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This article appeared in Philosophical Topics 28/1 (Spring 200): 177-192.

DIRECT AND VAGUE

In 1957, reporters asked maverick producer Mike Todd why he bought a huge 29-carat engagement ring for Elizabeth Taylor. Todd answered “Thirty carats would have been vulgar.” Todd’s quip absurdly implies he knew that 30 carats is the threshold for vulgarity. But most think stopping here misses the root of the joke. They think there is a more fundamental absurdity; that it is even possible for a single carat to make the difference between a vulgar ring and a non-vulgar ring. We epistemicists defend the possibility. The law of bivalence implies that discriminative terms (ones that apply to some but not all things) have sharp boundaries. Consequently, classical permits few degrees of freedom when solving the . One can follow the incoherentists (Wheeler 1967, Unger 1979) and conclude that nothing is vulgar. One can blaze a new path and conclude everything is vulgar. Or one could follow the epistemicists and say that there are some vulgar things, some non-vulgar things, and nothing lies between the Vulgar and the Non-vulgar. Epistemicism goes beyond conservatism about logic. Ordinary vocabulary must also be preserved. We epistemicists think that ordinary words apply to pretty much what ordinary folks assume they apply to. Thus our double conservatism obeys the time-honored dictum Loquendum enim est ut plures, sentiendum ut pauci (Augustine Niphus Comm. In Aristotelem de 1

Gen. Et. Corr., Bk. I, folio 29). “Think with the learned and speak with the vulgar.” Do epistemicists have a passion for precision? Consider the Victorian British bachelor, Phileas Fogg, protagonist of Mike Todd’s 1956 academy award winning film Around the World in 80 Days (based on the Jules Verne story). Fogg insists that his bath water be exactly one foot and three quarters inch deep, that his toast be precisely 83 degrees Fahrenheit, and that breakfast be served at 8:24, not a minute earlier or later. He goes through valets like Elizabeth Taylor went through husbands. was like Phileas Fogg. Frege thought vagueness was the product of slap-dash on the fly. Instead of properly defining terms, his fellow mathematicians opportunistically offered a necessary condition here, a sufficient condition there. Since these "definitions" lacked a condition that was both necessary and sufficient, their “concept” has semantic gaps. Frege did not take comfort in the thought that these borderline cases could be adjudicated later in light of future information and interests. This delay jeapordizes possibility proofs (for the possibility could be disqualified by an unanticipated necessary condition) and non- proofs (for the impossibility could be disqualified by an unanticipated sufficient condition). Consequently, Frege campaigned for the elimination of vagueness. Epistemicists have no special objection to vagueness and no special love of precision. They merely insist that vague statements conform to standard logic. However, most philosophers follow Frege in thinking that predicates inherit the incompleteness of linguistic intentions. 1

If there are singular , this distracting belief can be bypassed. A singular contains individuals, events, and other concrete things. For instance, `Mike Todd was Elizabeth Taylor’s third husband’ contains Mike Todd and Elizabeth Taylor. That seems self-evident. Except after philosophers ask embarrassing questions: Given that Mike Todd died in a 1958 plane crash, how can he now be a constituent of the proposition? The proposition still exists but he does not! And how old is the Elizabeth Taylor in the proposition? Is she at her current age or at the age when married to Mike Todd? Frege exposed further difficulties by embedding such sentences in modal contexts and belief contexts such as `Eddie Fischer believes that Mike Todd was Elizabeth Taylor’s third husband’. The behavior of these sentences led Frege to conclude that there are no singular propositions. Propositions are only indirectly related to concrete things via the senses that pick out those things. I think Frege went too far. I am amongst those (such as Paul Horwich 1998, 90-92) who believe that there are both singular propositions and general propositions. The case for singular propositions has been considerably strengthened by the work of , David Kaplan, and Ruth Marcus. Their seminal achievements have been amplified by Nathan Salmon and . I will not attempt to improve upon their case for the existence of some singular propositions. Nor will I present any direct arguments in favor of epistemicism. I merely aim to show that epistemicism provides the best 1 explanatory framework for vague statements that express singular propositions. The centerpiece of this project is a specific thesis: the vagueness of some statements arises from vagueness in how the speaker fixes reference. Thus the vagueness of a statement cannot always be traced to its descriptive content or to vagueness in the proposition it expresses (or to constituents of these propositions -- such as vague objects).

1. Vague reference fixing Saul Kripke (1972, 79n) uses the history of astronomy to illustrate the distinction between supplying a synonym for an expression and fixing the reference of an expression. Suppose Urbaine Leverrier introduced `Neptune' by saying "Let us give the name `Neptune' to the planet causing the perturbations in the orbit of Uranus". This ensures that `Neptune' denotes Neptune if it denotes anything at all. Although Leverrier introduced the name `Neptune' into the language by means of a definite description, the name is not synonymous with the definite description. Leverrier could well have believed that had Neptune been knocked off its course a million years ago, then it would not have caused the perturbations in the orbit of Uranus and even that some other planet might have caused the perturbations. As it turns out, Neptune exists. Encouraged by the success of Leverrier's method, astronomers turned their attention to the perturbations of Mercury. They said "Let us give the name `Vulcan' to the planet causing perturbations in the orbit of Mercury." However, their luck did not hold. There is no such planet. 1

Now consider a fanciful intermediate case. Astronomers detect perturbations in Jupiter's orbit and say `Let us give the name `Nepcan' to the planet causing the perturbations in the orbit of Jupiter'. They do find a heavenly body causing the perturbations but it is only a borderline case of a `planet'. The cautious discoverers assign a name to this body in a way that does not assume it is a planet. They say "Let us give the name `Runt' to the body causing the perturbations in the orbit of Jupiter". Reporters ask whether Runt is Nepcan. The Jovian astronomers shrug. They explain that since Runt is a borderline case of `planet', there is no way to tell whether `Nepcan' successfully denotes like `Neptune' or fails to denote like `Vulcan'. `Runt is Nepcan' is a vague identity. Notice there is no temptation to trace the vagueness to a vague object. Nor should there be a temptation to trace the vagueness to a vague sense. The vagueness of the identity `Venus is the loveliest planet' can be traced to the vagueness of the concepts involved in the proposition. But in `Runt is Nepcan' the vagueness is due to a term (`planet') that was merely used to fix the reference of `Nepcan'. `Nepcan' is not synonymous with `the planet causing the perturbations in the orbit of Jupiter'. The astronomer who introduced `Nepcan' could well have believed that if Nepcan had been knocked off its course a million years ago, then it would have caused no perturbations of Jupiter and even that some other object would have caused the perturbations. Since the statement `Runt is Nepcan' involves a singular term that is a borderline case of `empty term', there is a danger 1 that the indeterminacies associated with empty names might be confounded with the indeterminacies associated with vagueness. There are related difficulties when there is unclarity as to how many objects are involved in the situation. A complete theory of vague identities should eventually address these hard cases. But I shall avoid these complications by focusing on a simpler example of vague reference fixing. Suppose explorers say, "Let us give the name `Acme' to the first tributary of the river Enigma". When they travel up the river Enigma they finally reach the first pair of river branches. They name one branch `Sumo' and the other `Wilt'. Sumo is shorter but more voluminous than Wilt. This makes Sumo and Wilt borderline cases of `tributary'. Either Sumo is just a segment of the Enigma and Wilt feeds into the Enigma, or Wilt is a segment of the Enigma and Sumo is the tributary. The vagueness of `tributary' does not threaten to make `Acme' an because `the first tributary of the Enigma' denotes something on all admissible interpretations of `tributary'. `Acme' definitely refers to something even though it is vague whether it refers to Sumo and vague whether it refers to Wilt. Indeed, we know `Either Acme is Sumo or Acme is Wilt'. Since we also know that Sumo and Wilt both exist, we know that Acme exists. So although we know `Either Sumo is the first tributary of the Enigma or Wilt is the first tributary of the Enigma' there is no telling which. Hence, there is no telling whether `Sumo is Acme' or `Wilt is Acme'. We can know that `Acme is brackish' given that we know Sumo and Wilt are each brackish. Thus the vagueness of the 1 reference fixing does not prevent the name from being part of many knowable singular predications.

2. The completeness concern These claims about the knowability of `Acme is brackish’ presuppose that the exhaustiveness of the set of admissible interpretations is knowable. If there is no telling whether Wilt and Sumo are the only candidates for being Acme, then we cannot justifiably infer `Acme is brackish’ from `Wilt is brackish and Sumo is brackish’. Even if the inference is sound, it does not produce knowledge of the conclusion from knowledge of the premises. The reasoner must also know that either Acme is Wilt or Acme is Sumo. And he does not know this disjunction unless he knows that Wilt and Sumo are the only alternatives. Ambiguity always requires more than one interpretation. Not vagueness. Suppose that Wilt’s dwindling water sources have made it a borderline case of a `river branch’. The withered Wilt is no longer a clear candidate for being Acme. For all anyone can know, `Sumo is Acme’ comes out true on the sole admissible interpretation. Even so, `Sumo is Acme’ would be a vague identity. Not because there are further interpretations where it fails to come out true. Rather, because it is indefinite whether there are further interpretations. An identity statement could be vague without there being any alternative interpretations at all. It is good enough that it is indefinite whether there are alternative interpretations. The existence of admissible precisifications does not matter.Supervaluationists disagree. They say a statement is definitely true if it comes out true under all admissible interpretations. Period. Since supervaluationists do not picture 1 vagueness as an epistemological phenomenon, they are officially indifferent to whether there is any way to tell whether the list of candidates is complete. In particular, if it is indefinite whether there are further admissible candidates beyond Sumo, then supervaluationists must judge `Sumo is Acme’ as only indefinitely indefinite. In contrast, any right thinking epistemicist should judge `Sumo is Acme’ as definitely indefinite. Is Timothy Williamson a right thinking epistemicist? Almost always! But not on this issue. For his “logic of clarity” implies that epistemicism converges with supervaluationism on issues of higher order vagueness. Williamson’s basic premise is that since we do not know which concept we are using, we do not know which language we are using. This opens a structural analogy:

As a first approximation, for the supervaluationist, definiteness is truth under all sharpenings of the language consistent with what speakers have already fixed about its semantics (“admissible sharpenings”); for the epistemicist, definiteness is truth under all sharp interpretations of the language indiscriminable from the right one. In both cases, we hold everything precise constant as we vary the interpretation. (Williamson 1999, 128)

The idea is natural for the supervaluationist. An admissible sharpening completes a language in a way that does not contradict the past decisions of the speakers. If there are only ten sharpenings of `artwork’ and `All artworks are artifacts’ comes out true under each of them, then `All artworks are artifacts’ is definitely true. 1

At first blush, the idea also seems natural for an epistemicist . If there are only ten sharpenings of `artwork’ and `All artworks are artifacts’ comes out knowably true under all of them, then `All artworks are artifacts’ is knowably true. But there is a hitch: how do I know whether I have considered all the sharpenings of `artwork’? Each sharpening of `artwork’ may be known to be a sharpening that makes `All artworks are artifacts’ true. But if there is no way to know that this set of sharpenings is complete, then no one can know `All artworks are artifacts’. This uncertainty can arise from the vagueness of `admissible sharpening’. Some interpretations of `artwork’ are clearly admissible while others are clearly inadmissible. In between are interpretations that are borderline cases of `admissible sharpening’. Supervaluationism and epistemicism diverge in the case in which we cannot know (for reasons of vagueness) whether we have exhausted every sharpening but each sharpening can be individually known to make the statement true. In this circumstance, supervaluationism implies that the statement is definitely true while epistemicism implies the statement is true but indefinite. Given Williamson’s commitment to epistemicism, his account of definiteness is too broad. A statement can fail to be definitely true even though it is true “under all sharp interpretations of the language indiscriminable from the right one”. Let me restate the completeness concern in a way that is less abrasive to supervaluational sensibilities. Suppose it is indefinite whether `All artworks are artifacts’ comes out true under all admissible sharpenings. The supervaluationist says a 1 statement is definitely true when and only when it comes out true under all admissible sharpenings. So the supervaluationist would say that it is indefinite whether `All artworks are artifacts’ is definitely true. In contrast, the epistemicist should say that `All artworks are artifacts’ is definitely indefinite. The epistemicist should not require that all indefinite statements have a sharpening under which the statement comes out false Supervaluationism implies a Barcan formula for the definite operator (read `Dp’ as `It is definite whether p’):

(p)Dp - > D(p)p

If an epistemicist insists on having a logic of clarity, he should insist that this not be a theorem. After all, everyone agrees that the corresponding Barcan formula for knowledge fails:

(p)<>Kp > <>K(p)p

The supervaluationist believes vagueness is a purely semantic phenomenon and so sees little resemblance between the two Barcan formulae. However, any theorist who regards vagueness as a species of ignorance should see a damning resemblance. Williamson might hope that knowledge of completeness is never needed in the evaluation of sharpenings. After all, universal generalizations can sometimes be known without complete enumeration. I know that all men are mortal on the basis of induction. I know all men are male human beings on the basis of definition. Maybe vagueness creates a special situation in which access to the completeness of the enumeration is never 1 necessary. As long as each admissible sharpening is such that the speaker can know `All artworks are artifacts' is true under that sharpening, then he can know `All artworks are artifacts'. He may not be able to know that he knows. But that's irrelevant because the KK principle (if you know, you know you know) is false. The failure of KK is crucial for any epistemicist account of higher order vagueness. A genuinely indefinite definite statement can be known even though it cannot be known to be known. The vagueness of `know’ guarantees that there will be many such cases. But when the completeness concern is not satisfied, the knowledge attribution fails at the first level, not the second. I have no interest in jacking up standards for attributing knowledge. The completeness concern conforms to our ordinary practice of knowledge attributions. When strangers see me with my two boys, they sometimes ask whether all of my children are boys. The observer sees all of my children. He sees that each child is a boy. Yet he must ask me whether all of my children are all boys. Does the stranger merely fail to know that he knows that all my children are boys? Williamson assigns each predicate infinitely many interpretations. This magnitude of alternatives would stop us finite beings from knowing `All artworks are artifacts’ by enumeration. Our knowledge of the generalization would have to be explained in the same way we know arithmetic truths that have infinite domains. Whatever the details, such knowledge is just as sensitive to completeness concerns. Here is an illustration. A perfect number is a number whose positive divisors (except for itself) sum to 1 itself. For instance, 6 is perfect (1+2+3=6) as are 28 (1+2+4+7+14=28) and 496 (1+2+4+8+16+31+62+124+248=496). All and only even perfect numbers have the form: 2p-1(2p-1) where p is a Mersenne prime. (A Mersenne prime is a prime of the form 2n-1 such as 2, 3, 5, 7,13, etc). Consequently, there are exactly as many even perfect numbers as Mersenne primes. Mathematicians conjecture that the number of Mersenne primes is infinite. If this conjecture is correct, then there is also an infinity of even perfect numbers. More ancient is the conjecture that there are no odd perfect numbers. (It has been proved that any odd perfect number must exceed 10300 and must be divisible by a prime power exceeding 1020.) If there are no odd perfect numbers, then each perfect number is such that it is known to have the form 2p-1(2p-1) and yet it is not known that all perfect numbers have this form. If both conjectures are true, then there are infinitely many perfect numbers each of which is known to have the form 2p-1(2p-1) even though it is unknown that all perfect numbers have the form. Infinity does not inhibit the logicians’ completeness concerns. There are deductive systems in which any statement of the form `n = n' is provable but there is no proof that `For all n, n = n'. This “omega incompleteness” can be remedied by introducing a rule that would entitle the inference from a set of infinitely many premises. But this omega rule is a significant departure from the classical conception of a proof which is restricted to derivations of a finite length. The completeness concern is a rugged, universal feature of knowledge and demonstration. 1

Supervaluationism overgeneralizes from one kind of intellectual predicament -- having too many possibilities. This “embarrassment of riches” is the most vivid obstacle. Witness how epistemologists dwell upon skeptical “counter-possibilities”. But inquiry can be blocked by a wide variety of phenomena: circularity, unbelievability, inexpressibility, Gettier defeaters and on, and on. True, these obstacles do not normally suffice for vagueness. But the same goes for counterpossibilities. The inquiry resistance that constitutes vagueness must be absolute. But no epistemic obstacle has a monopoly on recalcitrance. Any adequate theory of vague identity must be sensitive to sources of vagueness beyond the existence of alternative interpretations. Supervaluationism is constitutionally narrow. It cannot provide a sufficiently broad theory of vague identity.

3. Direct reference Supervaluationism encourages the assumption that the referent is reached by means of a description contained in the statement to be evaluated. For instance, in `The president is puckish', the definite description within this statement seeks out the referent. Although referents can be selected in this Fregean manner, they can also be reached in a collateral fashion. According to Saul Kripke and Keith Donnellan, the referent of the utterance `Clinton is left-handed' is selected by an historical relation between the name `Clinton' and the bearer of that name. The name (or Clintonian sense) does not appear in the proposition; the bearer himself is in the proposition. The proposition does not seek out the individual. The individual is delivered to the proposition by external means. 1

These external means could be descriptive. As we have seen, Kripke explicitly allows names to be introduced by ad hoc description. David Kaplan (1989, 520n) emphasizes that the description can also be durable and standardized. The referent of `I' is secured by the semantical rule that the referent of `I' is the utterer. Everybody knows the descriptions that fix the reference of indexical terms. However, these well-known rules are not Fregean senses because they do not work from within the proposition itself. `I am not the person who is uttering this sentence' would be necessarily false if `I' were synonymous with `the person who is uttering the sentence'. But the proposition expressed by that sentence is contingent. The rule for fixing the referent of `I' is "off the record" because it is not part of the proposition itself. Consequently, the properties of the reference fixing description are not transmitted to the proposition itself.

4. The argument against vague identities Saul Kripke is committed to vague identity statements, such as `Acme is Sumo', in which the flanking terms are directly referring terms. However, there is a simple argument that such identity statements are impossible (Gareth Evans 1978 and Nathan Salmon 1982, 243-6). If `a = b' is vague, then b has a property that a lacks, namely being vaguely identical to a. Thus by Leibniz's law that identicals share all their properties, `a = b' would be false. But if `a = b' is provably false, then it is not vague. This argument is troublesome to Kripke because he defends the soundness of an analogous argument (Kripke 1971) against 1 contingent identities. Indeed, his celebrated endorsement of this simple refutation is the probable inspiration for the argument against vague identities. If `a = b' is contingent, then b has a property that a lacks, namely being only contingently identical to a. Thus by Leibniz's law that identicals share all their properties, `a = b' would be false. Kripke allows that `Benjamin Franklin is the inventor of bifocals' is a contingent identity statement. The non-rigid designator `the inventor of bifocals' picks out different individuals in different possible worlds. However, when the singular terms are both directly referential, the statement cannot acquire contingent status via the means by which the proposition selects the referent. So Kripke concludes, contrary to the received opinion of the era, that there is no contingency when only names are involved. All identity statements with rigid designators are free of contingency; all true identity statements are necessary truths. Similarly, Kripke should cheerfully concede that `Benjamin Franklin is the wisest American revolutionary' is a vague identity statement. The non-rigid designator `the wisest American revolutionary' picks out different individuals under different precisifications of `the wisest American revolutionary'. However, when the singular terms are both directly referential, vagueness cannot enter via the means by which the proposition selects the referent. So it seems that Kripke should conclude, contrary to what `Acme is Sumo' appears to illustrate, that there is no vagueness when only names are involved. But vague reference fixing establishes vague names within Kripke's own framework. So if Kripke accepts vague names, he apparently must reject the 1 argument against vague identity -- despite its strong resemblance to the argument against contingent identity. This would not be the first logical analogy to vex Kripke. Critics quickly compared the argument against contingent identity with the argument against unbelieved identities: If `a = b' is unbelieved, then b has a property that a lacks, namely being believed to be identical to b. Thus by Leibniz's law that identicals share all their properties, `a = b' would be false. Once again, there is no problem when one of the terms is non-rigid. This explains why the description theory of names seems to easily handle unbelieved identities. Kripke's (1976) reaction to unbelieved identities is painstakingly circumspect. In "A Puzzle about Belief", he acknowledges that it is natural to think that unbelieved identity statements show that special maneuvering around Leibniz's law is sometimes needed. To keep the treatment of identity uniform, some go on to reject Kripke's account of modal rigidity. Those more tolerant of discontinuity urge a compromise solution: accept that names are rigid in modal contexts but not in belief contexts (Plantinga 1978). The compromise is unavailable if rigid designators are defined generically. For instance, one popular idea is that names are rigid by virtue of their scopelessness (Peacocke 1975). If names, by nature, fail to support a de dicto/de re distinction, then they pick out the same individual in all contexts. Kaplan explains the rigidity of names as a consequence of them being directly referential: 1

If the individual is loaded into the proposition (to serve as the propositional component) before the proposition begins its round-the-worlds journey, it is hardly surprising that the proposition manages to find that same individual at all of its stops, even those in which the individual had no prior, native presence. The proposition conducted no search for a native who meets propositional specifications; it simply `discovered' what it had carried in. In this way we achieve rigid designation. (Kaplan 1989, 569)

Kaplan had assumed that Kripke intended direct reference to be the deep structure undergirding rigid designation. However, Kripke prefers a shallower characterization of rigid designation: "a designator d of an object x is rigid, if it designates x with respect to all possible worlds where x exists, and never designates an object other than x with respect to any ." (quoted by Kaplan 1989, 569). Kripke motivates the definition as a way of avoiding the question of whether a rigid designator can designate an object at a world in which the object does not exist. The narrowness of the definition also permits Kripke to underscore the consistency of restricting rigidity to modal contexts. However, he concedes that a compromise would ill accord with the spirit of the direct reference tradition inaugurated by . The theme of Kripke's "A Puzzle about Belief" is that the advantages of the description theory of names have been over- estimated. Kripke unveils the Pierre puzzle to show that many of the difficulties that are commonly traced to Leibniz's law can be 1 generated without that law. This centerpiece is supported by a wealth of smaller points about the relationship between belief and identity. Kripke concludes by warning of the dangers of drawing conclusions about this difficult issue. Some of Kripke's followers have taken the uniformitarian position that seems most natural for a Millian. For instance, Nathan Salmon (1986) and Scott Soames (1987b) accept the argument against unbelieved identities. They sharply distinguish between the and semantics of belief attribution. Given their defense of an unblinking application of Leibniz's Law, Salmon and Soames should also unblinkingly accept the argument against vague identities. And indeed, Salmon, as co-discoverer of the argument against vague identities, has explicitly denied the possibility of vague identities.

7. Rigid designators are unproblematic when descriptive David Lewis has presented Gareth Evans’s version of the argument against vague identities as a powerful but misunderstood objection to the view that vagueness is in the world. Lewis's own example of a vague identity statement is `Princeton = Princeton Borough'. It is unsettled whether `Princeton' designates Princeton Borough or the surrounding Township, or some larger region. Lewis treats `Princeton' as "analogically speaking non- rigid" by assuming that it denotes different things on different precisifications. According to Lewis, Evans’s point is that this attractive diagnosis cannot be extended to vague identity statements that owe their vagueness to the object rather than our way of describing the object. The vague-objects view 1

says that a name like `Princeton' rigidly denotes a certain vague object. In fact, the vague-objects view does not afford any diagnosis of the fallacy, so it is stuck with the unwelcome proof of an absurd conclusion, so it is in bad trouble. (Or better, what is in trouble is the vague-objects view combined with the view that vague identity yields identity statements with indefinite truth value.) (Lewis 1988, 129)

Lewis' interpretation does fit Evans’s analogy between `definitely' and `necessarily'. Precise names are, as it were, rigid designators. If the identity statement comes out definitely true under each admissible way of interpreting the designator, then it is definitely true. As indicated in section 2, an epistemicist should reject Evans’s modal analogy. Coming out true under each admissible precisification is a sufficient condition for truth but is not a sufficient condition for definite truth. A variety of epistemological hitches could render the statement indefinite even though it is true under all precisifications. Evans’s modal analogy should also be reined in with the observation that analogical non-rigidity is compatible with literal rigidity. There are unproblematic vague identities with flanking rigid designators. Suppose 43 is a borderline case of `large natural number'. Then the following definite description either picks out 1 in every possible world or it picks out 0 in every possible world: 1

R. The n such that [(if 43 is large, then n = 1) and (if it is not the case that 43 is large, then n = 0)].

Since R is a rigid designator under every admissible precisification of `large', R is a rigid designator. (Indeed R is a strongly rigid designator because it has a referent at every possible world.) Thus the following two statements are examples of vague identity statements that are composed solely of rigid designators:

R0. The n such that [(if 43 is large, then n = 1) and (if it is not the case that 43 is large, then n = 0) = 0]. R1. The n such that [(if 43 is large, then n = 1) and (if it is not the case that 43 is large, then n = 0) = 1].

Each of the identity statements is indefinite. However, their disjunction is definite. The supervaluationist can handle these vague identity statements. After all, R is just as analogically non-rigid as any other definite description; there are alternative precisifications of the descriptive content of R. Thus there is vagueness in the proposition expressed by `R = 1 or R = 0'. More natural examples can be constructed with a rigidifying term such as `actual'. For instance, `the actual inventor of bifocals' picks out Benjamin Franklin in every possible world. Hence, we can easily construct vague identity statements with flanking rigid designators such as `Benjamin Franklin is the actual wisest American revolutionary'. Some descriptive theories of names systematically incorporate rigidifying terms. Such a 1 theory could define `Acme' as `the actual first tributary of the river Enigma'. Substituting for `Acme is Sumo' yields `The actual first tributary of Enigma is Sumo'. The supervaluationist can then point out that `the actual first tributary of Enigma' picks out Sumo under one precisification of `tributary' and Wilt under the other. However, if the rigid designators in `Acme is Sumo' are directly referential, then the expressed proposition cannot be vague about which individual it selects. For the proposition does not select an individual. The individual is pre-selected. So the only problematic cases are vague identity statements with directly referential terms. These terms are rigid designators but rigid designation is not the source of the difficulty. An identity statement with directly referring terms has no internal descriptive aspect. Thus the "vagueness in describing view" cannot get a grip on these identities. Contrary to Lewis, the vague-objects theorist seems to have better explanatory prospects at this juncture. The proposition expressed with a directly referring term contains the referent itself. If the referent is a vague object, the proposition might inherit this vagueness.

6. Direct reference to vague objects Some vague object theorists say that propositions always inherit the vagueness of their referents. Michael Tye uses this principle to defend the vagueness of `m = Mount Everest' where "`m' is a name of a more precise mountain that differs from Everest only in that it lacks certain chunks of matter that are indefinite constituents of Everest" (1990, 556). According to Tye, the 1 argument against vague identity successfully shows that the identity statement is false. If m has the property of definitely containing those chunks of matter and Everest lacks this property, then m is not Everest. Nevertheless, Tye thinks we can still say `m is Everest' is still vague:

For one can grant that the argument demonstrates that identity statements (in which the identity sign is flanked by rigid names) cannot be indefinite in truth- value without admitting that such statements cannot be vague. To say that an identity statement is vague, on my view, is to say that it has a vague . This will be the case, I maintain, if either of the singular terms flanking the identity sign is vague. (1990, 556)

This sufficient condition for vagueness commits Tye to counting obvious truths such as `Everest is Everest' as vague. Ditto for `That [pointing at Everest] is that [pointing at Everest]'. Anaphoric pronouns will catch the vagueness from names: `Everest is big. It is hard to climb'. Therefore, even `Everest is itself' will be vague! Anaphoric pronouns behave like bound variables and demonstratives behave like free variables. Kaplan (1989, 571- 72) touts free variables as paradigms of direct reference. To evaluate `Fx' at a world w, we ask which value was assigned to x prior to the evaluation. Without this prior assignment, we do not have a proposition to evaluate. We do not need to how the assignment was made to evaluate the proposition. Only the result of the assigning process is relevant. The origin of the assignment 1 is off the record. In particular, one commits the genetic fallacy if one infers that the product of the assignment is vague because the process of assignment was vague. The referent of a variable is just the value assigned to it. Hence, if the vagueness of the referent is inherited by its singular term, there are vague variables. For instance, if Everest is assigned to x, then Tye must count `x = x' as vague. Everest's ability to make statements vague is even stranger when it does so simply in virtue of being in the domain of discourse. Relative to the domain {1, Everest} the statement `(x)(x = x)' is equivalent to `(1 = 1) and (Everest = Everest)'. Since Tye counts the conjunction as vague, he would need to count `(x)(x = x)' as vague. The point holds for any larger domain containing Everest. For the variable x will still range over Everest, hence, the vague `Everest is Everest' is a satisfaction instance of `(x)(x = x)'. The vagueness of `(x)(x = x)' is surprising because it contains nothing but logical words. But actually all quantified formulas will be vague relative to domains containing at least one vague object. Since logical statements must range over everything, all of the quantified formulas of logic will be vague. The vague object theorist might reject the principle that directly referring terms always inherit the vagueness of their referents. However, the principle is false only if some precise propositions contain vague objects. But now there is pressure to move from some to all. If `Everest = Everest' is precise despite the vagueness of Everest, then how could any identity proposition be vague in virtue of the vagueness of the object? What would this other vague object have that Everest lacks? 1

The last possibility is that the vagueness of the referent never suffices to make the proposition vague. Under this non- interactionist view, `Everest is vague' and `"Everest" is vague' are independent statements. The property of being vague is then like the property of being interesting or useful; although both words and things can have the property, one cannot infer that the word has the property on the grounds that the thing has the property or vice versa. However, if the vagueness of objects does not interact with the vagueness of language, then vague objects become philosophically uninteresting. They will not bear on the sorites or puzzles apparently involving indefinite identity. The proponent of classical logic will have no objection to them other than them being misleadingly labeled as vague. Can the vague-objects view be insulated from argument against vague identities by rejecting all direct reference? Granted, many theorists deny that names are directly referential and some deny that demonstratives are directly referential. But none deny that variables are directly referential. Just as (1982) strengthened the argument against unbelieved identities by recasting it in terms of variables, we can recast the argument against vague identities in terms of variables. The strengthened argument can proceed from the compositionality of propositional content: if two sentences are constructed in the same way from corresponding constituents having the same content, then those sentences have the same content. It is never vague whether x = x. Hence by the compositionality of propositional content, when x = y, it is never vague whether x = y. 1

The argument is most vivid for singular propositions containing individuals. But it only requires that the referring expressions be free of propositional content. For instance, Kripke's variables from his article "Semantical considerations in " are treated as primitive, contentless expressions having constant denotation. That suffices to satisfy the compositionality of propositional content.

7. Does the linguistic view do better? Lewis claims that the vagueness-in-describing view has diagnostic resources that the vague-objects view lacks. However, when the singular terms are understood as directly referential, both views have trouble finding any fallacy in the argument against vague identities. The vagueness-in-describing theorist could suggest a compromise view of direct reference: singular terms are directly referential in modal contexts but not in vagueness contexts. This compromise should be distasteful to Nathan Salmon because he explicitly rejects the compromise for unbelieved identities. Since Saul Kripke refuses to commit to a compromise to handle belief contexts, he should also refuse to commit to a compromise to handle vagueness contexts. Consequently, Kripke has no principled objection to the argument against vague identities. Kripke's diagnostic gap is inherited by more complex theories of names. For the theories that followed in the wake of Naming and Necessity typically concede that there are conditions under which names do behave as Kripke describes. And vague reference fixing scenarios can be tailored to those conditions. For instance, Gareth Evans developed a hybrid theory to deal with 1 apparent counterexamples to Kripke's theory. However, Evans concedes that names work in a simple Kripkean fashion when the speakers are using a name deferentially. Thus Evans inherits Kripke's diagnostic deficit when speakers have "the overriding intention to conform to the use made of [the vague names] by some other person or persons" (Evans 1973, 205). The reporters in the Nepcan scenario have this desire to conform to the astronomers who originated `Nepcan' and `Runt'. David Lewis quotes correspondence with Gareth Evans showing that Evans accepted Lewis's interpretation: the argument against vague identities is a sophism that the vagueness-in- describing view can diagnose but that the vague-objects view cannot. However, Evans’s mixed theory of names should have led him to reject Lewis's interpretation. After all, Evans agrees that names sometimes work like Kripke says. That is enough to commit Evans to singular propositions. In particular, if a speaker utters `Sumoe is Acme’ with “the overriding intention to conform to the use made of [the vague names] by some other person”, then the names operate as directly referring terms. Unlike Lewis (1986, 220-248), Evans is an haecceiticist. An haecceitist believes that distinct objects can be qualitatively identical. All haecceitists should accept singular propositions (Kaplan 1976). That includes identity statements with flanking directly referring terms. We haecceitists reject Lewis’s diagnosis as incomplete. It only addresses vague identities involving general propositions. Since Lewis denies that there are any singular propositions, he will view his analysis as complete. But the “vagueness in describing” view is less attractive if it carries a strong anti- 1 haecceticist commitment. Even if one denies that names always operate as Kripke pictures, they seem to sometimes operate that way. What goes for names, goes for demonstratives and indexicals. `Princeton is here' expresses a precise proposition even when there is no telling which place is denoted by `here'. Recall Kaplan's (1989, 535-6) example of the kidnapped heiress who is imprisoned in a car trunk. After being driven around for hours, the car is parked. When the heiress says `It is quiet here now', she picks out a place even though she is unacquainted with that place and cannot identify it. Her referential act would succeed even if her ignorance were irremediable. The thought must be distinguished from the cognitive significance of the thought. Just as ignorance is no barrier to securing a referent, vagueness is no barrier. For if vagueness were a barrier, directly referential vague identity statements would fail to express propositions. So would disjunctions of these identity statements such as `Either Acme is Sumo or Acme is Wilt’. These disjunctions express propositions. Therefore, their disjuncts do as well. Some vague identity statements express singular propositions. Consequently, they inherit the truth-value of their propositions. All propositions of the form `x = x' are truths. Hence, there are vague identity statements that express unknowable truths -- and necessary truths at that.

Roy Sorensen, Dartmouth College 1

Acknowledgments: An ancestor of this paper was read at the 1998 Mighty Midwestern Conference at Notre Dame. Observations from my commentator, Dean Zimmerman, prompted improvements. I also thank Ken Akiba, Chris Hill, Samuel Levey, and Ted Sider for their comments and suggestions.

REFERENCES Church, Alonzo (1982) "A Remark Concerning Quine's Paradox About Modality" Analisis Filsofico 2. Reprinted in Propositions and Attitudes ed. Nathan Salmon and Scott Soames (Oxford: Oxford University Press, 1988): 58-65. ______(1954) "Intensional Isomorphism and Identity of Belief" Philosophical Studies 5: 65-73. Reprinted in Propositions and Attitudes ed. Nathan Salmon and Scott Soames (Oxford: Oxford University Press, 1988): 159-168 Donnellan, Keith (1966) "Reference and Definite Descriptions" Philosophical Review 75: 281- 304. Evans, Gareth (1973) "The Causal Theory of Names" Aristotelian Society: Supplementary Volume 47: 187-208. ______(1978) "Can there be vague objects?" Analysis 38: 208. Garson, James W. (1984) "Quantification in Modal Logic" in Handbook of Philosophical Logic vol. II, ed. D. Gabbay and F. Guenthner (Dordrecht: Reidel): 249-307. 1

Horwich, Paul (1998) Truth (Oxford: Clarendon Press). Kaplan, David (1975) “How to Russell a Frege-Church” Journal of Philosophy 72: 716-729. Kaplan, David (1989) "Demonstratives" in Themes from Kaplan ed. Joseph Almog, John Perry, and Howard Wettstein (New York: Oxford University Press) 481-564. Kripke, Saul (1963) "Semantical Considerations in Modal Logic" Acta Philosophical Fennica 16: 83-94. Reprinted in Reference and Modality ed. Leonard Linsky (Oxford: Oxford University Press, 1971): 63-72. ______(1971) "Identity and Necessity" in Identity and Individuation (New York: New York University Press) 135-164. ______(1972) "Naming and Necessity" in Semantics of Natural Language ed. Donald Davidson and Gilbert Harman (D. Reidel) 253-355. Reprinted in book form as Naming and Necessity Harvard University Press (1980). Page are to the book. ______(1976) "A Puzzle about Belief" in Meaning and Use ed. Avishi Margalit (Dordrecht: D. Reidel). ______(1977) "Speaker's Reference and Semantic Reference" in Midwest Studies in Philosophy vol. II (Morris, Minnesota: University of Minnesota Press) 255-276. Lewis, David (1988) "Vague Identity: Evans Misunderstood" Analysis 48: 128-30. ______(1986) On the Plurality of Worlds (Oxford: Basil Blackwell). Peacocke, Christopher (1975) "Proper names, reference, and rigid designation" in Meaning, Reference and Necessity ed. Simon Blackburn (Cambridge: Cambridge University Press) 109- 132. 1

Plantinga, Alvin (1978) "The Boethian Compromise" The American Philosophical Quarterly 15 (April 1978): 129-138. Putnam, Hilary (1975) "The Meaning of `Meaning'" Language, Mind and Knowledge (Minneapolis, Minnesota: University of Minnesota Press) 131-193. Salmon, Nathan (1982) Reference and (Oxford, Blackwell). ______(1986) Frege's Puzzle (Cambridge, Massachusetts: MIT Press).] Soames, Scott (1987a) "Substitutivity" in On Being and Saying ed. Judith Jarvis Thomson (Cambridge, Massachusetts: MIT Press). ______(1987b) "Direct Reference, Propositional Attitudes, and Semantic Content" Philosophical Topics 15: 47-87. Reprinted in Propositions and Attitudes edited by Nathan Salmon and Scott Soames (Oxford: Oxford University Press, 1988): 197- 239. Sorensen, Roy (1988) Blindspots (Oxford: Clarendon Press) ______(1990) "Process Vagueness" Linguistics and Philosophy 13/5: 591-622. Unger, Peter (1979) "There are no Ordinary Things" Synthese, 4: 117-54. Van Inwagen, Peter (1990) Material Beings (Ithaca: Cornell University Press). Wheeler, Samuel C. (1967) "Reference and Vagueness" Synthese,30 no. 3/4: 367-79. Williamson, Timothy (1994) Vagueness (London: Routlege). ______(1999) “On the Structure of Higher- Order Vagueness” Mind 108: 127-143.