Transvaluationism makes two fundamental claims concerning vagueness.  First, vagueness is logically incoherent in a certain way: vague discourse is governed by semantic standards that are mutually unsatisfiable.  But second, vagueness is viable and legitimate nonetheless; its logical incoherence is benign rather than malevolent.[1] Just as Nietzsche held that one can overcome nihilism by embracing what he called the transvaluation of all values, transvaluationism asserts that vagueness, although logically incoherent, can and should be affirmed and embraced, not nihilistically repudiated.[2]

            In section 1 I argue that the logical incoherence of vagueness is a direct consequence of a feature that most everyone, myself included, thinks is essential to vagueness: boundarylessness.  I also argue that vague discourse, even if it is incoherent, nevertheless exhibits what I call logical discipline; and that this suffices to render vagueness viable and legitimate.  In section 2 I set forth a metaphysical consequence: viz., that there cannot be “ontological vagueness,” that is, vague objects or vague properties.  In section 3 I set forth a semantical consequence: viz., that truth, for vague discourse, is an indirect kind of language-world correspondence, rather than direct correspondence via direct connections between the referential apparatus of vague discourse on the one hand, and genuine objects and properties on the other hand.  In section 4 I consider a popular approach to vagueness that attempts to do full justice to boundarylessness.  I argue that this approach, which I call “iterated supervaluationism,” is really a version of transvaluationism rather than an alternative to it; and that the same will be true of any approach that makes a serious attempt to accommodate boundarylessness.  In section 5 I turn to epistemicism--a view that denies boundarylessness and treats vagueness as involving a certain kind of ignorance about determinate boundaries--and I briefly discuss its costs and benefits in comparison to those of transvaluationism.

            If indeed boundarylessness is both an essential feature of vagueness and also logically incoherent by virtue of generating mutually unsatisfiable semantic standards for vague discourse, then the received menu of principal positions concerning vagueness needs to be seriously rethought.  At a coarse-grained level of description, there are really only two viable competitors: transvaluationism and epistemicism.  Certain views that might initially appear to be alternatives to transvaluationism (e.g., iterated supervaluationism) are really just specific versions of the transvaluationist genus.  (Epistemicists could certainly accept this, and could reasonably regard it as providing further support for their own view.)  Moreover, anyone who resists the claim that a viable alternative to epistemicism must be a version of transvaluationism will have to take on the burden of showing that their own favored view really avoids logical incoherence--which  will not be easy, as my discussion of iterated supervaluationism in section 4 shows.  And in any event, whether or not  transvaluationism gets accepted as a superordinate category that subsumes all viable alternatives to epistemicism, transvaluationism certainly deserves a prominent place on the menu of live options.


1.         An Argument for Transvaluationism: Boundarylessness and Disciplined Logical Incoherence.

            It is widely believed that an essential attribute of vagueness is what Mark Sainsbury 1990 calls boundarylessness, a feature that can be characterized by reference to sorites sequences associated with vague terms.  Consider a vague term--say, ‘heap’--and consider a sorites sequence involving the given term in which the initial statement is true and the final statement is false--say, a series of statements successively predicating the vague term ‘heap’ first to pile of sand with 1 billion grains, then to an object produced by removing just one grain, then to an object produced by removing yet another single grain, and so forth down to a statement predicating ‘heap’ to a single grain of sand.  To say that vagueness is boundarylessness is to say that in such a sequence, (1) initially there are true statements (with each predecessor of any true statement being true); (2) later there are false statements (with each successor of a false statement being false); and (3) there is no determinate fact of the matter about the transition from true statements to false ones.  Condition (3) I call robustness.  It requires not only that there be no determinate abrupt transition from true statements to false ones, but also that the truth/falsity transition should involve no determinate semantic transitions at all; thus it also precludes, for instance, an overall true-to-false transition involving first a determinate abrupt transition from truth to the semantic status “indeterminate whether true or false,” and later another determinate abrupt transition from this in-between status to falsehood.[3]

            If one considers what it would take to fully accommodate boundarylessness--that is, accommodate it in a way that thoroughly eschews arbitrary semantic transitions of any kind--one finds that, for the successive statements in a sorites sequence, there are semantic requirements in play that cannot be simultaneously satisfied.  Boundarylessness has two conceptual poles.  On one hand there is an individualistic pole, applicable to individual pairs of adjacent statements in a sorites sequence: viz., for any pair of adjacent statements, the two statements must have the same semantic status (truth, falsity, indeterminateness, or whatever).  Otherwise there would a determinate semantic transition between them, contrary to the claim that there is no determinate fact of the matter about semantic transitions in the sequence.  On the other hand, there is also a collectivistic pole in the notion of boundarylessness, applicable globally with respect to a sorites sequence as a whole: viz., it is impermissible to iterate indefinitely the individualistic-pole requirement for successive adjacent pairs of statements, in the manner of paradoxical sorites arguments; in addition, there is simply no determinate overall assignment of semantic status to every one of the respective statements in a sorites sequence.  These individualistic and collectivistic requirements cannot be jointly satisfied; for, the only way that a sorites sequence could fully conform to the individualistic pole would be for every statement in the sequence to have the same semantic status.   (This is the lesson of the sorites paradox, which emanates directly from the individualistic pole of boundarylessness.) So boundarylessness is logically incoherent, in a specific way: it imposes mutually unsatisfiable semantic standards upon vague discourse.

            Although these incompatible semantic requirements are indeed in force insofar as vague discourse exhibits boundarylessness--no requirement is defeated by any others, in the sense of having defeasibility conditions that are satisfied by the presence of  the competing and incompatible requirements--they are not on a par with one another either.  The collectivistic-pole requirements dominate the individualistic-pole requirements without defeating them; that is, to the extent that the requirements conflict, truth is determined by the collectivistic-pole requirements.  In practice, this means that paradoxical sorites arguments are to be eschewed; it also means that one must not acknowledge the existence of any determinate semantic transitions (even unknown or unknowable ones) in a sorites sequence. (Semantic status still must conform partially to individualistic-pole requirements, however.  For instance, it is never the case, for any specific pair of adjacent statements in a sorites sequence, that the two statements differ in semantic status.)  Apparently, then, the semantic standards governing vague discourse are logically disciplined, in this sense: no logically incoherent statement is true, under those standards.

            Logical incoherence is usually a bad thing, of course.  In particular, it is a bad thing to affirm, or to be committed to affirming, logically incoherent statements.  But the specific kind of logical incoherence exhibited by boundarylessness evidently produces no such intolerable commitments.  On the contrary, even though there are mutually unsatisfiable semantic standards in play, the collectivistic-pole standards apparently exert enough control over the individualistic-pole standards that no logically incoherent statement employing vague discourse gets sanctioned as true.  There certainly is some incompleteness and indeterminacy with respect to the semantic status of statements in vague discourse; for instance, there is no correct, determinate, overall assignment of semantic status to each statement in a sorites sequence.  But there is nothing necessarily problematic about that; indeed, it is to be expected.  In short, although logical incoherence is usually a bad thing, logically disciplined logical incoherence need not be.  On the contrary, since (i) vagueness is presumably a useful--indeed  essential--feature of  human language and thought, (ii) boundarylessness is apparently an essential feature of vagueness, and (iii) boundarylessness essentially involves logically disciplined logical incoherence, evidently this latter feature is a good thing--in connection with vagueness, anyway. 

            All of this adds up to an argument for transvaluationism, the two-part view asserting (a) that vagueness is logically incoherent, by virtue of being governed by semantic standards that are mutually unsatisfiable, and yet (b) vagueness is viable and legitimate nonetheless.  In full deductive dress, the argument goes as follows:

(1)        If vagueness is boundaryless, then the semantic standards governing vague discourse are mutually unsatisfiable.

(2)        Vagueness is boundaryless.

(3)        If the semantic standards governing vague discourse are logically disciplined, i.e., no logically contradictory statement is sanctioned as semantically correct, then vagueness is viable and legitimate.

(4)        The semantic standards governing vague discourse are logically disciplined.

Hence,  the semantic standards governing vague discourse are mutually unsatisfiable; yet vagueness is viable and legitimate.

            Since each premise of this argument is highly credible, the transvaluationist conception of vagueness deserves to be taken very seriously, and to be articulated and investigated in further detail.  It is time to get over our Victorian hangups about the allegedly inevitable evils of logical incoherence, and to acknowledge the real attractions of sado-semantic dominance relations among mutually unsatisfiable semantic standards.  Arguably, this sort of kinkiness is what’s been going on all along in the case of vagueness, right under our noses; we should not be shocked.


2.         Vagueness as Non-Ontological.

            There is philosophical dispute about whether or not there is any such thing as ontological vagueness--that is, vagueness in the world itself, over and above whatever vagueness exists in language and in thought-content.   More specifically, the question is about whether (a) there are vague objects, and/or (b) there are vague properties.  The denial of ontological vagueness is probably the more common view in the philosophical literature on vagueness, but the issue is tendentious.

            A consequence of the transvaluationist conception of vagueness is that the negative position is right: there is not, and cannot be, ontological vagueness.  More specifically, there are not, and cannot be, vague objects or vague properties.[4]

Take vague objects first.  For concreteness, consider the putative object Mt. Whitney, which is vague in its spatial extent.  Suppose, for reductio, that Mt. Whitney really exists.  Consider a suitable first-order sorites sequence--say, a sequence of pins positioned vertically in the ground one centimeter apart, proceeding in a straight path from the peak down to the lowest point in Death Valley[5]; I  will call this the descent sequence.  Mt. Whitney, being  vague in spatial extent, is boundaryless in spatial extent: hence, with respect to the descent sequence, Mt. Whitney has features that are ontological analogs of the mutually incompatible semantic requirements that are in force in vague discourse and are all undefeated.  So the following conditions all obtain.  (1) Initially there are pins on the mountain (with each predecessor of a pin on the mountain also being on the mountain).  (2) Eventually there are pins not on the mountain (with each successor of any pin not on the mountain also not on the mountain).  (3) No pin in the sequence is both on the mountain and not on the mountain.  (4) For each pair of  adjacent pins in the sequence, either both pins are on the mountain, or else both pins lack this status.  (This is the ontological analog of the individualistic requirements imposed on vague language by boundarylessness.  It must obtain, because otherwise Mt. Whitney would not be vague in its spatial extent.)   Now, certain other conditions supposedly obtain too: an ontological analog of the collectivistic requirements, and an ontological analog of domination without defeat among mutually incompatible semantic requirements.  But it makes no difference how such further conditions might get spelled out, because  conditions (1)-(4) already entail that the last pin in the descent sequence both is, and is not, on Mt. Whitney--which is impossible.  Therefore, Mt. Whitney does not exist.  By analogous reasoning, no vague objects exist.  And, since suitable  sorites sequences are always constructible  for any putative vague object in any possible world, vague objects cannot exist.[6]

The argument that there cannot be vague properties is parallel in structure, and so I will not I will not rehearse it.  Both for objects and for properties, the reasoning is a straighforward adoption of the sorites paradox, an enormously powerful tool for drawing ontological conclusions. Although the individualistic aspects of boundarylessness can be dominated and controlled insofar as vagueness is a phenomenon of language and thought, they cannot be held in check for ontological vagueness.  If indeed vagueness is boundaryless, then, since boundarylessness is logically incoherent, the sorites paradox demonstrates that ontological vagueness is impossible.


3.         Truth as Indirect Correspondence.

            For those who hold both (a) that there is no ontological vagueness and (b) that vague discourse nevertheless is sometimes true, it is very natural to conceive of truth, for statements containing vague terminology, as involving some kind of indirect correspondence with the world, rather than direct correspondence.  A statement can directly correspond with the world only when its basic sub-sentential constituents--names, predicates, the apparatus of quantification--are referentially connected to real objects and real properties.  (Likewise, there can be direct non-correspondence--direct falsity--only when (i) a statement’s basic subsentential constituents are all referentially connected to genuine objects and properties, but (ii) the objects and properties referred to are not as the statement says.[7])  But if there are no vague objects and properties in the world, then vague singular terms and vague predicates presumably are not referentially linked to real objects and properties, and quantification over putative vague entities presumably does not involve genuine ontological commitment to such items.  Rather, for vague discourse, truth is an indirect kind of language/world correspondence--indirect enough that there need not be real objects or real properties answering to vague terms.  (Falsity, likewise, will be an indirect kind of non-correspondence.)  If I say, for instance, that Mt. Whitney is tall, I am not attributing a real property (tallness) to a real object (Mt. Whitney).

            Supervaluationism, when combined with the denial of ontological vagueness, provides an example. Instead of construing truth directly, by appeal to a single intended interpretation that assigns vague objects as the referents of singular terms and assigns vague properties (or vague extensions) to predicates, the supervaluationist construes truth simpliciter in an indirect way: viz., as truth in all permissible interpretations--there being more than one permissible interpretation, each of which “precisifies” the statement’s vague vocabulary in a specific way.  The language/world correspondence relation is now indirect because truth is not grounded in straightforward referential links between the sub-sentential vocabulary on one hand, and genuine vague objects and vague properties on the other hand.  (Falsity, likewise, is indirect non-correspondence, for the same reason.)  In the case of the statement that Mt. Whitney is tall, for instance, the indirect correspondence that constitutes truth is essentially this: under all ways of precisifying the expressions ‘Mt. Whitney’ and ‘tall’ that are consistent (apart from their precision) with the semantic standards governing these terms, the object assigned to ‘Mt. Whitney’ instantiates the property assigned to ‘tall’.[8]

            Although supervaluationism provides one possible implementation of the notion of indirect correspondence, the core distinction between direct and indirect correspondence can be formulated more generically.  We can construe truth as semantically correct assertibility, under contextually operative semantic standards (and falsity as semantically correct deniability, under such standards.)[9]  The semantic standards conspire with goings-on in the mind-independent, discourse-independent, world to yield truth.  In the limit case, the applicable standards are referentially strict: they require referential linkages connecting a statement’s basic subsentential constituents to real objects and real properties.  In this limit case, truth (i.e., semantically correct assertibility) is direct correspondence.  But the contextually operative semantic standards can also work in such a way that the requisite goings-on in the world need not involve entities or properties answering directly to the statement’s basic subsentential constituents. In such cases, truth is indirect correspondence.[10]

            The conception of truth as indirect correspondence is certainly natural, for someone who denies ontological vagueness and affirms that statements employing vague terms are sometimes true.[11]   But under transvaluationism, it is more than merely natural: it is mandatory.  Truth is semantic correctness under the operative semantic standards; moreover, there are indeed truths involving vague discourse.  So, since the semantic standards governing vague discourse exhibit a kind of logical incoherence, truth for vague discourse cannot be a matter of direct language-world correspondence; for if it were, then the world itself would be logically incoherent, which is impossible.

Let me add that there are independent reasons, apart from matters of vagueness, to think that for much of our discourse, truth is indirect correspondence.  Consider, for example, the following statements:

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

(b)        Mozart composed 27 piano concertos.

(c)        There are more than 20 regulatory agencies in the U.S. Federal Government.

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

Although no plausible-looking way of systematically paraphrasing such statements into a more austere idiom is even remotely in sight, nevertheless is it plausible that the mind-independent, discourse-independent, world does not include among its real constituents such putative entities as universities (as distinct from  buildings, classrooms, professors, computers, etc.), abstract piano-concertos (as distinct from concrete performance-events, concrete copies of sheet music, etc.), governments and government agencies (as distinct from buildings, bureaucrats, etc.), and book-types (as distinct from concrete books).  Among the reasons this is plausible are considerations of ontological parsimony and conformity with a broadly naturalistic approach to metaphysics.  Yet statements (a)-(d) are all true, even if there are no genuine entities answering to terms like ‘The University of Memphis’, ‘piano concerto’, or ‘regulatory agency’.  Plausibly, then, the truth of such statements is a matter of indirect, rather than direct, language-world correspondence.

So transvaluationism has the corollary that truth, for vague discourse, must be indirect language-world correspondence.  And there are independent reasons, apart for vagueness, for thinking that truth is often indirect correspondence anyway.  These facts reinforce the prima facie viability of the transvaluationist approach.  If the semantic standards governing vague discourse require only indirect correspondence with the world, then there is no apparent reason why these standards cannot impose mutually unsatisfiable requirements on the successive statements in a sorites sequence--provided that this logical incoherence is sufficiently well disciplined, with the collectivistic-pole aspects of boundarylessness  exerting firm domination over the individualistic-pole aspects.  Truth, as indirect correspondence, is semantic correctness under the dominant semantic standards (and falsity, as indirect non-correspondence, is semantic incorrectness under the dominant semantic standards).


4.         Case Study: Iterated Supervaluationism as a Version of Transvaluationism.

If vagueness is really boundaryless, as most of us believe it is, then, since boundarylessness involves disciplined logical incoherence, an adequate treatment of vagueness will have to be some version of transvaluationism.  Moreover, as so far characterized, transvaluationism is still a fairly generic approach, potentially open to further development and articulation in a variety of different ways; numerous questions about the logic and semantics of vagueness remain open within the generic conception, and might get handled differently in different versions.  But regardless how the details go, any account of vagueness that seriously comes to grips with boundarylessness inevitably must be a version of transvaluationism--whether its proponents acknowledge this fact or not.

As a case study to illustrate this fact, let me consider a version of supervaluationism that does attempt to accommodate full-fledged boundarylessness:  iterated supervaluationism, as I will call it.  Standard supervaluationist approaches to vagueness are set forth in a metalanguage governed by classical logic and classical semantics; for that reason, they flout boundarylessness, because they inevitably end up committed to sharp semantic transitions in sorites sequences--e.g., a sharp transition between truth and the in-between status “neither true nor false,” and another sharp transition between the latter and falsity.[12]   But the core idea of iterated supervaluationism is that the metalanguage too is vague, and thus it too is subject to a supervaluationist semantical treatment in a meta-meta-language; and so on, all the way up the metalinguistic hierarchy.  More specificially, the metalinguistic expression ‘permissible interpretation’ is itself vague, and thus has various permissible interpretations itself in a meta-meta-language; and so on up.  The thought is that this endless reiteration of supervaluationist semantics, through successively higher metalanguages, is a version of supervaluationism that can fully respect the boundarylessness of vagueness--the fact that there is no determinate fact of the matter at all about semantic transitions in a sorites sequence.[13]

            Iterated supervaluationism might initially appear to be not a version of transvaluationism, but rather an alternative to it--a competitor view.  After all, its proponents do not normally assert or acknowledge that boundarylessness involves mutually unsatisfiable semantic requirements for the successive statements in a sorites sequence.[14]  But a closer look reveals that it does not really eliminate the logical incoherence of boundarylessness, but instead offers us one way of implementing the dominance of the collectivistic pole in the notion “no fact of the matter about semantic transitions” over the individualistic pole.

            This can be seen by focusing on how the approach deals with sorites arguments.  Let the expression ‘B(n)’ abbreviate the open sentence ‘A man with n hairs on his head is bald’.  Let the baldness sequence be the sorites sequence of statements B(0), B(1), ..., B(1017).  Consider the following sorites argument involving the baldness sequence, which I call a conditional sorites argument:


B(0) É  B(1)

            B(1) É  B(2)




            B(1017-1) É  B(1017)

\         B(107)

Under supervaluationism (both the standard and the iterated versions), this argument gets rejected by asserting the negation of the conjunction of all its conditional premises:

(1)        ~{[B(0) É B(1)] & [B(1) É B(2)] & ...  & [B(1017-1) É B(1017)]}.

Since classical logic remains in force within supervaluationism, statement (1) is logically equivalent to:

            (2)        {[B(0) & ~B(1)] v [B(1) & ~B(2)] v ... v [B(1017-1) &  ~B(1017)]}.

In each permissible interpretation of the predicate ‘B’, some specific disjunct  in (2) comes out true; thus, (2) itself is true, since it comes out true in every permissible interpretation.  Moreover, since different disjuncts are true in different permissible interpretations, and since ‘permissible interpretation’ is itself vague under the iterated version of supervaluationism, no sharp semantic transitions are posited.  So far, so good.

But suppose we do a forced march (as I call it) left to right through the respective disjuncts in statement (2), asking with respect to each disjunct in turn, “Is it true?”  Each of these questions is perfectly meaningful.  And for none of these questions could the answer be “Yes,” under iterated supervaluationism.  For, to say of any disjunct in statement (2) that it is true would be to commit oneself to a sharp semantic transition in the relevant sorites sequence, contrary to the vagueness and boundarylessness of ‘bald’.  Thus, an advocate of iterated supervaluationism is committed to affirming statement (2), and under forced-march querying would be forced to say, of each respective disjunct in statement (2), that it is not true.   This being so, iterated supervaluationism does not avoid the logical incoherence of boundarylessness.  The individualistic pole in the notion of boundarylessness remains operative--a fact that comes to the surface when we consider a forced march through the respective disjuncts in statement (2).

            Essentially the same point emerges when we focus on how iterated supervaluationism handles quantificational sorites arguments, like the following:

            (n)[B(n) É B(n+1)]


\         B(1017)

Under  supervaluationism (both the standard and the iterated versions), this argument gets rejected by asserting the negation of the quantificational premise:

            (3)        ~(n)[B(n) É B(n+1)].

Since classical logic remains in force under supervaluationism, statement (3) is logically equivalent to

            (4)        ($n)[B(n) & ~B(n+1)].

In each permissible interpretation of the predicate ‘B’, some specific instantiation of (2) comes out true; thus, (2) itself is true, since it comes out true in every permissible interpretation.  Moreover, since different instantiations of (2) are true in different permissible interpretations, and since ‘permissible interpretation’ is itself vague under the iterated version of supervaluationism, no sharp semantic transitions are posited.  So far, so good.

But once again, if we do a forced march through a sequential series of statements of the form [B(i) & ~B(i+1)], beginning with B(0) and ending with some enormous number of hairs that undoubtedly suffices for thorough non-baldness--e.g., 1017--then an advocate of iterated supervaluationism is committed to affirming statement (4), and under forced-march querying would be forced to say, of each respective instance of statement (4), that it is not true.  The individualistic pole of boundarylessness remains operative, even though it logically conflicts with the collectivistic pole.

Now, the right thing for the advocate of iterated supervaluationism to say, when confronted with the prospect of forced-march querying, is to refuse steadfastly to play that game.  Resolutely fall back upon the collectivistic-pole aspects of  the notion of boundarylessness, and say with a confident tone of voice, “I cannot, will not, and need not answer these questions; there is just no determinate fact of the matter about semantic transitions in a sorites sequence, and that’s all there is to say.”  This is indeed true; for, it reflects the dominance of collectivistic-pole semantic requirements over individualistic-pole semantic requirements.  But although this refusal to take the forced march is entirely appropriate as a tactic for avoiding committing oneself to any logically contradictory statements, it would be self-deception to think that such an avoidance tactic somehow eliminates the logical incoherence of boundarylessness.  The individualistic-pole requirements are still in force, even though they are dominated by the logically incompatible collectivistic-pole requirements; for, the respective queries in the forced march are all still meaningful and all still demand a negative answer, even though it is proper and respectable to duck those individualistic semantic requirements by refusing to take the forced march. But this kind of semantic respectability is not the Victorian, logically straight-laced, variety.  Instead it is sado-semantic discipline in action--the dominatrix cracking her whip.  Iterated supervaluationism is not an alternative to transvaluationism, but is rather a species of it

This moral can be generalized, to cover any approach to vagueness that accommodates boundarylessness rather than positing determinate semantic transitions in sorites sequences.  To respect boundarylessness is to take on board the mutually unsatisfiable semantic requirements it imposes on the statements in a sorites sequence--while also providing some kind of logico-semantical implementation of the domination of collectivistic requirements over individualistic ones.  The underlying transvaluationist nature of the approach will reveal itself with respect to forced-march querying: inevitably it will be necessary to duck the respective queries, even though they are each meaningful and they each demand the same answer.  Even if the approach is dressed up so as to create an appearance of impeccable Victorian logical coherence, beneath that dress is a leather boustier.  Victoria’s Secret: Accommodating boundarylessness inevitably involves disciplined logical incoherence, whether one admits it or not.

Since the generic transvaluationist position leaves open many questions about the logic and semantics of vagueness, in principle there could be various different versions of it.  Indeed, much of the recent philosophical literature on vagueness, especially the literature that explicitly attempts to accommodate boundarylessness (often under the rubric ‘higher-order vagueness’), can be appropriately viewed as  falling within the transvaluationist framework.[15]  Let different approaches be developed in detail; only then can their comparative costs and benefits can be assessed.  But let it also be realized, both in developing different approaches and in assessing them comparatively, that insofar as they do justice to boundarylessness, they all will be versions of transvaluationism.


5.         Transvaluationism vs. Epistemicism.


            Given that boundarylessness is logically incoherent, at a coarse-grained level of description there are really just two viable competing positions concerning vagueness: transvaluationism and epistemicism.[16]  Epistemicists can, and I think should, accept the claim that boundarylessness is logically incoherent; and they can cite this fact as a reason, over and above the other reasons they have on offer, in favor of their own view.  The case for the position is thereby strengthened, surely.

            In terms of comparative benefits and costs, the two approaches are largely mirror images of one another.  Transvaluationism has the benefit of accommodating boundarynessness, which initially appears to be an essential feature of genuine vagueness.  But it does so at the substantial cost of positing logical incoherence in the semantic standards governing vague discourse.  Another significant cost is the complications in logic and semantics that result from deviating from classical logic and semantics.

Epistemicism has the benefit of retaining classical logic and semantics, thereby avoiding those complications and also avoiding the need to posit any logical incoherence in vagueness.  But it does so at the cost of denying that vagueness really is boundaryless (as it appears to be); of positing determinate but inherently unknowable boundaries; and of propounding either deep metaphysical mysteries about how the determinate boundaries get fixed where they are, or else surd unexplainable supervenience connections linking specific patterns of language usage to specific determinate boundaries for vague terms.[17]

Reasonable people can differ in how they weigh the comparative costs and benefits of these two approaches, and hence can differ in their overall assessments about which is more likely true. Both approaches deserve to be further developed and further explored, and informed comparative assessment may well depend partly on the details of articulations yet to be carried out.  In the meantime many of us, myself included, find epistemicism just impossible to believe, despite its theoretical virtues.  My money is on transvaluationism, with its nihilism-transcending affirmation of  disciplined logical incoherence.[18]


Terence Horgan

University of Memphis

Burgess, J. A. 1990:  “The Sorites Paradox and Higher-Order Vagueness,” Synthese (85), 417-74.

Fine, K. 1975:  “Vagueness, Truth, and Logic,” Synthese (30), 265-300.

Horgan, T.: 1986a:  Psychologism, Semantics and Ontology,” Nous (20), 21-31.

_____  1986b:  ”Truth and Ontology,” Philosophical Papers (15), 1-21.

_____  1990:  Psychologistic Semantics, Robust Vagueness, and the Philosophy of Language,” in S. L. Tsohatzidis (ed): Meanings and Prototypes: Studis in Linguistic Categorization.  London: Routledge.

_____  1991:  “Metaphysical Realism and Psychologistic Semantics,” Erkenntnis (34), 297-322.

_____  1994: “Robust Vagueness and the Forced-March Sorites Paradox,” Philosophical Perspectives (8), 159-88.

_____  1995: “Transvaluationism: A Dionysian Approach to Vagueness,” Southern Journal of Philosophy (33), Spindel Conference Supplement on Vagueness, 97-126.

_____  in press:  “Brute Supervenience, Deep Ignorance, and the Problem of the Many,” Philosophical Issues.

Horgan, T. and Timmons, M.:  “Metaphysical Naturalism, Semantic Normativity, and Meta-Semantic Irrealism,” Philosophical Issues (4), 180-203.

McGee, V. and McLaughlin, B. 1995:  “Distinctions without a Difference,” Southern Journal of Philosophy (33), Spindel Conference Supplement on Vagueness, 203-51.

Sainsbury, R.M. 1990:  Concepts without Boundaries.  Inaugural Lecture, King’s College London.

Tye, M. 1994:  Sorites Paradoxes and the Semantics of Vagueness,” Philosophical Perspectives (8): 189-206.

Unger, P. 1980:  “The Problem of the Many,” Midwest Studies in Philosophy (5), 411-67.

Williamson, T. 1994:  Vagueness.  London and New York: Routledge.


[1] See Horgan 1994, 1995.  In Horgan 1994 I defended the incoherence thesis, and I combined it with a proposed nonclassical logic for handling sorites arguments.  In Horgan 1995 I wedded the incoherence thesis to the legitimacy thesis, named the resulting position ‘transvaluationism’, and articulated and defended it within the framework of a general approach to language/world relations I call contextual semantics.  I now think that a strong case can be made for transvaluationism on the basis of quite minimal assumptions; doing so is one principal goal of the present paper.

[2] Williamson 1994, section 6.2, gives the name ‘global nihilism’ to the view that “a typical sorites argument is sound, its conclusion strange but true” (p. 166).  He cites various authors as discussing views “close to global nihilism” (p. 298), including Horgan 1994.  This way of classifying of the view I began staking out in my 1994 paper is part of what prompted me subsequently to name my position transvaluationism.

[3] For convenience of exposition, here and mostly throughout I discuss boundarylessness metalinguistically, in terms of statemements and their semantic status.  But the same core idea applies equally well at the first-order level of description.  Consider a sorites sequence consisting of the respective sand-conglomerations themselves.  To say that heaphood is boundarylessness is to say that (1) initially initially in this sequence there are heaps (with each predecessor of a heap being a heap); (2) later there are non-heaps (with each successor of a non-heap being a non-heap); and (3) there is no determinate fact of the matter about the transition from heaps to non-heaps.  In section 2 below I engage in semantic descent to the first-order level, in addressing the issue of ontological vagueness.

[4] Here and throughout, I use the word ‘property’ broadly to include relations.

[5] The top of Mt. Whitney is the highest point in the contiguous portion of the United  States; the lowest point is in Death Valley, which is visible from the top of Mt. Whitney.

[6] Objection: “But surely this conclusion is lunatic!  Most every kind of object we talk about and quantify over, both in science and in everyday life, is vague in some respects.  Even persons are vague--in the extent of their diachronic existence, for example, and in their synchronic physical composition.”  Reply: Even though there is not, and cannot be, ontological vagueness, statements about vague entities--including existence claims--are often true nonetheless.  See section 3.

[7] I ignore complications arising from cases involving nonreferring singular terms that are used in a way that purports to be directly referential.

[8] Similarly, in the case of the statement that Mt. Whitney exists, the indirect correspondence that constitutes truth is essentially a matter of this statement’s coming out true under all ways of precisifying the expression ‘Mt. Whitney’ that are consistent (apart from their precision) with the semantic standards governing this term.

[9] The relevant notion of semantic correctness has nothing to do with matters of etiquette.  A statement can be semantically correct, in the relevant sense, even if it would be impolite, impolitic, or otherwise inappropriate to utter it.  Semantic correctness is also distinct from epistemic warrant; a statement can be epistemically warranted but semantically incorrect, and can be semantically correct but epistemically unwarranted.

[10] I call this general approach to truth “contextual semantics.”  I have explored and developed it in a number of papers, some co-authored with Mark Timmons; see Horgan 1986a, 1986b, 1990, 1991, 1995; and Horgan and Timmons 1993.  (For awhile I called the view “language-game semantics, then “psychologistic semantics.”)  The approach allows for contextual variation in semantic standards not only across different domains of discourse, but even within a given domain.  In philosophical contexts of inquiry where serious ontological issues are at issue, for instance, typically the standards governing a local mode of discourse become referentially strict--or anyway, closer to referentially strict than they normally are.  (In the penultimate paragraph of section 2 above, this happens for discourse employing the term ‘Mt. Whitney’.)  One way to signal such a shift of score in the language game is by means of emphasis terms, thus:  “Does Mt. Whitney exist?  Of course!  Does it really exist?  No!”.

[11] Another option, perhaps, is to say that vague expressions are directly referentially connected to precise objects and properties.  But as a general construal of vague terms, this approach is not plausible.  One important reason why not is that normally there will be too many eligible candidate-referents for a given vague term, with no principled basis for choosing one candidate over any of the others.  This is a version of what Peter Unger 1980 calls “the problem of the many.”

[12] The same problem arises for standard “degrees of truth” approaches.  These too are formulated in a metalanguage governed by standard logic and semantics, and hence are inevitably committed to sharp semantic transitions in sorites sequences--for instance, a sharp transition between the last statement that is true to degree 1 and the first statement that is not true to degree 1.

[13] For versions of of this general approach, see Fine 1975 pp. 287-98, Burgess 1990, and McGee and McLaughlin 1995.  See also the discussion in Williamson 1994 section 5.6.

[14] Brian McLaughlin, in conversation, has suggested to me that the approach in McGee and McLaughlin 1995 accommodates my own view that there is logical incoherence in vagueness.  Several features of their treatment are relevant here.  First, they introduce a metalinguistic ‘definitely’ operator, appendable to the truth predicate.  Second, they claim that the pre-theoretic notion of truth bifurcates into two distinct notions with respect to vagueneness, which they express as ‘true’ and ‘definitely true’, respectively.  Third, they maintain that the former notion obeys the disquotation principle, which “tells us that any adequate understanding of truth ought to give us the [Tarskian] (T)-sentences and (F)-sentences” (p. 214); whereas the latter notion obeys the correspondence principle, which “tells us that the truth conditions for a sentence are established by the thoughts and practices of the speakers of a language, and that a sentence is true only if the nonlinguistic facts determine that these conditions are met” (p. 214).  McLaughlin’s claim to me is that, in effect, this approach treats the pre-theoretic notion of truth as a logically incoherent amalgam of the two distinct truth-notions that he and McGee express as ‘true’ and ‘definitely true’ respectively.  But even if they are right that the notion of truth needs to be thus bifurcated, I maintain that the logical incoherence of vagueness cuts deeper.  As I am about to argue, logical incoherence is still present within iterated supervaluationist treatments of vagueness--including theirs, despite its explicit distinction between two notions of truth.

[15] In Horgan 1994 sections 2 and 3, I sketch one possible approach to the logic and semantics of vagueness that accommodates boundarylessness.  Another is proposed by Michael Tye 1994.  Tye does not acknowledge that boundarylessness is logically incoherent, and he is a defender of ontological vagueness.  But it is striking that in the final footnote (addressing forced-march problems I had raised) he says, “The full question and answer game Horgan describes is one which an astute defender of the semantics must, by the very principles of the semantics, refuse to play.  This is not to say, of course, that if I am asked only whether B(0) and B(1), say, have the same truth-value, it is incorrect for me to say “Yes”.  Nor is it to claim that the this reply is not to be taken at face value.  The point is rather that if I am a cogent defender of the position, I should not allow myself to be drawn into answering questions about every member of a sorites sequence  (pp. 205-6).  Just so; the dominatrix cracks her whip.

[16] For articulation and defense of epistemicism, see Williamson 1994 chapters 7 and 8, and the additional sources cited there.

[17] I describe these costs in more detail in Horgan 1994 section 5, and Horgan in press.

[18] I thank Brian McLaughlin, Matjaz Potrc, John Tienson, and Mark Timmons for helpful discussion and comments.