Published in Philosophical Studies 81/2-3 March 1996: 193-214
Semantic indeterminacy is the ether of philosophy of language. It fills the interstices of our intentions and pervades accounts of presupposition, tense, fiction, translation, and especially, vagueness. Yet semantic indeterminacy is as impossible as ectoplasm. Indeed, more so! The demonstration need only borrow a few assumptions used elsewhere in widely accepted impossibility results. Since an impossibility is never a necessary condition for anything actual, semantic indeterminacy must be superfluous. Language is no more explained by semantic indeterminacy than calculus is explained by (pre-Robinsonian) infinitesimals.
So why does semantic indeterminacy seem like a relevant alternative rather than a non-starter? I trace this appearance of viability to a double standard; we are more permissive about words than things. However, once we view words as things, impossibility results for linguistics take on the same gravity as impossibility results for number theory. I shall argue that even deviant logicians are committed to infinitely many theorems about impossible predicates. Since these theorems parallel ones that classical logic has for vague predicates, I conclude that deviant logic is ontologically ad hoc.
Semantic indeterminacy seems inevitable because meaning is anchored in conventions. Conventions are in turn dependent on our intentions and these intentions can be incomplete. For example, suppose Mr. Incomplete introduces `dommal' by stipulating that all dogs are dommals and that all dommals are mammals (Williamson 1990). That leaves the case of cats undefined. The indeterminist claims that familiar borderline cases are just undefined cases.
If Mr. Incomplete's two clauses for `dommal' are the only semantically relevant meaning makers, then Mr. Incomplete has failed to define a predicate. Compare Mr. Incomplete's exercise with Mr. Grelling's definition of `heterological'. The particular moral of Grelling's paradox is that there can be no predicate that applies to all and only those predicates that do not apply to themselves.
The use/mention distinction separates two kinds of "impossible predicates". A word that means something impossible can itself be possible. Adding `round square' to the axioms of geometry leaves the system consistent. A contradiction would only follow if we added the further assumption that round squares exist. Geometers can exploit this contradiction to obtain a reductio ad absurdum. This proof refutes the existence of round squares, not the existence of the term `round square'. Indeed, the reductio relies on `round square' being a well defined predicate.
I define `super-stretcher' to cover exactly those expressions that are longer than themselves. The logic of comparatives ensures that it is impossible for any term to be a super-stretcher. `Super-stretcher' is not itself a super-stretcher. `Super-strecher' is a possible word about impossible words. Ditto for `impossible term'. If `impossible term' were impossible, its impossiblity would show that there is an instance of `impossible term'.
J. F. Thomson extracted an interesting lesson from Grelling's paradox thirty years ago:
Thomson's theorem: Let S be any set and R any relation defined at least on S. Then no element of S has R to all and only those S-elements which do not R to themselves. (Thomson 1962, 104)
If we let S be the collection of men, then this set contains no man who bears the relation of shaving all and only those men who not shave themselves. That dissolves the Barber paradox. Thomson goes on to show how his "small theorem" is at the root of Richard's paradox and Bertrand Russell's paradox about the class of all classes which are not members of themselves. Russell (1903, 366-8) himself made the connection between his paradox, Grelling's, and Cantor's diagonal argument. Indeed, the attempt to dispel puzzles by reining in the powers of stipulation goes back to the fourteenth century logician John Buridan (1966, chapt. 6) and continues today (Goldstein 1985).
We should emulate Thomson's modesty about the depth of his theorem. It has the same status as the observation that no word means something other than what it means. Thomson's theorem is less problematic than the impossibility of a word being red all over and green all over or the impossibility of a word containing itself as a sub-word. Any impossibility theorem will have a counterpart concerning a predicate. Since many of these theorems are surprising, the pseudo-status of many "predicates" will also be surprising.
The subtlety of an inference is largely an accident of our psychology. Human beings find it obvious that there are no predicates that are both primitive and not primitive. A Martian who lacked negation but did have the Scheffer stroke, l, would need to tease out the impossibility from a complex sentence; the Martian translation of -(p & -p) is
[[pl(plp)]l[pl(plp)]l[pl(plp)]l[pl(plp)]].
Words must conform to classical logic just as much as barbers, voting systems and time travelers. Words are only limitless in the way that numbers are limitless. True, there are infinitely many possible words and infinitely many properties. But not anything goes. There are also infinitely many non-words and infinitely many non-properties.
In addition to impossible predicates, there are impossible connectives. Arthur Prior (1978, 38) defined `tonk' by the rule that from P one can deduce P tonk Q, and from P tonk Q, one can deduce Q. By the transitivity of deduction, tonk will let us infer anything from anything. To avoid inconsistency, we must deny that tonk forms genuine propositions. `P tonk Q' is not itself a contradiction because a contradiction is a genuine proposition. Rather `P tonk Q' functions as a pseudo-lemma for generating genuine contradictions such as `Q and not Q'.
It is standard mathematical practice to reject definitions on the grounds that they lead to inconsistency. Benson Mates (1965, 188-89) assembled nice examples: It is intuitive to define division in terms of multiplication:
(x)(y)(z)(x/y = z <-> x = yz).
But then we can derive `0/0 = 1' and `0/0 = 2', therefore `1 = 2'. This definition of division must be rejected even though it is just as common sensical as the perfectly acceptable definition of subtraction in terms of addition:
(x)(y)(z)(x - y = z <-> x = y + z).
Structural analogies with acceptable definitions do not suffice for acceptability. Surprises should be expected because a definition's absurdities emerge only when the definition is conjoined with the principles of the theory in which it is embedded. Absurdities are hard to detect when buried several inferential steps below the surface.
Two possible predicates may not be co-possible. It is possible for there to be a predicate that can be used to define any other predicate. And it is possible for there to be a predicate that cannot be defined in terms of any other predicate. But both kinds cannot co-exist as distinct predicates in the same system.
When background assumptions are meager, a definition might produce an absurdity other than an inconsistency. Suppose we define a binary operation symbol `G', for the theory of groups, as follows:
(x)(y)(z)(Gxy = z <-> x = y).
A consequence is that everything is identical to everything, `(x)(y) x = y'. This Parmenidean theorem restricts the theory's applicability to models that have just one element in their domains. This shows that the acceptability of predicates can be influenced by theoretical desiderata other than sheer consistency. After all, group theorists could have written `~(x)(y) x = y' into the axioms.
These hazards may make us leery of ever adding new terms. However, logicians routinely allay fear of runaway deductions by showing that the new definition yields a "conservative extension" of the old system. That is, they show that the new definition only makes a verbal difference to the system's theorems. If a geometer stipulates that `borp' applies to any and all triangles that are neither isosceles nor right triangles, then he can deduce that no borp is a 3, 4, 5 triangle. But the geometer can also translate the new theorem into the old vocabulary and show that he could have derived the theorem without the new term. However, adding `heterological' or `tonk' to nearly any interesting system yields theorems that are more than artifacts of abbreviation. Very many of these theorems are false and so unwelcome.
But even a creative definition that only introduced true novelties would be lamentably misleading. Definitions are supposed to explain meaning, not assert substantive matters of fact. Definitions that sneak in synthetic propositions amidst analytic packing win converts by passing off substantive claims as mere matters of convention. The only value of "creative" definitions is sophistical: the indirection is unnecessary because the synthetic portion can always be explicitly asserted. Thus Belnap (1993, 124-6) persuasively argues that eliminability and conservativeness are individually necessary and jointly sufficient for being a good definition.
Now consider a definition of a vague predicate: a time is noonish if and only if it is near noon. Promoters of the sorites paradox commence interrogation. They draw our attention to the finite sequence <12:01 PM, 12:02 PM, . . . , 10:00 PM>. Does the definition of `noonish' imply the following sufficient condition?
(Paradigm) One minute after noon is noonish.
Obviously. Someone who sincerely denied the paradigm condition would be suspected of linguistic incompetence! Well, does the definition of `noonish' imply that some differences are too small to make a difference? Say, one minute?
(Inductive principle) If n minutes after noon is noonish, then so is n + 1.
The inductive principle is not as obvious but there is a persuasive consideration in its favor. The negation of the inductive principle implies there is an n such that n minutes after noon is noonish but n + 1 is not noonish. `Noonish' seems too rough to be sensitive to minute differences.
So far, our answers are consistent. There are predicates that satisfy both a paradigm condition and an inductive principle, such as `is noon or a time before or after noon'. This predicate will not yield a sorites paradox because it applies to all times. However `noonish' is a discriminative predicate. It applies to one minute yet
(Foil) It is not the case that six hundred minutes after noon is noonish.
The promoter of the sorites correctly observes that no predicate can satisfy all three conditions given classical logic. He soundly generalizes:
(Sorites theorem) There are no predicates that are both inductive and discriminative.
There are at most four ways to reconcile with this theorem. (Don't worry about a four way tie; classical logic ultimately disposes of three ways.)
First, the linguistic incoherentist holds that vague predicates would have to be both inductive and discriminative, and so concludes that there are no vague predicates. He thinks `noonish' is a pseudo-predicate like `heterological'. Thus the linguistic incoherentist dismisses the sorites argument as meaningless on the grounds that its premises and conclusion use a pseudo-predicate.
Second, the metaphysical incoherentist accepts the existence of vague predicates but views them as empty. `Noonish' is treated like `round square'. Thus, he says the base steps of positive sorites arguments are false. However, the metaphysical incoherentist accepts the soundness of negative sorites arguments. (A positive sorites argues from a paradigm to the negation of a foil, a negative sorites from a foil to the negation of a paradigm).
A third option is to reject the foil condition and say that each vague predicate applies to everything. Although this outrage to common sense has the precedent of the outrage committed by tenured incoherentists (Unger 1979), no one has ever adopted this position.
And no one ought to. Although these three options respect the sorites theorem, classical logic eliminates them in another way. Each of the positions makes essential use of the vague predicate `vague' and so can only be true if not true. (For a proof of the vagueness of `vague' see Sorensen 1988, 227-8). Linguistic incoherentism says `There are no vague predicates' and so can only be true if `vague predicate' is a genuine predicate. But if `vague' is a predicate, it has borderline cases and so is vague. `There are vague predicates' is undeniable; negating it is as self-defeating as asserting `There are no words'. If there are worlds without words, then `There are words' is something short of a tautology. One could say of this other world that there are no words. But it is self-defeating to say this about one's own world. Hence, linguistic incoherentism is ruled out by the applicability of logic to the actual world.
Metaphysical incoherentism says `There are vague predicates but they are all empty'. Yet if `vague' is vague, then there are no vague predicates. Therefore, metaphysical incoherentism is self-contradictory. The third option states that `Vague predicates exist but do not have any negative instances'. However, the complement of a vague predicate must itself be vague, hence, `nonvague' is vague. However if `nonvague' lacks any negative instances, then all predicates are nonvague. This yields the contradiction `There are vague predicates that are nonvague'.
The collapse of these alternatives leaves only the fourth option. Epistemicism rejects the inductive principle but affirms the paradigm and foil conditions (Sorensen 1988, Williamson 1994). The epistemicist regards `noonish' as a genuine predicate that only looks like it obeys the inductive principle. In particular, he thinks that for any finite sequence that begins with Fs and ends with non-Fs, there will be a threshold at which the Fs end and the non-Fs begin. This differs from saying that the thresholds are part of the meaning of the term. Since the intension of the term F typically comprises infinitely many Fs, the epistemicist allows that `last noonish time' might be as empty as `last fraction less than one'. These predicates only yield a threshold when relativized to a finite sequence -- as in the sorites. Nevertheless, the epistemicist is committed to the extraordinary thesis that an F can be arbitrarily close to a non-F. He must deny that there is any buffer between the Fs and nonFs. The epistemicist can take cold satisfaction in the spectacle of those who insert an intermediary such as a truth value gap or a degree of truth. The exotic neighbor crowds the last F as intolerably as the conservative old neighbor. A newer type of buffer is then enlisted to insulate the Fs from old buffer-Fs. The regress is vicious because each successive buffer is no more satisfactory as a neighbor than the preceding buffer.
Words are useful because they unshackle us from what is immediately perceivable. Instead of confining ourselves to things currently in our perceptual range, we describe things that are out of sight and refer to things that do not currently exist. This is the phenomenon of "displaced speech". However, freedom from our surroundings makes us peculiarly susceptible to existential errors. As Ponce de Leon searched through Florida, he became more and more worried about whether `The Fountain of Youth' denoted.
Descriptions of words do not arouse the same ontological anxiety. Normally, we can immediately produce the word itself and so the existence of the word is incontestible. Occasionally, one is forced to construct a description of a word. Uncertainties about pronunciation may lead me to carefully describe what my Japanese guest uttered after dinner. I may then learn that there is no such word; he just belched. Word games such as "Scrabble" provoke some controversy over whether a string of letters is really a word. But even in Scrabble, such disputes are rare. Native speakers have a strong consensus about the vocabulary of their language.
Well, maybe not the French. Cultural pride has led them to create an official linguistic authority, the French Academy, to defend the integrity of the French language. This institution has underwritten patriotic debate about whether loan words are French, especially "franglaise". Professional linguists refuse to participate in this debate on the grounds that they are experts in descriptive linguistics rather than normative linguistics.
Nevertheless linguists do make impossibility claims about words. One rule of English pronunciation is that the only consonantal letter that can precede a t at the beginning of a word is an s. Hence `stun' is possible, `stum' is possible but not actual, but `rtun' and `ltab' are phonetically impossible. Such constraints are compatible with the principle that words are determined by convention. The result only shows that selection of words must be from a limited range of possible words. We feel unconstrained because the impossible combinations do not occur to us. Nevertheless, there are practical implications for historical linguists. They try to trace the lineage of languages by comparing overlaps in vocabulary. However, limits on which sounds can be articulated ensure that some overlaps will be coincidences. Thus an historical linguist must estimate how much of the overlap is due to chance.
I intend my metaphysical investigation into words to be on the same continuum as the impossibility results of linguists. To say that a word is impossible is to say that it is precluded by a set of laws. `Rtun' owes its impossibility to the law about consonantal letters. Perhaps, this law can be reduced to physiological laws governing the organs composing the speech tract. Other words are impossible relative to physics, say, those that take longer to utter than the duration of the universe. Stronger impossibilities are obtained by relativizing to weaker and weaker laws. At the limit are the laws of logic.
Scientific interest in a law does not completely correlate with its strength along the continuum. Linguists were initially intrigued by Zipf's law that a word's frequency varies inversely with its length. This suggested that there is a deeper psychobiological law from which Zipf's law could be derived. But then Benoit Mandelbort proved that Zipf's law holds for texts produced by a wide variety of processes. Even if we take an English text and designate `e' as the space, we find that the resulting "words" also obey Zipf's law. Noam Chomsky concludes that Zipf's law tells us nothing about human language:
Thus the fact that actual words come fairly close to this predicted distribution is of virtually no interest. It is a fact that can be explained on quite general grounds, in terms of assumptions that could hardly fail to be true. (1971, 46-47)
Likewise, proof that a word is logically impossible cannot show the linguist anything special about our language. The logician's results hold equally well for a Martian language. Nor can logical impossibility results interest linguists by showing an interesting connection with a linguistic law; the logician is not using any empirical premises. Nevertheless, logical results have interested linguists by showing that some "alternatives" are inconsistent and by revealing unanticipated alternatives.
Epistemicism purports to be of interest chiefly as an alternative excluder. By ruling out inductive yet discriminate words, epistemicism narrows the possible behavior of words. In particular, vague words must have discriminative powers that far exceed the recognitional powers of speakers. `The last noonish minute' selects the last noonish minute even though no one can know which minute that is.
So the epistemicist claims to be making logical discoveries about language. He is not echoing Gottlob Frege's call to avoid semantic indetermincy. The kinds of underdefined words Frege found objectionable cannot exist. Avoiding them is inevitable! The epistemicist's metaphysics of words is descriptive rather than revisionary: "Descriptive metaphysics is content to describe the actual structure of our thought about the world, revisionary metaphysics is concerned to produce a better structure" (Strawson 1959, 9). The epistemicist aims to "to lay bare the most general features of our conceptual structure". This is consistent with the postulation of hidden thresholds because conceptual structure "does not readily display itself on the surface of language, but lies submerged" (Strawson 1959, 10).
Words appear ontologically inert because we rarely need displaced speech to refer to them. True, we do need displaced speech about speech when quoting. But concern about whether someone said something is not doubt about whether the word types exist.
People are frequently incredulous when I assert that there is no predicate corresponding to Grelling's definition of `heterological'. After all, did I not just mention the word? If I quoted it, it must be there to be quoted!
Suppose someone tries to prove that `heterological' exists by writing HETEROLOGICAL on a blackboard. This existence proof is reminiscent of Samuel Johnson's "refutation" of George Berkeley's idealism. Kicking a stone does not refute Berkeley because idealism predicts that Johnson's visual and tactile ideas of his making contact with the stone will be followed by ideas associated with the stone's motion. Berkeley admitted that there are stones; he just denied that stones are material objects. Berkeley believed that material objects are inconceivable. Since he sided with common sense in believing that there are stones, Berkeley concluded that stones have a different nature than the materialists ascribe.
The immediacy of an entity may seem to immunize it against metaphysical disorders. Some phenomenalists thought they could cut through skeptical doubts about sense data by conducting private demonstrations. Here is their instant recipe for making a sense datum: Close your eyes. Put your thumb on one eyeball. Apply mild pressure. Wait a few seconds. Note the resulting blob of color in your subjective visual field. Presto, there is your own personal sense datum.
Nevertheless, many contemporary philosophers deny that there are sense data -- at least in the sense needed by phenomenalists. Phenomenalists require that sense data be purely subjective: all of the properties of a sense datum must self-intimating and our beliefs about sense data must be incorrigible. However, mental imagery lacks these epistemological features. For instance, Roderick Chisholm (1942) drew attention to the indeterminacy of one's mental image of a speckled hen. How many speckles does it have? Ten, Eleven, . . . . ? The possessor of the mental image cannot tell. Thus one begins to wonder whether other features of mental images might be overlooked. Given that self-intimation is an essential feature of sense data, one might then deny that one has any sense data. The critic of phenomenalism admits he has mental images, twinges, aches, and so on. He just denies that any of these qualify as sense data.
Mind the illusions of direct reference. When G. E. Moore "proves" the existence of material things by presenting his hand as a sample, he portrays the skeptic as committed to asserting a self-defeating negative existential: `That does not exist'. We can see how `That does not exist' might be true if `that' is being used indirectly: `A monopole is a magnet without a contrast between its north and south pole; that does not exist'. That `that' is just a pronoun of laziness. If I point to a picture of the philosopher's stone, I can say `That does not exist' meaning that the philosopher's stone does not exist. But it is self-defeating to assert that the picture does not exist. For then I am implying that a precondition of my demonstrative use of `that' is not satisfied.
Foundationalists try to parlay demonstrative self-defeat into a form of metaphysical immunity. For example, Russell's principle of acquaintance requires that descriptions be analyzed into statements that make ultimate reference to objects of immediate awareness. That way, one cannot coherently doubt that one's subject matter exists. If `heterological' is defined as an idea that applies to all and only those ideas that do not describe themselves, then it seems to indubitably exist. It is a creature of thought; to deny the word one would have to mentally point to an object of consciousness and say `That does not exist'.
However, it is uncharitable to interpret the skeptic as committed to self-defeating negative existentials. He is not pointing to something and saying `What I am pointing at is not there to be pointed at'. The skeptic is saying that the object as described does not exist.
One must be wary of equivocation between thin and thick readings of `word' (and of other representational terms such as `idea'). Thinly construed, `heterological' is just a concatenation of letters and therefore plainly exists. Thickly construed, the word is composed of a symbol and certain relationships. If Grelling's definition of `heterological' is essential to it, then there can be no such word. If thinly construed, the impossibility is relocated from ontology (the theory of what there is) to cosmology (the theory of how they work).
The thin/thick distinction applies to larger linguistic units. W. V. Quine's `yields a truth when appended to its own quotation' is a phrase in the thin sense of being a string of genuine words composed in accordance with ordinary grammar. However, the string is not a phrase in the thick sense of words collectively composing a coherent satisfaction condition.
Words are things and so pose the standard sort of metaphysical dilemmas. With the sorites, a conflict has been revealed between our beliefs about the existence of a class of things (vague predicates) and our beliefs about their properties (that vague predicates are inductive and yet discriminative).
Sometimes we solve an existence versus property dilemma by retaining our beliefs about the properties (Unger 1982). This choice lies behind the inference to the non-existence of phlogiston, perpetual motion machines, and the arbitrary individuals of early quantification theory. Much more frequently, we "save the appearances" by changing beliefs about the properties of the troubled entity. `Atom' was traditionally defined as the smallest unit of matter. After nuclear fission was discovered, scientists spared atoms by redefinition. `Gene' was saved by postulating an ambiguity between the smallest unit of mutation, the smallest unit of function, and the smallest unit of recombination. The resolutions of other conflicts between existence beliefs and nature beliefs are controversial. Most biologists oppose the extinction of `species'. But some think that species must be immutable and conclude that Charles Darwin was right to epitomize his revolution as a denial of species.
Epistemicism conforms to the pattern of preferring to revise property beliefs rather than existence beliefs. It is far more plausible that there are vague predicates and that we have misunderstood their nature than that we have correctly understood their nature and so must infer their non-existence. Indeed, the previous section shows that the preference for changing property beliefs about vagueness is over-determined by the self-defeating character of sacrificing the existence of vague predicates.
Ontological conservatism even offers a second chance to `heterological'. Instead of flatly denying that the predicate exists, conservatives say that `heterological' means something slightly different from Grelling's explicit definition. Consider the idea of language levels popularized by Alfred Tarski's theory of truth. Tarskians interpret `heterological' as a systematically ambiguous predicate; it always means `heterological in language L' where L is a lower level language. This makes the question `Is `heterological' heterological?' incomplete. Nevertheless, the Tarskian account of `heterological' is still a mixed solution because it does exclude the existence of some predicates. ("But if we have to say this in the end, why not say it at the beginning?" (Thomson 1962, 114)) For example, Tarskians deny that there can be a predicate that is true of any heterological predicate of any level.
Thomson himself tries to save appearances about `heterological' by emphasizing that people normally understand words irreflexively. When they hear that `Maurice loves every lover' they rarely infer that Maurice loves himself. If `heterological' is understood irreflexively then it is not ruled out by Thomson's theorem.
Indeed `heterological' has acquired some currency in academic English. We could interpret this use of `heterological' as a non-literal use. An astronomer who says `There are a jillion stars' conveys the idea that there are stupendously many stars even though the utterance is made meaningless by the non-word `jillion'. Just as we can communicate truths via false statements (as in sarcasm, irony, and metaphor), we can convey truths via meaningless utterances. Xlikeyz xthisyz xoneyz.
We need not argue the merits of how much of `heterological' should be salvaged by which method or combination of methods. It suffices that we acknowledge that all the methods entail some predicate restriction. I intend this `we' to be maximally inclusive. Even a deviant logician must limit himself to predicates that really exist. Denying classical theorems cannot summon non-existent predicates into being. A deviant logic is not a seance. It is a rival theory of what follows from what. This rivalry presupposes agreement on the "what" part. Deviant logic and classical logic have had no debates on vocabulary. Indeed, deviant logicians have been content to follow customary formalization procedures: stating the vocabulary, the connectives, the conditions for a well formed formula, and the rules of inference.
Deviant logicians have recognized that reliance on a classical meta-logic will prevent them from recognizing higher order vagueness (Kamp 1981, 254-5). In order for there to be a borderline `borderline F', the meta-logic must contain whatever device the deviant uses for representing borderline cases -- truth value gaps, degrees of truth, etc. Classical logic lacks these innovations. Nevertheless, some deviant logicians have been content to just formalize first order vagueness:
Our models are typical purely exact constructions, and we use ordinary exact logic and set theory freely in their development. This amounts to assuming we can have at least certain kinds of exact knowledge of inexact concepts. (When we say something, others may know exactly what we say, but not know exactly what we mean.) It is hard to see how we can study our subject at all rigorously without such assumptions. (Goguen 1969, 327)
Goguen's point is that the deviant should be willing to settle for half a loaf (a representation of only first order vagueness) in the interest of rigor.
However, the use of a classical meta-logic will also preclude the deviant from representing first order vagueness. Any predicate that can be used at the object level must be mentioned at the meta-level when stating the object language's vocabulary. If the meta-logic implies that the predicate does not exist, it cannot declare that the predicate is an element of the object language. Classical logic would still have a filtering effect if deviant logicians were to devise a deviant meta-language but kept classical logic as the meta-meta-language.
A deviant logician might be tempted to cede this apparently modest degree of ontological hegemony to classical logic. After all, the deviant wanted to agree on what the propositions are in order to disagree about what the propositions entail. However, the concession would constitute capitulation on the particular application to vagueness. For the applicability of minimal classical logic yields:
Threshold theorem: Vague predicates exist and are discriminative but not inductive.
The point of applying deviant logic to vagueness was that it seemed unrealistic to postulate thresholds. Deviant logic was recruited because it is supposed to be better suited to "genuinely vague" predicates. But now we find that these purported predicates have been expurgated by the meta-language used by the deviant logician. The very process of setting up the apparatus needed to recognize "genuine vagueness" precludes it.
Supervaluationists and many-valued logicians emphasize that their systems converge with classical logic when the predicates are precise. They both attempt to co-opt classical logic as being accurate for (but only for) the limiting case of complete precision. Therefore, deviant logic can aspire to be a rival logic of vagueness only if its placeholders for predicates can be filled by some predicates that are both inductive and discriminative. If the system lacks this potential for interpretation, then it is either intensionally isomorphic with classical logic or a bare mathematical structure. Either way, the "deviant" logic turns out to be a pseudo-rival.
The situation is reminiscent of the deviant logician's dilemma. W. V. Quine (1970, 80-83) argued that the criteria governing the translation of the speaker's logical connectives guarantee that they must mean the same as the translator's. If an anthropologist reports that a tribe uses `p or q' to cover only the situation in which both p and q are true, then we will reply that he should interpret the tribe as meaning `p and q'. Similarly, the deviant logician cannot be interpreted as disagreeing with us about the meaning of the logical connectives. The very attempt to disagree just changes the topic.
My conclusion extends beyond the logical words and does not rely on premises about translation. It is a metaphysical argument about words:
1. Classical logic implies that there are no predicates that are both inductive and discriminative.
2. If a higher order language implies the impossibility of a predicate, then no such predicate can appear in one of its lower order languages.
3. All deviant logics use classical logic in a higher order language.
4. A deviant logic can be a rival account of vagueness only if its vocabulary contains some predicates that are both inductive and discriminative.
5. Therefore, no deviant logic is a rival account of vagueness.
To avoid irrelevance to the sorites, the deviant logician must do two things. First, he must abstain from classical logic at any meta-level. (More about abstinence in the next section.) Second, the meta-logic must disagree on the ontology of predicates without opening the floodgates. As a logician, the deviant accepts the established lore about the need to distinguish grammatical from logical form. For example, he is appreciative of circumspection about whether `exists' is a predicate. Should his interests encompass induction, he will wish to treat `grue' and `non-raven' differently from `green' and `raven'.
But most crucially, the deviant logician must rule in inductive yet discriminative predicates while still ruling out `heterological' and `tonk' and their anarchical brethren. Even the father of relevance logic, Nuel Belnap (1978), bans `tonk' -- and `plonk' and `plink' -- to avoid inconsistency. Thomson's theorem is provable in relevance logic, so Belnap should also ban `heterological'. The question is not whether to ban words but which words to ban. There is a powerful analogy between the infinitely many words that the deviant wishes to ban and the infinitely many the epistemicist wishes to ban. Thus it would be ad hoc of the deviant logician to exempt the inductive yet discriminative predicates.
At this point, a deviant logician might just thank me for showing that it is pointless to settle for half a loaf. He might agree that any logic of indeterminacy must be a meta-logic of indeterminacy. And indeed, some deviant logicians (Fine 1975, Tye 1994) have proposed schemes for making vagueness "go all the way up". None of these outlines purports to be a fully developed meta-theory. Unlike the ordinary language character of ancient Indian mathematics, these informal discussions are fortified with formalisms. Nevertheless, deviant "meta-theories" are merely sketches that resemble formal systems.
Outlines are fine as long as there is a prospect of them being filled in. However, complete specifity is incompatible with indeterminacy -- and only complete specificity satisfies the exacting standards of meta-theory. Meta-theory is a study of sentences used to express the truths of logic. These sentences must be formulas of a formal language. By definition, a formal language must be completely specified without any reference to the meaning of the formulas. This requirement of uncompromising definitude is the essential difference between informal discussion of logic and the formal enterprise that is meta-theory.
Suppose I attempt a vague meta-theory by specifying my well-formed formulas. I say the symbols of this language are the roundish letters of the English alphabet (such as o, a, q but not w, t, z, with r, s, c as borderline cases). The one formation rule is that any short string of symbols is a formula. Thus `oa' is a definite formula, `wtzwtzwtzwtz' is a definite non-formula, and `oaooao' is a borderline formula. The deductive apparatus consists of the axiom oa and the rule of inference that from any pretty formula one can infer any other pretty formula.
This is a parody of a formal system because it violates the necessary condition of rigor. One is free to say that an informal account is potentially a formal system just as an acorn is potentially an oak tree. But potential rigor is not itself rigor. Hence, the supervaluationist cannot say that a supervaluational meta-theory meets the requirement of rigor by virtue of its potential refinements. Nor can one say that this is just rigor deferred to a higher order system. For a rigorous meta-system would exert the restrictive hegemony described above for classical logic. Nor can one say that a vague formal system is just rigor delayed. For deferring to a future rigorous language has the same predicate restricting effect. Ditto for any hypothetical system.
If the object language defers to an endless chain of indeterminate meta-languages, then there is an infinite regress of definitions. If it does not, then the definition is incomplete.
`Indeterminate logic' strikes many people as an oxymoron. Usage of `logic' may be too loose to directly substantiate this impression. However, the more technical term `meta-theory' is well enough defined to make `indeterminate meta-theory' a contradiction in terms. Given that `indeterminate logic' implies `indeterminate meta-theory', the contradiction in `indeterminate logic' emerges indirectly.
If I am begging the question against the deviant logician, then he should be able to sensitize me to the injustice by begging the question against me. However, this role reversal cannot be effected simply by noting that a precise language can have a vague meta-language. When students discuss standard logic in English, they are using a language that permits any term needed by that object language. The predicate restricting effect will only be exerted by a vague meta-language that also restricts English. Since deviant logicians wish to model the permissiveness of natural language, they will have independent reason to reject such meta-languages as overly repressive.
One can discuss a two-valued logic with a three-valued logic. However, the many-valued logic will not restrict the predicates available to the two-valued logic. For it is a weakening of two-valued logic -- less follows in many-valued and supervaluational systems, not more. Since there are fewer valid arguments, there are fewer impossibility results. Any logic (properly) stronger than classical logic would also preclude inductive yet discriminative predicates. Thus a successful rival logic of vagueness would need to be weaker in one respect (to preserve the "vague" predicates) and stronger in another (to preclude predicates needed by the epistemicist's solution to the sorites). Intuitionism may seem like a candidate. However, Crispin Wright (1994, 140-141) has shown that the epistemicist's meta-sorites against tolerant predicates is intuitionistically valid.
The logician is in the habit of separating vocabulary from inference rules and theorems. Deviant logicians follow this pattern. They have not quarreled with the expressive power of classical logic. Indeed, that is why they have been content to use classical logic to set up their deviant logics. However, if the deviant opens a debate about the existence of inductive yet discriminative predicates, he must portray classical logic as expressively incomplete. The deviant must say that classical logic restricts the content of our thoughts rather than merely mis-codifying our inferential practices.
Indeed, the deviant logician should maintain that a genuine epistemicist can neither agree nor disagree with the deviant's thesis that there are vague predicates. For if there are inductive yet discriminative predicates, then the vagueness of `vague' would ensure that `vague' is inductive yet discriminative. So the deviant will picture the epistemicist as unwittingly banning `vague' itself.
The epistemicist will insist that he has not banned any vague predicates. He will assure the deviant that they agree that vague predicates exist and that their disagreement is restricted to the nature of vague predicates. Thus the epistemicist contrasts himself with a word banner such as Rudolph Carnap. Carnap (1950, chapter 1) proposed that the sorites be avoided by expurgating all vague predicates from the language in which one philosophizes.
Or did he? Carnap's proposal uses the vague term `vague'. Hence his proposal cannot be stated in the expurgated language. Nor can he rephrase his proposal as requiring that all of the vocabulary be precise. For this use of `precise' is the complement of `vague', and the complement of a vague predicate must itself be vague.
Carnap's proposal also shares the linguistic shortcomings of George Orwell's proposal about thoughtcrime. In his disutopian novel 1984 totalitarian authorities are developing "Newspeak", a language designed to prevent wrongdoing by preventing bad intentions. Bad intentions require concepts of freedom, rights, and so forth. Since the expurgated language "Newspeak" lacks the words to express these concepts, the rulers hope that thoughtcrime will become impossible. The basic problem with the Newspeak proposal is that it assumes the primacy of language over thought. However, linguists have demonstrated that concepts can exist without any words for them (Pinker 1994, 82). Children who are only exposed to a simplified hybrid language (a pidgin) will spontaneously inject a rich syntax and attach new senses to old words thereby developing a language (a creole) that is as expressive as any natural language.
Creolization is not youthful rebellion. The children are trying to defer to their parent's language. However, an instinct embodying universal grammar inadvertently sophisticates the pidgin. Therefore, even if Newspeak were developed, the speakers would still think about freedom and still form politically incorrect intentions. Unless the mind control occurs pre-linguistically (say through pediatric brain surgery), their children or grandchildren will creolize Newspeak and undo the expurgation.
Carnap's regimented language will be equally ineffective against the sorites. Although the speakers may not be able to formulate the paradoxes of vagueness in Carnapspeak, the paradoxes will still bedevil their thinking. Unless Carnapspeak is constantly policed, it will suffer the fate of Newspeak and become "corrupted" by the creativity of our innate mental organ for language.
Some deviant logicians have emulated Carnap's restrictivism by banning classical connectives. However, given that their mission is to more closely model natural language, they should permit us to add new words. So what stops us from reintroducing the connectives that re-ignite the sorites? You cannot solve the paradox of the heap by hiding your head in the sand.
Carnap's key error is his assumption that there are genuine predicates to ban. This error is motivated by his latitudiniarism about verbal matters. We all take diplomatic satisfaction in showing that a metaphysical issue is just a verbal dispute. And we all endorse tolerance over how things are described given that the meaning stays the same. Carnap, in this spirit, pictured semantic ascent as an escape from metaphysics. Instead of talking about things, one talks about words. But the substitution is only an escape from metaphysics if words do not raise ontological and cosmological issues. For if they do, then semantic ascent will not work for the special case of words themselves. The strategy behind semantic ascent is that, for philosophical purposes, words are easier to study than what they represent. However, words cannot be easier to study than words.
Carnap tends to view words as metaphysically neutral. This attitude lies behind his distinction between internal and external questions (Carnap 1947, 206). Internal questions arise from within a linguistic framework. For example, when one asks `Do numbers exist?' from within standard arithmetic, the answer is yes. But one could be asking the question external to any such a background. Then, says Carnap, the issue is the practical one of whether one should adopt a linguistic framework that permits talk about numbers. Carnap diagnoses much metaphysics as a conflation between internal and external questions.
However, the distinction between internal and external questions presupposes that the alternatives are genuine linguistic frameworks. Witness the triviality of `Is there a linguistic framework in which there are linguistic frameworks?'. The external question of whether a certain type of discourse is expedient assumes that the vocabulary of that discourse is consistent. Carnap's anti-metaphysical snake cannot eat its own head.
Often, semantic ascent clarifies. It is, in general, salutary to separate words and from other things. For we do have a tendency to reify, to fall into verbalism, and to revert to word magic. But it is equally superstitious to treat the de-mystifying medium of words as forming a privileged realm immune to ontological investigation. Once we break the positivistic spell, we are more likely to apply as stringent standards for words as for other things. We will then be no more inclined to locate indeterminacy in predicates than in what they represent.
* This paper has been improved by words of wisdom from Kent Bach, Keith DeRose, Carsten Hansen, Roy Mash, and Peter Unger.
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