TRANSVALUATIONISM: A DIONYSIAN APPROACH TO VAGUENESS

TRANSVALUATIONISM: A DIONYSIAN APPROACH TO VAGUENESS

Terry Horgan, University of Memphis

I advocate a two-part view concerning vagueness. On one hand I claim that vagueness is logically incoherent; but on the other hand I claim that vagueness is also a benign, beneficial, and indeed essential feature of human language and thought. I will call this view transvaluationism, a name which seems to me appropriate for several reasons. First, the term suggests that we should move beyond the idea that the successive statements in a sorites sequence can be assigned differing truth values in some logically coherent way that fully respects the nature of vagueness--a way that fully eschews any arbitrarily precise semantic transitions.[1] We should transcend this impossible goal by accepting that vagueness harbors logical incoherence. Second, just as Nietzsche held that one can overcome nihilism by embracing what he called the transvaluation of all values, my position affirms vagueness, rather than despairing in the face of the logical absurdity residing at its very core. This affirmation amounts to a transvaluation of truth values, as far as sorites sequences are concerned. Third, the term 'transvaluationism' has a nice ring to it, especially since one of the principal philosophical approaches to vagueness is called supervaluationism.

I will call the first claim of transvaluationism, that vagueness is logically incoherent, the incoherence thesis. I will call the second claim, that vagueness is benign, beneficial, and essential, the legitimacy thesis. The legitimacy thesis, taken by itself, seems overwhelmingly plausible; anyone who denies it assumes a heavy burden of proof. But prima facie, it seems dubious that the legitimacy thesis can be maintained in conjunction with the incoherence thesis. For, there is reason to doubt whether there is any cogent way to embrace the incoherence thesis without thereby becoming mired in what Williamson (1994) calls global nihilism about vagueness--the view that vague terms are empty (i.e., they do not, and cannot, apply to anything). Global nihilism, Williamson argues, has such destructively negative consequences that it does not deserve to be taken seriously--for instance, the consequence that vastly many of our common sense beliefs are false, and the consequence that these beliefs are not even useful (since the constituent terms in 'Common sense beliefs are useful' are vague and hence this statement turns out, given the incoherence thesis, to be false itself).[2] In short, the idea that one can adopt the incoherence thesis and then somehow transcend nihilism might initially seem hopelessly optimistic; transvaluationism would then be an unattainable, chimerical, goal rather than an intelligible and conceptually stable position concerning vagueness.

Given certain widely held philosophical views about how language and thought must map onto the world in order for statements and the beliefs they express to be true--views that fall appropriately under the label 'referential semantics'--transvaluationism probably is a chimerical goal. From the perspective of referential semantics, if we attempt to combine Frege's view that vagueness is logically incoherent with Wittgenstein's view that vagueness is a legitimate and essential feature of language, we seem to produce a conceptual monster--a Fregenstein monster.

But I myself favor an alternative general theoretical framework concerning language/world relations--an orientation I have been attempting to develop in a series of papers, some co-authored with Mark Timmons.[3] Originally I called it "language-game semantics," then "psychologistic semantics," but I no longer like either label; Timmons and I now call the orientation contextual semantics. I maintain that contextual semantics has some important theoretical advantages over referential semantics, and thus is independently credible quite apart from matters of vagueness. But I also believe that the incoherence thesis fits plausibly and naturally within contextual semantics, in a way that renders transvaluationism not only intelligible but also quite viable.

My principal goal in the present paper is to articulate this package-deal version of transvaluationism, and to argue that this position treats the logical, semantical, and metaphysical aspects of vagueness in a very attractive way.[4] Although I do advocate a kind of Fregenstein view, I think that the Fregenstein option concerning vagueness is not really a conceptual monster at all; it is merely misunderstood.

1. Contextual Semantics.

I begin by describing the broad approach to language/world relations that Timmons and I call contextual semantics.[5] This framework has been evolving and developing in a series of papers; articulating and exploring it in further detail is a large-scale, long-term, research project of mine. The overall framework includes theses not only about truth and falsity per se, but also about meaning, ontology, thought, and knowledge.

Contextual semantics, as I think of it, is intermediate between two prevalent orientations toward language, truth, and ontology in recent analytic philosophy--between (i) a position viewing truth as direct correspondence between language and the mind-independent, discourse-independent, world; and (ii) a position viewing truth as radically epistemic (e.g., as warranted assertibility, or as ``ideal'' warranted assertibility (Putnam), or as "superassertibility" (Wright)).[6] (Radically epistemic construals of truth often are wedded to global metaphysical irrealism, according to which there is no such thing as a discourse-independent, mind-independent, world at all.) These two perspectives might be called, respectively, referential semantics and neo-pragmatist semantics (or, referentialism and neo-pragmatism).

In articulating the distinctive claims of contextual semantics, and for related expository purposes throughout the paper, I will borrow from Hilary Putnam the device of sometimes capitalizing terms and phrases like `object', `property', and `the world'; this makes it unambiguously clear when I mean to be talking about denizens of the mind-independent, discourse-independent, world--the world whose existence is denied by global irrealists. (Global irrealists typically regard as perfectly legitimate various everyday uses of the uncapitalized expressions, and some of their philosophical uses as well. The capitalization convention guarantees that claims which I intend to be incompatible with global irrealism will be construed as I intend them, rather than receiving a "compatibilist" reading.)

I will set forth contextual semantics as a list of theses, interspersed with commentary:

(1) The semantic concepts of truth and falsity are normative. Truth is correct assertibility; falsity is correct deniability.

Since we deny statements by asserting their negations, a statement is correctly deniable just in case its negation is correctly assertible. So henceforth I will usually speak only of "correct assertibility."

(2) Contrary to neo-pragmatism, truth is not radically epistemic; for, correct assertibility is distinct from warranted assertibility, and even from ``ideal'' warranted assertibility and from "superassertibility."[7]

This thesis says, in effect, that the kind of semantic normativity that makes for truth and falsity is not reducible to epistemic normativity.

(3) Standards for correct assertibility are not monolithic within a language; instead they vary somewhat from one context to another, depending upon the specific purposes our discourse is serving at the time.

Not only do assertibility standards often vary from one mode of discourse to another, but they also often vary within a given mode of discourse. For instance, what counts as flat surface is subject to contextually variable parameters within a given discourse. Similarly, what counts as the contextually eligible referent of a definite description like 'that guy we were talking with awhile ago', in a situation where several distinct entities in the relevant domain of quantification are eligible referents, is subject to contextually variable parameters. (Such parameters determine what David Lewis (1979) calls "the score in the language game.")

(4) Contrary to global metaphysical irrealism, correct assertibility is normally a joint product of two factors: (i) the relevant assertibility norms; and (ii) how things actually are in THE WORLD.

I will say that the operative semantic standards in a given discourse context are maximally strict provided they have this feature: under these norms a sentence counts as correctly assertible (i.e., as true) only if there are OBJECTS and PROPERTIES in THE WORLD answering to each of the sentence's constituent singular terms, constituent assertoric existential quantifications, and constituent predicates.[8] The next two theses employ this notion.

(5) Contrary to referentialism, our discourse often employs standards of truth (i.e., correct assertibility) that are not maximally strict.

I.e., even though truth does typically depend upon how things are with THE WORLD, often this dependence is not a matter of direct correspondence between the constituents of a true sentence and OBJECTS and PROPERTIES. When the assertibility norms are not maximally strict, the dependence is less direct.

Under contextual semantics, there is a whole spectrum of ways that a sentence's correct assertibility can depend upon THE WORLD.[9] At one end of the spectrum are sentences governed by assertibility norms, in a given context of usage, that are maximally strict (and thus coincide with those laid down by referentialism); under these norms a sentence is true only if some unique constituent of THE WORLD answers to each of its singular terms, and at least one such entity answers to each of its unnegated existential-quantifier expressions. (Sentences asserted in order to make serious ontological claims--like the sentence 'There exists an all-powerful, all-knowing, perfectly good God', as asserted by a conventional theist--are plausible candidates for this status.) At the other end of the spectrum are sentences whose governing assertibility norms, in a given context, are such that those sentences are sanctioned as correctly assertible by the norms alone, independently of how things are with THE WORLD. (Sentences of pure mathematics are plausible candidates for this status.) Both ends of the spectrum are limit cases, however. Various intermediate positions are occupied by sentences whose correct assertibility, in a given context, does depend in part on how things are with THE WORLD, but where this dependence does not consist in direct correspondence between (i) the referential apparatus of the sentence (its singular terms, quantifiers, and predicates), and (ii) OBJECTS or PROPERTIES in THE WORLD.[10]

As a plausible example of a statement that normally would be governed by semantic norms falling at an intermediate point in the spectrum just described, consider:

(B) Beethoven's fifth symphony has four movements.

The correct assertibility of (B) probably does not require that there be some ENTITY answering to the term `Beethoven's fifth symphony', and also answering to the predicate `has four movements'. Rather, under the operative assertibility norms, (B) is probably correctly assertible (i.e., true) by virtue of other, more indirect, connections between the sentence and THE WORLD. Especially germane is the behavior by Beethoven that we could call ``composing his fifth symphony.'' But a considerably wider range of goings-on is relevant too: in particular, Beethoven's earlier behavior in virtue of which his later behavior counts as composing his fifth symphony; and also a broad range of human practices (including the use of handwritten or printed scores to guide orchestral performances) in virtue of which such behavior by Beethoven counts as ``composing a symphony'' in the first place. Further plausible examples of statements governed by semantic norms that are not maximally strict include:

(a) The University of Memphis is a public institution.

(b) Mozart composed 27 piano concertos.

(d) Quine's Word and Object is an influential book.

Although contextual semantics asserts that the operative semantic standards governing truth (correct assertibility) can vary from one context to another, it also asserts that contextually operative metalinguistic semantic standards normally require truth ascriptions to obey Tarski's equivalence-schema T:

(6) Even in discourse contexts where the operative semantic standards are not maximally strict, typically these standards sanction as true (i.e., as correctly assertible) instances of Tarski's equivalence-schema:

(T) "P" is true if and only if P.[11]

Thesis (6) says, in effect, that normally the contextually operative semantic standards governing the truth predicate operate "in tandem" with those governing first-order discourse; as I put it in Horgan (1990 b), truth talk is assertorically consistent with first-order talk.

If contextual semantics is right, so that truth is intimately bound up with assertibility norms, then meaning too is intimately bound up with these norms.[12] Intuitively and pre-theoretically, meaning is what combines with how THE WORLD is to yield truth. Thus, if truth is correct assertibility under operative assertibility norms, then the role of meaning is played by the assertibility norms themselves. So matters of meaning are, at least in large part, matters of operative assertibility norms.[13] Contextual semantics makes the following nonreductionist claim about matters of meaning:

(7) In general, if a statement S is correctly assertible under certain frequently operative semantic standards, but S is not correctly assertible under maximally strict semantic standards, then S is not equivalent in meaning to--or approximately equivalent in meaning to, or "intensionally isomorphic" to, or "regimentable" into--a statement that is correctly assertible under maximally strict assertibility standards.

Thesis (7) is one I came to believe after pursing for some time the project of trying to systematically paraphrase ("regiment," in Quine's terminology) statements whose surface grammar embodies an apparent commitment to ontologically dubious entities, into a more austere idiom that eschews reference to such entities. Although the paraphrase strategy can sometimes be carried through piecemeal for certain local segments of discourse, very often it evidently will not work. (Trying to implement the strategy for statements like (B) and (a)-(d) caused me to lose faith in it.)

Under contextual semantics the issue of ontological commitment becomes much more subtle than it is under referential semantics, because whenever the contextually operative assertibility standards are not maximally strict, the so-called "referential apparatus" of our discourse need not connect directly to OBJECTS and PROPERTIES in the world in order for our statements to be true. Here then are several theses concerning ontology:

(8) It is necessary to distinguish between regional ontology, which concerns the range of putative entities overtly posited by a given mode of discourse; and ultimate ontology, which concerns the range of entities posited by statements which are correctly assertible under maximally strict assertibility standards.

(9) Quine's well known criteria of "ontological commitment" are directly relevant only to regional ontology, not to ultimate ontology.

(10) Determining the ultimate ontological commitments of our scientific and non-scientific discourse is a methodologically subtle matter, in which we inquire what THE WORLD is like IN ITSELF in order to be correctly describable, under various contextually operative assertibility standards, by those statements that are true in everyday life and in science.

Whatever exactly the right story is about ultimate ontology, it seems quite plausible that a complete and accurate accounting of what there really IS in the WORLD need not include entities like the State of Tennessee, the U.S. Federal Government, Mozart's 27th piano concerto, or Quine's book Word and Object. In terms of ultimate ontology, such entities are artifacts of our conceptual scheme; they are not mind-independently, discourse-independently, REAL. Although THE WORLD does normally contribute to the truth or falsity of statements that are regionally ontologically committed to such entities, it does so quite indirectly. As one might put it, such statements provide a trace--a trace of THE DING as it is AN SICH.

Although contextual semantics rejects the epistemic reductionism of neo-pragmatism, it also acknowledges something importantly right that is reflected in that approach, viz.:

(11) Contextually operative standards for correct assertibility are typically intimately linked to prototypical evidential conditions for statements.

We all know quite well, for instance, what sorts of evidence are relevant to claims like (B) and (a)-(d); and the kind of evidence we would look for has rather little to do with the philosophical question whether ultimate ontology should include entities like SYMPHONIES, PIANO CONCERTOS, BOOKS, or a FEDERAL GOVERNMENT. Under the assertibility standards operative in ordinary discourse contexts, it is quite appropriate that the relevant epistemic standards should bypass the issue of ultimate ontology; for, the semantic standards themselves are not maximally strict. There is a comparatively small "conceptual gap" between the epistemic standards for warranted assertibility and the semantic standards for correct assertibility (even though semantic standards are not reducible to epistemic ones). There is a gap though, in part because of the holistic aspects of evidence:

(12) Our attributions of truth and falsity usually are defeasible even under prototypical evidential conditions; for, the correct assertibility of any given statement normally depends, in part, on the correct assertibility of various other statements which are assumed, in a given evidential situation, to be correctly assertible themselves.

As Quine and Duhem stressed long ago, our statements really face the tribunal of empirical evidence jointly, not singly.

Contextual semantics also includes a psychologistic dimension (which, as John Biro has urged on me, might better be called psycho-social):

(13) Which assertibility standards are the operative ones, in any given context of discourse, depends largely upon the contextually attuned, socially coordinated, truth-judging and falsity-judging dispositions of competent speakers.

The interconnections between the judgment dispositions of competent speakers and the contextually operative assertibility norms are typically fairly subtle; surely no crudely reductive account will work. (For one thing, even competent speakers often exhibit linguistic performance errors. For another, normally a competent speaker's judgement dispositions are more directly indicative of what is warrantedly assertible given available evidence; and sometimes this diverges from what is correctly assertible under contextually operative semantic norms.) Nonetheless, such socially coordinated psychological dispositions do figure importantly in determining the contextually operative assertibility standards.

Contextual semantics has various points of contact with the views of other philosophers on language/world relations. It seems to me a natural and plausible extension, for instance, of the treatment of contextually variable discourse parameters in Lewis (1979). Likewise, it seems to me a natural further step in a direction already taken by advocates of philosophical projects of "regimentation": viz., the direction of denying that the surface ontological commitments of true statements always constitute ultimate ontological commitments. I have already mentioned that it accommodates certain motivating ideas in neo-pragmatism (and in verificationism), but without the mistake of embracing epistemic reductionism. There are echoes of Carnap's famous contention (Carnap 1950) that a "linguistic framework" can automatically sanction existence claims concerning the entities posited by the framework, and that such existence claims are ontologically innocent. The approach is somewhat similar to the treatment of truth and ontology in Sellars (1963, 1968).[14] Finally, contextual semantics seems to me rather similar in spirit to the general approach to truth, and to philosophical debates about realism and irrealism concerning various forms of discourse, in Wright (1992).[15]

There are further theses to be added to the 13 mentioned in this section: theses concerning vagueness. But before turning to those, let me briefly mention some considerations in favor of contextual semantics as thusfar articulated.[16]

Among the advantages of this general approach to semantics are the potential resources it provides for accommodating various forms of discourse within a naturalistic worldview. Take sentences like (B), for example. Evidently, an adequate semantics for sentences like (B) should be semantically nonreductionist; for, no plausible-looking way of systematically paraphrasing such sentences into a more austere idiom is even remotely in sight. If the notion of truth works in the way just characterized, then even though semantic reductionism evidently won't fly, we can still accommodate symphony discourse as literally true, and can accommodate assertions like (B) as knowable, without being forced to populate THE WORLD with SYMPHONY TYPES.

On the other hand, if we try construing (B) in terms of referentialism, and also accept that (B) is true, then we must try accommodating SYMPHONY TYPES, tokenable by concrete performance-events, within a naturalistic metaphysics; and we must face the correlative task of accommodating them in a manner that allows for genuine knowledge about such ENTITIES. This is no small task, especially since there will be strong theoretical pressure to consign these putative, abstract, ENTITIES to Plato's non-spatio-temporal HEAVEN--which in turn will seriously exacerbate the task of giving a naturalistically acceptable account of how humans can know about them (and can refer to them).

2. Vagueness and Contextual Semantics.

I turn now to my proposed way of treating vagueness, within the framework of contextual semantics. In principle, various approaches to be vagueness could be situated within this framework, some perhaps quite different from others.[17] But I will cut to the chase directly, focusing only upon the kind of approach which seems to me the one that is most plausible and most theoretically attractive.

Here is the basic picture I advocate, in a nutshell: Vagueness harbors a certain sort of logical incoherence. This logical incoherence does have nihilistic consequences for discourse conducted under maximally strict semantic standards (which means that there can be no vague OBJECTS or PROPERTIES in THE WORLD). But this does not amount to nihilism tout court. For, contexts in which the operative semantic standards are maximally strict are an extreme, and in fact highly unusual, limit case. Furthermore, in the vast majority of typical contexts of usage, the operative semantic standards work in such a way that (1) certain statements that make vague predications are true and others are false (correctly assertible, correctly deniable); (2) the underlying logical incoherence gets quarantined rather than exerting a malignant and destructive effect on language and thought (as I put it below, vagueness-sanctioning discourse is logically disciplined, even though it is not logically coherent); (3) sorites arguments are effectively blockable; and (4) not only are the benefits of vagueness in language and thought not undermined by the logical incoherence, but (on the contrary) the very feature of vagueness that largely generates its benefits--a feature I call robustness--is also the feature that harbors incoherence.

2.1. Vagueness: Robust, Discriminatory, and Logically Incoherent.

By robustness I mean the idea that there is no precise fact of the matter about semantic transitions among the respective statements in a sorites sequence. Robustness, I maintain, is an essential feature of genuine vagueness; if there are semantic transitions at all in a sorites sequence (as common sense supposes there is), then there is no precise fact of the matter about what they are. Furthermore, robustness does not simply mean that there is no precise point of transition between truth and falsity; it means that there are no precise semantic transition-points of any kind in a sorites sequence. This rules out, for instance, a precise transition point between truth and non-truth; a precise transition point between non-falsity and falsity; and (if we suppose that truth comes in degrees) precise transitions between specific degrees of truth. So I hold this thesis:

(14) Vagueness is robust--i.e., in a sorites sequence that exhibits semantic transitions, there is no precise fact of the matter about those transitions.[18]

When I say that vague terms are discriminatory I mean that for typical sorites sequences, if some of the statements in the sequence are true then others are false.

(15) Vagueness is discriminatory--i.e., for typical sorites sequences, if some of the statements in the sequence are true then others are false.[19]

In ordinary discourse contexts, we use vague terms in a way that purports to be discriminating (as I will put it)--i.e., non-vacuously discriminatory, so that some statements in a typical sorites sequence are true and others are false.

Vagueness is essentially robust. It is also essentially discriminatory: vague predicates would lose their very point and purpose if they applied to everything, both actual and merely possible, to which they can be sensibly predicated.

But when one considers carefully the robustness of vagueness, the notion "no precise fact of the matter about semantic transitions," it turns out that vagueness is logically incoherent, in the following sense: the robustness condition and the discriminatoriness condition jointly generate semantic requirements, for the respective statements in a sorites sequence, which cannot be simultaneously satisfied if any of the statements in the sequence are true. (The only way to assign truth values to a sorites sequence without violating either condition is a degenerate way: assign falsity to every statement in the sequence.) Robustness is the real culprit in this logical incoherence, as it turns out; the role played by discriminatoriness is to rule out the degenerate assignment in which all the statements in the sequence are assigned truth.

I will now describe the structure of this incoherence. The description I will give is also intended to work as an argument in support of the incoherence thesis: I will be pointing to apparent features of vagueness which are apparently essential to it, but which also appear to jointly generate mutually unsatisfiable semantic requirements for statements in a sorites sequence.

The notion of robustness exhibits a certain conceptual bi-polarity, and its two poles are in tension with one another whenever the contextually operative semantic standards impose a discrimination requirement (as I'll put it)--i.e., whenever those standards render some of the statements in a sorites sequence true (so that the discriminatoriness condition applies non-vacuously rather than vacuously). One pole is individualistic: it involves statements in a sorites sequence considered singly, in relation to their immediate neighbors in the sequence. The semantic requirements of the individualistic pole, and the fact that these requirements are in conflict with the discrimination requirement, rise to the surface in sorites reasoning, as follows. Consider a typical sorites sequence of statements. By the discrimination condition, statements early in the sequence are true and statements late in the sequence are false. Consider any true statement S_i. Given that S_i is true, S_i+1 must also be true; for otherwise there would be a precise semantic boundary between S_i and S_i+1, contrary to the robustness of vagueness. So S_i+1 is true. By iteration of this reasoning, each subsequent statement in the sequence must be true too--a requirement directly in conflict with the discrimination condition (since no statement in the sequence can be both true and false).

The other pole in the notion of robustness is collectivistic: it involves the statements in a sorites sequence considered collectively, rather than individually. This pole requires the repudiation of the semantic requirements generated by the individualistic pole--and likewise, requires the repudiation of sorites reasoning. The notion "no precise fact of the matter about semantic transitions" is applied to a sorites sequence as a whole, specifically with the purpose of rejecting the idea that there is any determinate, correct, assignment of semantic statuses to the statements considered individually.

So the notion of robustness is at odds with itself: the individualistic and collectivistic poles are both present, and are in conceptual tension with one another whenever the discrimination condition is also operative (i.e., whenever the discriminatoriness condition applies non-vacuously). On one hand, with respect to the statements in the sequence taken collectively, it is required by robustness that there be no correct, determinate, assignment of truth values to the statements. On the other hand, with respect to the statements individually, it is required by robustness that each statement have the same truth value as its immediate neighbors. Since the requirements of the individualistic pole conflict with the discrimination requirement, but those of the collectivistic pole do not, the two poles thereby conflict directly with one another, given the discrimination requirement. Thus,

(16) Vagueness is logically incoherent, in the following way: it is not possible for the statements in a typical sorites sequence to fully satisfy, in such a way that some of these statements are true, the semantic requirements imposed by the robustness and the discriminatoriness of vagueness.

2.2. Vagueness and Dionysian Discourse.

If indeed vagueness is logically incoherent in the manner just described, then in any discourse for which the contextually operative semantic standards are logically coherent, vague terms will be necessarily empty--i.e., the semantic standards will guarantee that vague predications are never true (and are always false). Conversely, in any discourse in which vague predications are sometimes true, the contextually operative semantic standards themselves must be logically incoherent, in that they generate mutually unsatisfiable semantic requirements for the statements in typical sorites sequences. Now, it seems undeniable that much of our actual discourse is governed by semantic standards that sanction vague predications; and I myself certainly think so:

(17) Much actual discourse is vagueness-sanctioning: i.e., under the contextually operative semantic standards, many vague predications are true.

Thus, I am also committed to claiming that much actual discourse is logically incoherent:

(18) Vagueness-sanctioning discourse is logically incoherent, in the following way: the contextually operative semantic standards impose, on the statements in a typical sorites sequence, semantic requirements that cannot be mutually satisfied.

I will call discourse that is logically incoherent in this particular way, Dionysian discourse. By contrast, I will say that a discourse is Apollonian when the contextually operative semantic standards are not logically incoherent.[20] Given theses (18) and (19), I am committed to the view that a workable, vagueness-sanctioning, discourse can (and must) be Dionysian.[21] So the key task I face is to explain how this could be--how, that is, Dionysian discourse can be efficacious rather than nihilistically self-destructive.

Consider, as a suggestive model, a kind of situation that sometimes arises in the sphere of morals: a person finds himself with two conflicting moral obligations; both obligations remain in force, even though they conflict; yet the person is morally required to uphold one of these obligations specifically, and to violate the other one specifically. Here is a plausible example of this sort of moral conundrum, for concreteness. A philosopher promises to write a referee letter for someone's tenure/promotion file. The philosopher believes that the tenure candidate's philosophical work, though sparse, is truly excellent; that the candidate fully deserves tenure; that a carefully written letter by the philosopher himself will greatly enhance the candidate's chances; and that without such a letter from the philosopher himself, the candidate's chances of tenure are virtually nil. (The philosopher believes that the candidate's department needs very strong persuasion, because of the sparsity of the candidate's output; and that none of the candidate's other tenure referees are likely to help the candidate's case.) The philosopher also believes that preparing such a letter will require quite a lot of time--including much time spent carefully reading through the candidate's written work. Shortly thereafter, the philosopher gets invited to give a paper at a very prestigious philosophy conference, so prestigious that giving a paper there is bound to enhance the philosopher's professional reputation substantially. So the philosopher accepts, promising to prepare and present a paper at the conference. The conference, some six months away, will occur at about the same time that the tenure letter is due. Five months pass by quickly, without the philosopher's attending to either task. (As usual, the philosopher has been playing catchup on prior commitments, and has been behind on everything. Had he thought the matter through six months ago, he would have realized then that he was likely to end up in this dilemma.) Suddenly he finds himself realizing that he can't possibly keep both promises. (He believes that compromise corner-cutting just isn't possible here. Corner-cutting on the letter writing will result in a letter that lacks the detailed documentation necessary to make a persuasive case for the candidate's tenure. Corner-cutting on the paper preparation will result in a paper so poor that it does not meet an understood condition involved in his promise to write that paper, viz., that produce a paper that he himself considers intellectually respectable.)

In this situation, I suggest, both obligations are still in force; the philosopher faces a genuine moral dilemma. As I will put it, neither obligation is defeated--where by defeated I mean that an obligation has defeasibility conditions that are presently satisfied. In particular, neither obligation defeats the other one. (Although each of the two promises, like virtually any promise, does have certain implicit defeasibility conditions, neither promise's defeasibility conditions are here satisfied; in particular, neither's are here satisfied by the existence of the competing promise-keeping obligation.) However, given what is at stake, the philosopher is morally required to honor his promise to prepare and write a tenure letter, and he is morally required to violate his promise to prepare and present a paper at the conference. As I will put it, the former promise dominates the latter one. Thus, the first promise dominates the second without defeating it.[22]

This kind of moral-dilemma situation, I suggest, is importantly similar to what happens in Dionysian discourse when we are confronted with instances of the sorites paradox.[23] In those situations, the conflicting semantic requirements imposed by the contextually operative semantic standards come right to the surface--hence the paradox. In asking about the semantic status of the various individual pairs of adjacent statements in a sorites sequence, considered as individual pairs, we realize that there is a contextually operative semantic requirement that for each pair, the two items must have the same semantic status, and therefore that the universally quantified premise of the corresponding quantificational sorites argument be true--and thus a requirement that all the conditional premises in the corresponding conditional sorites argument be true. (I will call these individualistic semantic requirements.) However, in asking about the semantic status of the items in the sequence, considered collectively, we realize that we are obliged--because of "no precise fact of the matter about semantic transitions"--to reject a universally quantified sorites premise as not true, and also to reject its classical-logic contradictory as not true. Likewise, mutatis mutandis, for the conjunction of conditional premises in a conditional sorites argument, and the various classical-logic contradictories of this conjunction. (I will call these collectivistic semantic requirements.) Both kinds of requirements are present, applicable, and nondefeasible. Since neither is defeasible, neither is defeated by the other (even though they conflict); i.e., neither has defeasibility conditions that are satisfied by the presence of the competing requirement. The paradox is quite real, because there really are conflicting, nondefeasible, semantic requirements in play.

Even so, however, the competing semantic requirements are not on a par within Dionysian discourse. Instead, the collectivistic requirements semantically dominate the individualistic ones without semantically defeating them, in much the same way that one moral obligation sometimes morally dominates a competing one without morally defeating it. Truth (correct assertibility) is a matter of what is semantically proper to assert according to the semantically dominant requirements generated by the operative semantic standards:

(19) Truth is correct assertibility under semantically dominant, contextually operative, semantic standards.

Therefore, some vague predications are indeed true, within Dionysian discourse.

Pragmatic factors are behind these dominance relations, factors involving the point and purpose of vagueness in our language and thought. If the individualistic requirements were semantically dominant, or if there were no semantic dominance relations at all, then language and thought would evidently self-destruct under the pressure of sorites arguments: numerous statements of the form "P and not P" would turn out true, when P is a vague predication.

With collectivistic semantic dominance relations in operation within the semantic standards, this kind of radical logical self-destruction need not occur. Instead, Dionysian discourse can be logically well behaved--or logically disciplined, as I will put it--despite being logically incoherent:

(20) Dionysian discourse is semantically disciplined, in the following way: there are semantic dominance relations among nondefeasible, mutually unsatisfiable, semantic requirements.

(21) Dionysian discourse is logically disciplined, by virtue of being semantically disciplined; i.e., truth conforms to determinate logical principles.

So what we have with Dionysian discourse is sado-semantics (as one might put it). The semantic standards generate conflicting, nondefeasible, semantic requirements; but the standards also generate semantic and logical discipline in the discourse. This discipline arises through dominance relations: certain semantic requirements play the role of dominatrix, relative to other incompatible semantic requirements. The principles that systematize this kinky discipline comprise what I will call transvaluationist logic. I next turn briefly to that.

2.3. Dionysian Logical Discipline: Transvaluationist Logic.

Consider a familiar sorites argument like the following, where 'Bn' abbreviates the statement 'a man with n hairs on his head is bald'.

(Q) (1) (n)(Bn É Bn+1)

(2) B(0)

\ (3) B(10⁷)

We want a logic of vagueness under which it is possible to reject premise (1) without becoming committed to sharp semantic transitions in the sequence of statements B(0), B(1),..., B(10⁷). It is beyond doubt that the logic we seek must somehow differ from classical logic. For, this is a logical truth in classical logic:

(a) (n)(Bn É Bn+1) v ($n)(Bn & ~Bn+1).

Yet the left disjunct of (a),

(b) (n)(Bn É Bn+1),

is the major premise for the quantificational sorites argument (Q), whereas the right disjunct of (a),

asserts the existence of a sharp boundary between the bald and the not-bald. Within Dionysian discourse, statements (b) and (c) should both turn out to be not true. Yet under classical logic, one of the disjuncts (and only one, since (b) and (c) are contradictories) must be true. So transvaluationist logic has to differ somewhat from classical logic.

In principle, there are various potential systems of non-classical logic that might be candidates for systematizing the logical discipline of Dionysian discourse. The approach to vagueness I am describing in this paper is not officially committed any specific system. Officially, then, by 'transvaluationist logic' I mean whatever nonclassical logical principles reflect truth-preserving inference within Dionysian discourse.[24] Thus,

(22) In Dionysian discourse, truth conforms to the principles of transvaluationist logic.

In an earlier paper (Horgan 1994b) I did sketch a particular approach to the logic of vagueness which seems to me along the right general lines; Michael Tye (1990, 1994) has been working in a very similar vein. Let me briefly mention a few key features of this approach, with respect to how it handles sorites arguments like (Q).

First, the approach introduces two kinds of negation, strong and weak. Strong negation works in the manner of negation within classical logic: The strong negation ~S of a statement S is true (i.e., correctly assertible) iff S itself is false (i.e., correctly deniable, in the strong way). The weak negation ¬S of a statement S is true iff S itself is not true (i.e., not correctly assertible). (I appropriate the phrase 'it's not the case that' to express weak negation.) Some statements, for instance statements (b) and (c) just above, are neither true nor false, i.e., neither correctly assertible nor correctly deniable (in the strong way). Likewise, the strong negations of statements (b) and (c) are also neither true nor false. What are true, then, are the weak negations of each of these statements:

(d) ¬(n)(Bn É Bn+1)

(I.e., it's not the case that for any n, if an n-haired person is bald then an (n+1)-haired person is bald.)

(e) ¬($n)(Bn & ~Bn+1).

(It's not the case that there is some n such that an n-haired person is bald but an (n+1)-haired person is not bald.)

(f) ¬~(n)(Bn É Bn+1).

(It's not the case that not every n is such that if an n-haired person is bald then an (n+1)-haired person is bald.)

(g) ¬~($n)(Bn & ~Bn+1).

(It's not the case that there is not an n such that an n-haired person is bald but an (n+1)-haired person is not bald.)

Quantificational sorites arguments like (Q) thus get blocked, without commitment to any sharp semantic transitions in a sorites sequence. (Statement (a) gets the same treatment as (b) and (c).)[25]

A Tarski-style truth characterization can be given for a simple formal language with the two kinds of negation. This truth characterization yields the desired results for statements like (b) and (c), provided that the metalanguage is governed by the same non-classical logic that is operative in the object language. Truth itself is vague (in a way that directly reflects vagueness in object-language predictions); this means that metalinguistic discourse about object-language statements is itself Dionysian, so that appropriate metalinguistic reasoning conforms to the same logical principles that govern the object language. And so on ad infinitum, all the way up the metalinguistic hierarchy.

A transvaluationist logic along these lines blocks sorites reasoning: it allows us to reject statement (b), and to do so without thereby becoming committed to the truth of statement (c) (or to the truth of any other statement, at the level of either object language or metalanguage, which posits some sharp semantic boundary). The fact that truth conforms to transvaluationist logic does not eliminate the sorites paradox, however. For, although this logic reflects the dominance of the collectivist aspect of robustness over the individualist aspect, the individualist aspect is still present, generating nondefeasible (though dominated) semantic requirements upon the statements in a sorites sequence. It still makes perfectly good sense to ask, of any true statement in a sorites sequence, what the semantic status of its immediate successor could be; it's still the case that the only allowable answer to each such question, given the robustness of vagueness, is that the immediate successor itself must be true; and this reasoning still seems to be iterable, across the entire sorites sequence.

To be sure, the semantically correct thing to do, when queried about the semantic status of the items in the sorites sequence considered individually and successively, is to steadfastly refuse to play the game (cf. Tye 1994, pp. 205-6). Refuse to answer such questions in the form they are posed. Refuse to take up the challenge of explaining what specifically is wrong with stepwise sorites-style reasoning that focuses individually and sequentially on the successive statements in a sorites sequence. Resolutely fall back on the collectivist aspect of robustness, saying "There is simply no fact of the matter about semantic transitions in the sequence, and that's all there is to say." But although this obstinate head-in-the-sand stance is indeed semantically correct (since it accords with the semantic dominance of the collectivist aspect of robustness over the individualist aspect, and thereby reflects the workings of truth itself), it is not, and cannot be, fully satisfying intellectually. The individualist aspect of robustness still asserts itself, generating the nondefeasible requirement that all the statements in the sorites sequence have the same semantic status. This requirement persists, nondefeasible and therefore undefeated, even though it is dominated by other semantic requirements with which it clearly conflicts. There is no theoretically acceptable account of how the requirement gets either satisfied or defeated by contextually operative semantic standards, because it doesn't get satisfied or defeated--but only dominated. So although transvaluationist logic systematizes the semantic discipline at work in Dionysian discourse, this kind of logic does not, and cannot, exhibit full-fledged semantic coherence--because the contextually operative semantic standards generate mutually unsatisfiable semantic requirements. As one might put it, transvaluationist logic has the feature of depthlessness; there is no theoretically deep account that reveals coherent, non-conflicting, semantical principles underlying this non-classical logic.

Dionysian discourse works, though. Vagueness is ubiquitous in our language and thought, even though it is logically incoherent. Moreover, vagueness in language and thought appears to be not only very useful but absolutely essential, for creatures like ourselves with finite cognitive and discriminatory capacities. Hence, to insist upon complete logical coherence in the semantic standards governing one's discourse is to make an unrealistic and unlivable demand; logical discipline is enough. To realize this fact and accept it, thereby taking a step toward becoming a philosophical Ubermensch, is to embrace transvaluationism:

(23) Although vagueness is logically incoherent, it is also a benign, beneficial, and indeed essential feature of human thought.

Let me make a final point about logic and Dionysian discourse. Although the need to block sorites reasoning without committing ourselves to sharp semantic boundaries effects a certain de-centering of classical logic--i.e., it forces upon us the realization that truth in Dionysian discourse does not fully conform to classical logic--nevertheless the logical discipline exhibited by Dionysian discourse does largely approximate conformity to classical logic. Most of the time when we are operating within Dionysian discourse, vagueness does not directly intrude into our reasoning, and can be safely ignored. Thus, most the time we can and do make truth-preserving inferences, in Dionysian discourse, by resorting to classical logic. This being so, it remains appropriate for most purposes to insist upon full-fledged logical coherence as a legitimate constraint upon inference, belief fixation, and the like.

But the fact remains that thoroughgoing logical coherence cannot always be had. When we come face-to-face with a sorites argument, we find ourselves confronted by a genuine paradox, stemming directly from the fact that mutually unsatisfiable semantic requirements apply to the statements in the sorites sequence. The paradox does more than just force us to abandon classical logic in favor of a non-classical logic that allows us to reject the argument without incurring a commitment to precise semantic boundaries. The paradox also forces us to settle for logical discipline as a feature of vagueness-sanctioning discourse, and to give up the hope of full-fledged logical coherence. Complete logical coherence is unattainable, because such discourse is Dionysian, not Apollonian.

2.4. Vagueness and Apollonian Discourse.

Earlier I defined Apollonian discourse as the kind of discourse in which the contextually operative semantic standards are logically coherent, rather than merely being logically disciplined. Given that vagueness is logically incoherent, we get this result:

(24) In Apollonian discourse, vague terms are necessarily empty; i.e., the contextually operative semantic standards guarantee that vague predications are never true.

It seems quite clear that in the vast majority of actual discourse contexts (scientific contexts included), the contextually operative semantic standards are vagueness-sanctioning. Thus, we are led to conclude that Apollonian discourse is highly unusual:

(25) Apollonian discourse is quite rare, even in contexts of scientific inquiry.

On the other hand, as I stressed late in section 2.3, the logical discipline exhibited in Dionysian discourse does largely approximate full-fledged logical coherence. Vagueness normally does not intrude directly into our reasoning, and hence normally our inferences are truth-preserving when we employ classical logic. Logical coherence is thus a regulative ideal within Dionysian discourse, even though sorites arguments show that this ideal cannot be fully respected.

The only way to fully respect it is to employ semantic standards under which vague terms are necessarily empty. For virtually all our purposes in language and thought, this is far too high a price to pay--especially since logical discipline, in combination with approximate logical coherence, suffices to render Dionysian discourse workable. No wonder, then, that Apollonian discourse occurs so extremely rarely. Its principal usage, evidently, is in certain philosophical contexts when ontology is at issue. This brings us to ontology.

3. Vagueness and Ontology.

As I pointed out in section 1, within contextual semantics an important distinction arises between regional ontology and ultimate ontology (cf. theses 8-10). Ultimate ontology concerns questions of what OBJECTS, PROPERTIES, or other ENTITIES are denizens of THE WORLD. Regional ontology is a matter of (i) what Quinean "ontological commitments" are incurred by various statements, and (ii) which statements, which their associated regional ontological commitments, are true (correctly assertible) under contextually operative semantic standards. Maximally strict semantic standards are those under which there is direct language/WORLD correspondence. Thus, a statement carries ultimate ontological commitments to certain entities if (a) it carries regional ontological commitment to them, and (b) it is contextually governed by maximally strict semantic standards.

THE WORLD itself surely cannot be logically incoherent (although it could certainly turn out to be unintelligible to humans). So, since maximally strict semantic standards involve direct language/WORLD correspondence for true statements, and direct language/WORLD non-correspondence (as one might put it) for false statements, such semantic standards are not logically incoherent either. Thus, maximally strict semantic standards are Apollonian, not Dionysian. The discourse of ultimate ontology is Apollonian discourse. This means, given the ubiquity of vagueness in most discourse (even scientific discourse), that the discourse of ultimate ontology is quite unusual:

(26) Maximally strict semantic standards are Apollonian, not Dionysian.

(27) Discourse governed by maximally strict semantic standards is quite rare, even in contexts of scientific inquiry.

Let me now consider in turn the two broad kinds of ontological issue about vagueness that can be distinguished, within the framework of contextual semantics: issues about regional ontology and about ultimate ontology, respectively.

3.1. Regional Ontology: There is Vagueness in the World.

Dionysian discourse is up to its neck in regional ontological commitments to vague objects and properties. Mountains are vague--with respect to their spatial boundaries, for instance. Most middle-sized dry goods (desks, tables, coffee cups) are somewhat vague too, with respect to their precise spatio-temporal boundaries, for instance, and with respect to their microphysical composition. As Quine has written:

Who can aspire to a precise intermolecular demarcation of a desk? Countless minutely different aggregates of molecules have equal claims to being my desk... Vagueness of boundaries has sparked philosophical discussion in the case of desks because of their false air of precision. Mountains meanwhile are taken in stride: the thought of demarcating a mountain does not arise. At bottom the two cases really are alike: our terms delimit the object to the degree relevant to our concerns.... [The] cases differ only in degree. (Quine 1985, 167-8)

Similar remarks apply to the molecules that compose the desk, and to virtually all other things we talk about in ordinary life and in science. In particular, similar remarks apply to persons: human beings too are vague with respect to their spatio-temporal boundaries and their molecular composition.

These kinds of regional ontological commitments are quite legitimate, according to the view I have been developing here. They occur within Dionysian discourse, and statements that bear such ontological commitments are very frequently true within Dionysian discourse--i.e., they are correctly assertible, under the contextually operative assertibility standards. On the other hand, I also maintain that the way the WORLD contributes to correct assertibility, under Dionysian standards, is sufficiently indirect that these regional ontological commitments to vague entities do not constitute ultimate ontological commitments:

(28) Statements that are ontologically committed to vague objects and properties are often true, under contextually operative semantic standards (viz., Dionysian semantic standards).

(29) Under Dionysian semantic standards, statements that posit vague objects, or that predicate vague properties, do not carry ultimate ontological commitments to these entities; such ontological commitments are only regional.

Is there vagueness in the world? Certainly. There are mountains, tables, molecules, and persons, for instance, and these are all vague objects. There are properties like baldness, tallness, and heaphood, for instance, and these are vague properties. But when I say these things I am speaking under Dionysian semantic standards, and my remarks do not carry ultimate ontological commitment.

3.2. Ultimate Ontology: There is No Vagueness in THE WORLD.

The language of ultimate ontology is language governed by maximally strict semantic standards, and is Apollonian. Because vagueness is logically incoherent, vague terms are necessarily empty under Apollonian semantic standards. This means that there are not, and cannot be, vague OBJECTS or vague PROPERTIES.

The same conclusion can be reached by sorites reasoning, which works within Apollonian discourse to yield reductios of claims about vagueness in THE WORLD. Here is an example: Suppose there are OBJECTS that have vague spatio-temporal boundaries. Consider some putative vague OBJECT S, and some distance-interval measure that is very small--say, 10^-100 centimeters. Let P₁, P₂, ..., P_n be a sequence of space-time points with these features: (1) P₁ falls within S's spatio-temporal boundaries; (2) all the other points have the same temporal coordinate at t; (3) they are all positioned in a straight line emanating outward from P₁ in some specific spatial direction; (4) each point P_i in the sequence is 10^-100 cm closer to P₁ than is the successive point P_i+1; and (5) P_n does not fall within S. Since S is spatio-temporally vague, there are no successive points P_j and P_j+1 such that P_j falls within S and P_j+1 does not. So for each i, 1 £ i £ n, if P_i falls within S then so does P_i+1. Hence P_n falls within S, contradicting (5). Therefore, no OBJECTS are spatio-temporally vague. By analogous reasoning, one can show argue that no OBJECTS are vague in any other respect, that there are no vague PROPERTIES, and no vague ENTITIES of any kind.

So there are no MOUNTAINS, TABLES, MOLECULES, or PEOPLE. There are no such PROPERTIES as BALDNESS, TALLNESS, or HEAPHOOD. Whatever the correct ultimate ontology is, it does not include any vague denizens of THE WORLD.

(30) Statements that are ontologically committed to vague objects or properties are never true under maximally strict semantic standards.

(31) There are no vague OBJECTS or PROPERTIES in THE WORLD. I.e., the correct ultimate ontology does not contain any vague entities.

3.3. Ontological Double-Talk: The Dionysian/Apollonian Zig-Zag.

Needless to say, it sounds enormously bizarre--even lunatic--to say that there are no such things as mountains, tables, and people. The capitalization convention does little to mitigate this bizarreness; indeed, since this convention is not part of everyday discourse or of scientific discourse, it seems quite bizarre itself.

But the bizarreness is to be expected, given what I have been saying about vagueness and about vagueness-sanctioning discourse. Since humans employ vague terms ubiquitously, they almost always speak within Dionysian discourse; they almost never speak within Apollonian discourse, the discourse of ultimate ontology. Moreover, even when one does deliberately speak under maximally strict semantic standards, with the specific purpose of making claims about ultimate ontology, the introduction of vague terms inevitably makes one feel the "pull" of Dionysian, vagueness-legitimating, semantic standards. Thus, even though a statement like 'There are no persons' is true when asserted as a claim about ultimate ontology, it is bound to sound very peculiar anyway.

So although we can, when we choose to, shift the score in the language game from the ordinarily operative Dionysian standards into maximally strict standards, this shift into Apollonian discourse of ultimate ontology is not a smooth transition but a jarring zig-zag. When we employ vague terms under Apollonian standards, we find ourselves speaking and writing under erasure (as one might put it); i.e., we are speaking and writing under semantic standards radically at odds with ordinary, vagueness-legitimating, standards.[26] Our linguistic intuitions inevitably feel the strain, since they are so strongly tethered to Dionysian, vagueness-sanctioning, semantic standards. It is bound to sound peculiar to claim that are no such entities as desks or persons, even when this claim is deliberately intended as a remark about ultimate ontology; for, as soon as the terms 'desk' and 'person' are used, we feel strongly the tendency to slide back into the Dionysian semantic standards in which these terms have their ordinary use.

Once the zig-zag nature of discourse about ultimate ontology becomes thematized in this way, I think it becomes clear that the extreme linguistic oddness of statements like 'There are no desks or persons' does not necessarily provide good grounds for questioning the truth of such statements, construed as claims about ultimate ontology. Apollonian discourse is very rarefied indeed.

My use of Putnam's capitalization device is intended as a way of explicitly marking the zigzag into Apollonian discourse. Another such device is the ultimate-ontological use of the term 'really'. Once when Wilfrid Sellars had just presented a philosophical lecture in which he resorted heavily to talk of common roles played by intertranslatable terms in different languages, Michael Tye pointed out to Sellars that although he seemed to be ontologically committed to roles, his official ontology evidently did not include such entities. Sellars replied to Tye with an overt ontological zigzag. He said: "Are there roles? Of course! Are there really roles? No!" My own position about vagueness and ontology can be succinctly expressed by employing the same linguistic device that Sellars used. Are there mountains, desks, molecules, and persons? Of course! Are there really mountains, desks, molecules, and persons. No!

4. The Virtues of Transvaluationism.

Contextual semantics, in my view, is quite attractive as a general theoretical approach to language/world relations; I briefly rehearsed some of its attractions at the end of section 1. Transvaluationism, as situated within the framework of contextual semantics, likewise seems to have important theoretical attractions. I will conclude by mentioning three broad virtues.

First, this approach evidently does quite well at theoretically accommodating many of our strong pre-theoretic beliefs about matters involving vagueness, like the following:

Many vague predications are true.

Many things in the world are vague in one way or another (e.g., vague with respect to their spatio-temporal boundaries, and/or their physical composition.)

Vagueness in thought and language is useful, legitimate, and essential.

Sorites arguments are not sound.

Vagueness is robust: there is no precise fact of the matter about semantic transitions in a sorites sequence.

Vagueness does not undermine logic or reasoning.

Second, although transvaluationism does embrace certain intuitively odd-seeming claims (as any philosophical approach to vagueness probably must do), this approach has substantial explanatory resources for explaining why those claims sound odd. Bullet-biting becomes theoretically more palatable when the theory itself explains why the bullet should feel difficult to bite.

Third, transvaluationism exhibits a thorough theoretical respect for the sorites paradox, by treating it as a genuine paradox. Instead of claiming that the paradox is only apparent--perhaps the product of some kind of subtle intellectual confusion--transvaluationism locates the source of the paradox directly within the semantic standards that govern vagueness-sanctioning discourse: there are conflicting, nondefeasible, semantic requirements for the statements in a sorites sequence, and hence the paradox is quite real. Less respectful approaches, which seek to dissolve the paradox as illusory or to resolve it as though it were an intellectual puzzle that has some straightforward solution waiting to be discovered, tend to be dissatisfying and unconvincing. Upon reflection, those approaches just don't seem to give vagueness its due. The ancient sorites paradox, which has been too much neglected in philosophy and in logic, demands a deeper kind of philosophical respect.[27]

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