second, Schwartz (2002: 270-1) raises the challenge: If rigidity is so rampant, then why isn’t the necessary a posteriori more prevalent? In the course of developing an objection to an example of LaPorte’s, Schwartz compares the following:

(7) Water is H2O.

(8) The honeybee is apis mellifera.

He says (2002: 270):

In order for this [view—i.e., a kind term is rigid iff it designates the same kind in every possible world] to be effective [8] must be not only necessarily true but a posteriori.

Two preliminary comments: First, the necessary a posteriori status of (7) is far from uncontroversial, even among Kripke’s allies and followers.22 Second, I do not think that (8) is the proper analogue to (7). The relevant sort of statement to compare with (7) is not (8) but rather something essence-identifying, along these lines:

(9) The honeybee is S(n). (where S(n) is a bio-chemical formula that singles out a species via its DNA sequence)

Given that ‘honeybee’ is used as a natural kind term, then, quite plausibly, (9) is as strong a candidate as (7) for necessary a posteriori status.

Regardless of those fussy points, though, there is a deep mistake underlying this line of thought. Directly contrary to Schwartz’ contention, (this account of) rigidity, per se, says absolutely nothing about epistemic status. If all of the relevant terms are rigid, then (7)-(9) all three express necessary truths; that is compatible with a wide range of positions regarding their respective epistemic statuses. Many premises are required, to get from the rigidity of the designators to the a posteriority of any one of these, let alone of all of them. So, even if there is this epistemic difference between some of the likes of (7)-(9), this does not entail anything having to do with which designators are rigid. With respect to the question of rigidity, a posteriority is a red herring.23

One motivation for some of these desiderata for a general definition of rigidity is a (latent) intuition along the following lines: The thesis that proper names are rigid designators was at the forefront of a revolution in the philosophy of language; and so something similarly monumental is likely to come from rigidity in the case of kind terms. Such expectations are bound to go unfulfilled.24 The reason why the notion of rigidity initially caused such a splash is that some of the things Frege and Russell said about certain singular terms entail that they are nonrigid (crucially, because the relevant terms are denoting expressions in disguise—their meaning, and hence their possible-worlds extension, is given via a semantically structured denoting expression); so that if these singular terms are rigid, then some pillars of the seminal views of Frege or Russell must be altered or rejected.25 However, for the reasons given, rigidity—i.e., relative variance in possible-worlds truth-conditions—does not provide a promising way to characterize any interesting subset of the set of semantically unstructured kind terms. So while something monumental may well fall out of a definitive understanding of the semantics of natural kind terms, there is good reason to doubt that it will have anything to do with rigidity.

Another thing motivating these prevalent desiderata is the idea that the concept of rigidity must do some specific sorts of work, in order to earn its keep.26 However, the concept of rigidity earned its credentials as a significant and worthwhile notion long before the onset of these subsequent debates. If the notion of rigidity cannot solve every subsequent problem to which Kripke or his followers have tried to apply it, this would not render it incoherent or useless. This would point to problems for the subsequent theses, not with the notion of rigidity. In particular, rigidity was not introduced to explain anything about mind-independent essences, epistemic status, or differences between the likes of ‘gold’ and ‘bachelor’.

Contrary to the triviality objection, the thesis that all unstructured kind terms are rigid designators affords a sense of historical continuity, of the steady progress in philosophical logic, from Mill to Frege and Russell through to Kripke and Kaplan. Each of these thinkers has made a significant, convergent contribution toward an understanding of the distinctive semantic properties of unstructured referring expressions. The thesis of rigidity did not come from out of nowhere, but is rather a consequence of Russell’s (1905) notion of referring (as distinct from denoting) once we take modal matters seriously.27 That all unstructured terms are rigid does not trivialize Kripke’s (brilliant, seminal) work—it is also trivial, in a sense, that water is H2O, or that the rising sun is not new every morning, but always the same. Indeed, the thesis of rigidity is in some ways akin to Kripke’s examples of necessary a posteriori statements: it is an important and hard-won discovery about the modal status of the terms of our thought and talk, by no means self-evident, but nonetheless constitutive of the way we use our terms. A language in which ordinary terms were nonrigid would be rather strange indeed.

Appendix A: more on kinds

Schwartz (2002) raises a number of challenges to LaPorte’s (2000) general definition of rigidity. As my view is close to LaPorte’s, some of these challenges apply to it. I think that many of them are met, directly or indirectly, in the body of the paper; however, some are not. In this Appendix I consider two questions raised by Schwartz: the first concerns the metaphysics of kinds, the second concerns the possible-worlds truth-conditions of the sorts of kind terms that many have been tempted to classify as nonrigid.

First, Schwartz (2002: 267) argues as follows:

Unfortunately LaPorte is only able to float his preferred solution [i.e., a kind term is rigid iff it designates the same kind in every possible world] on some dubious metaphysical claims. LaPorte’s claim that ‘the insect species typically farmed for honey’ picks out different species in different worlds whereas ‘the honeybee’ picks out the same species in every world is based on his position that the honeybee is a genuine kind whereas the insect species typically farmed for honey is not. … [This] account will be trivialized if we allow what he calls the abstruse kind insect species that is typically farmed for honey.

According to Schwartz, LaPorte begs a key metaphysical question by assuming that, for example, there is a kind to which ‘honeybee’ refers, while there is no genuine kind referred to by ‘the insect species typically farmed for honey’. On what grounds, asks Schwartz, can we deny that the latter rigidly designates a genuine kind (i.e., the kind of insect that is typically farmed for honey, which, although actually co-extensive with ‘honeybee’, has a different extension across possible worlds)? If we can find no such grounds, does this not undermine the view that these latter sorts of designators are nonrigid? The notion of rigidity would thus be thoroughly trivial, as even terms that denote kinds (including the likes of ‘Dan’s favorite color’, ‘our most precious resource’, and so on) would come out as rigid designators of abstruse kinds.

Laporte (2000: 300-2) sketches one line of response to this worry. In brief outline, exactly the same problem can be posed for singular designators—i.e., one could argue that such terms as ‘the Prime Minister of Canada’ or ‘the inventor of bifocals’ are (contra Kripke) rigid designators of offices, or roles, which may be constituted by different individuals in different contexts.28 On this view, all singular designators—whether referring or denoting—turn out rigid, the only difference being that what otherwise seem to be nonrigid denoting expressions turn out to rigidly designate abstract offices or roles. So, as LaPorte explains, even if this objection does pose a problem, it is not specifically a problem for kind terms, for “the metaphysical issue about kinds behind the objection pertains as well to concrete objects” (2000: 300).

Even further, though, I do not see that this is much of a problem anyway. This objection strikes me as another instance in a long line of errors caused by being insufficiently attentive to Russell’s distinction. Denoting and referring are very different semantic phenomena. True, denoting expressions, like all other expressions, have one constant intension, regardless of which context of evaluation is under consideration. However, that intension is decidedly not a kind—i.e., a function that takes arguments to truth-values—but is rather a compositionally determined complex condition—i.e., a function that takes contexts as input and outputs an individual or kind. Expressions that refer to kinds and expressions that denote kinds are semantically associated with very different sorts of function, and one result of this is a significant difference in extensions across logical space. There is a clear sense in which the likes of ‘water’ or ‘bachelor’ has a univocal extension (i.e., it designates a single kind of thing) across possible worlds while the likes of ‘our most precious resource’ or ‘the element most highly prized by local jewelers’ does not (i.e., they complex condition expressed picks out different kinds in different contexts).

Unstructured kind terms, like ‘honeybee’, are semantically linked to one characteristic or feature; hence have a relatively homogenous extension across possible worlds. In contrast, expressions that denote kinds, like ‘the insect species typically farmed for honey’, semantically express a compositionally determined condition that can be satisfied by different kinds in different possible worlds; hence their possible-worlds extension is relatively motley. So, this objection is met by paying careful heed to the difference between kind-dependent and kind-independent propositions. This approach to rigidity for kind terms need presuppose no capricious or dubious metaphysical doctrines.

Turning now to a second objection, Schwartz presses questions about possible-worlds truth-conditions for certain sorts of kind terms:

[This view holds that] ‘bachelor’ denotes the same marital status from world to world … But marital customs may vary from world to world in subtle and not so subtle ways. So is it the exact same marital status if the marital customs differ? How much do they have to vary before it is different? (2002: 269)

This sort of challenge does not apply to natural kind terms, on the Kripke-Putnam view, as the view has it that we are able to discover the essences of natural kinds, and thereby a precise possible-worlds extension. However, the challenge does apply to various varieties of non-natural kind terms (perhaps to varying extents). (How different from an actual pencil (or hunter, or philosopher, or sofa …) can some merely possible object be, and yet still count as a pencil (or hunter, or philosopher, or sofa …)?)

It seems to me that the indeterminacies pressed by Schwartz are exactly as things should be. The charge applies only to the extent that we, the bachelor-experts, have not defined a precise extension across logical space for this term. We have not yet decided how different marital conventions could be, whilst there still are bachelors, because it has not yet mattered to us to do so. (I suspect that most terms are like this; if so, it is a virtue, not a problem, for a semantic theory to reflect this.) If the need for that level of precision were to arise, and we had a big referendum (or whatever) to decide exactly what should count as a bachelor, that would yield a more precise extension across logical space. Until that referendum, there is some open-texture. (To the extent that this charge applies) given all the facts about all ‘bachelor’ thought and talk, up to present, the possible-worlds extension of the term is vague.29

Be very clear, however, that vagueness is distinct from nonrigidity. These differences do not by any means entail that there are contexts of evaluation in which our actual use of the term ‘bachelor’ (‘pencil’, ‘philosopher’, ‘sofa’, …) designates kinds that are distinct from its actual designatum, or in any way touch the fact that unstructured kind terms cannot satisfy a necessary condition for nonrigidity, and do satisfy a sufficient condition for rigidity.

Appendix B: an argument against rigidity

It might be thought that this approach to rigidity flies in the face of not just conventional wisdom but also conventional orthodoxy: that is, Lewis holds that such general terms as ‘pain’ and ‘heat’ are nonrigid designators. (Cf., e.g., 1994: 304). However, crucially, Lewis holds that these terms are semantically equivalent to denoting expressions.30 Lewis is explicitly wary of the anti-descriptivist, causal-historical arguments. (He is skeptical of the descriptive semantic claim that speakers use these terms as semantically unstructured referring expressions, and he is skeptical of the related but distinct metaphysics of essence championed by Kripke, Kaplan, and Putnam.) On Lewis’ view, these designators have semantic structure, just as surely as ordinary proper names do, on Russell’s view—i.e., their semantic content is given by a denoting expression. (‘Pain’ or ‘heat’, for Lewis, are semantically akin to ‘Bismarck’, for Russell.)

I will not argue against Lewis’ view here, but rather just underline that: (i) these terms’ turning out nonrigid depends on the premise that they are semantically equivalent to denoting expressions, and (ii) most contemporary philosophers reject this premise. Anyone who holds that ‘pain’ or ‘heat’ is a referring expression, as opposed to a denoting expression, is thereby committed to the view that it is a rigid designator.

There are real deep semantic and metaphysical issues bound up with the question of rigid vs. nonrigid denoting expressions. Many scientific and metaphysical disputes in some ways rest on, or turn on, how to rigidly denote a certain kind. A satisfactory answer to those questions can only be given to the extent that we have identified the essence of the kind; but surely introducing an expression to refer to the kind presupposes, or depends upon, no such knowledge.

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