FACING UP TO THE SORITES PARADOX
Terry
Horgan
The
ancient sorites paradox is traditionally attributed to Eubulides, a
contemporary of Aristotle and a member of the Megarian school, who is also
credited with inventing the liar paradox.
The sorites paradox figures centrally in most discussions of vagueness
in philosophy and in logic. In my view,
it has profound implications for metaphysics and semantics, as well as for
logic. In this paper I will briefly
explain why I think so, in a way that draws upon my other writings on
vagueness.[1] The paper also will constitute a brief,
opinionated, overview of the pertinent intellectual landscape, reflecting my
own philosophical perspective on the issues.[2]
1. Troubles for Standard Two-Valued Logic.
Let
‘B(n)’ abbreviate ‘a man with n hairs on his head is
bald’. Here is a familiar
quantificational version of a sorites-paradoxical argument:
(n)[B(n) É B(n+1)]
B(0)
\ B(1017)
Although the conclusion of this
argument is surely false, it is hard to find a satisfying diagnosis of how the
argument goes wrong if we confine ourselves to standard two-valued logic. For, since the inference is classically
valid, and the second premise is clearly true, the only apparent option is to
reject the first premise by asserting its negation:
(1) ~(n)[B(n) É B(n+1)].
But this statement is logically
equivalent to
(2) ($n)[B(n)
& ~B(n+1].
And this latter statement, as
interpreted within standard two-valued logic, asserts the existence of a
determinate one-hair transition point between baldness and non-baldness. Yet it seems essential to the vagueness of ‘bald’ that there
be no such transition point. How, then,
to diagnose such paradoxical sorites arguments?
One
approach, which has the virtue of preserving standard two-valued logic, is
epistemicism.[3] On this view, statement (2) is true, and this
is because there really is a determinate transition-point between baldness and
non-baldness; yet this transition point is unknowable by finite minds like
ours. The essence of vagueness, says the
epistemicist, is a particular kind of unknowability of determinate conceptual
boundaries; vague concepts always have such boundaries, but we can never know
where they are. But most philosophers
and logicians who have reflected on vagueness, myself
included, find epistemicism just impossible to believe.
So
for those who reject epistemicism, it appears hard to deny that an adequate
diagnosis of the sorites paradox will require some kind of repudiation of standard two-valued logic. What seems needed is a non-standard logic of
vagueness which does one or both of these things: (i) provides some way of
rejecting the first premise in the above sorites argument without commitment to
a sharp transition from baldness to non-baldness; or (ii) renders such sorites
arguments invalid. There are various
proposed approaches to the logic of vagueness, and there is no consensus about
which is best. Among philosophers, and
among logicians in philosophy departments, probably the most popular approach
is supervaluationism, which I will say more about below.[4] (Among computer scientists and mathematicians
who work in logic, probably the most popular view is so-called “fuzzy logic.”[5] More on this below too.)
2. The Metaphysics of Vagueness: Two Approaches.
There
are two broad metaphysical approaches to vagueness. One approach affirms so-called ontological
vagueness, i.e., vagueness in the mind-independent, discourse-independent,
world. More specifically, it affirms the
existence of genuine objects and properties that are vague. The other approach denies ontological
vagueness, and asserts that vagueness is entirely a matter of language (and of
thought-content). On the non-ontological
view, vagueness is a purely semantic affair, akin to phenomena like ambiguity
and semantic indeterminacy.[6] On this metaphysical issue, too, there is no
consensus. But probably the more popular
view is the non-ontological conception of vagueness.
If
one does adopt the non-ontological view, in combination with some alternative
to standard two-valued logic, then what happens to the notion of truth? In effect, truth (for vague discourse) gets
treated not as direct language-world correspondence, but rather as indirect
correspondence between vague language (and vague thought-content) on one hand,
and non-vague reality on the other.
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 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’.[7]
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. I myself, in a number of writings, have
proposed construing truth as semantically correct assertibility, under
contextually operative semantic standards (and falsity as semantically correct
deniability, under such standards.)[8] 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.[9]
In
short, approaches to vagueness that (i) repudiate ontological vagueness, (ii)
embrace some alternative to standard two-valued logic, and (iii) treat various
non-negative statements employing vague language as true, are
effectively construing truth as indirect correspondence. Moreover, indirect correspondence is
naturally understood as semantically correct assertibility, under assertibility
standards that are not referentially strict.
From this perspective, vague statements are frequently true, under
contextually operative semantic standards, but they do not carry genuine
ontological commitment to any vague objects or vague properties (under the
operative standards)—a good thing, given the denial of ontological vagueness.
Those
who posit ontological vagueness, on the other hand, are under no immediate
pressure to construe truth as indirect correspondence. Instead, they can regard truth as direct
correspondence between vague language (and vague thought-content) on one hand,
and vague reality on the other. Typically,
believers in ontological vagueness will also espouse some approach to the logic
of vagueness that departs from standard two-valued logic; for, there remains an
apparent need for such a departure in order to disarm the sorites paradox
without commitment to sharp transitions between categories like bald and
non-bald. So, for believers in
ontological vagueness, reality itself—the instantiation of genuine, vague,
properties by genuine, vague, objects—conforms to some kind of non-standard
logic.
One
way to implement this approach is to posit ontological vagueness and also embrace
some version of so-called “fuzzy logic.”
Here the key idea is that genuine vague properties can be instantiated
not merely in an all-or-nothing way, but also partially, in specific
numerical degrees from zero to one.
Likewise, genuine vague objects can possess spatial or temporal parts
not merely in an all-or-nothing way, but in specific numerical degrees. Language-world correspondence is still
direct, on this account, but comes in degrees.[10]
3. The Impossibility of Ontological Vagueness.
So
far I have described two broad, non-epistemicist,
approaches to vagueness. One approach
repudiates ontological vagueness, rejects standard two-valued logic, and treats
truth as indirect correspondence. The
other approach embraces ontological vagueness, rejects standard two-valued
logic, and treats truth as direct language-world correspondence. In my own view, careful attention to the
nature of vagueness settles this debate in favor of the first alternative. Matters of logic, closely connected to the
sorites paradox, now enter the discussion in a new and different way, and
establish the impossibility of ontological vagueness. Let me explain.
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 a 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, (i) initially there are true statements (with
each predecessor of any true statement being true); (ii) later there are false
statements (with each successor of a false statement being false); and (iii)
there is no determinate fact of the matter about the transition from true
statements to false ones. Condition (iii) 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.[11]
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. Two collectivistic requirements
apply. First, it is impermissible to
iterate indefinitely the individualistic-pole requirement for successive
adjacent pairs of statements, in the manner of paradoxical sorites
arguments. Second, there is simply no
determinate collective assignment of semantic status to all the 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.
The
logical incoherence of vagueness is generic, in the following sense: it
is not directly linked to, and does not presuppose, any particular approach to
the logic of vagueness. Debates about
the specific logical principles governing vague discourse are largely
independent of the generic logical incoherence of vagueness—a theme to which I
will return shortly. Moreover, the
specific kind of generic incoherence exhibited by vagueness needs to be
distinguished from a stronger, and highly malevolent, kind of generic logical
incoherence. Vagueness does involve weak
generic logical incoherence—viz., the presence of mutually unsatisfiable
semantic standards governing vague discourse (and vague thought-content). But this does not necessarily bring in its
wake strong generic logical incoherence—viz., commitment to individual
statements that are logically contradictory, such as statements of the form F & ~F. On the contrary, I maintain that vagueness
can, and does, possess weak generic logical incoherence without possessing the
strong kind—another theme to which I will return shortly.
So weak logical incoherence is a feature of the contextually
operative semantic standards governing vague discourse, in ordinary contexts of
usage. Insofar as these semantic
standards are not referentially strict, and thus only require indirect
language-world correspondence, there is no particular problem about this
(provided that the semantic standards are not also strongly logically
incoherent). However, the world
cannot be logically incoherent, even in the weak way: it cannot have features
that are the ontological analogues of mutually unsatisfiable semantic
standards.[12] Hence there cannot be ontological
vagueness. Therefore, barring
epistemicism, the only viable general approach to vagueness is the one that
conceives it non-ontologically, and construes truth as indirect correspondence
between vague language and non-vague reality.
The sorites paradox is at the heart of the matter in the reasoning that
has led up to this conclusion, since the paradox directly reflects the weak
generic logical incoherence of vagueness.
4. Logical Discipline and Weak Logical Incoherence.
How is it that the semantic
standards normally governing vague discourse can incorporate
weak generic
logical incoherence without the strong kind?
Briefly, my story goes as follows.
Incompatible individualistic and collectivistic semantic requirements
are indeed in force insofar as vague discourse exhibits boundarylessness. That is, 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.
But these competing requirements 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.[13]) So the semantic standards governing vague
discourse are logically disciplined, in virtue of the dominance (without
defeat) of
collectivistic-pole requirements.
Because of this logical discipline, no logically incoherent statement is
true, under those standards; strong generic logical incoherence is avoided.
5. Transvaluationism and Its Potential Implementations.
Transvaluationism
is my name for the general approach to vagueness I have been describing. Transvaluationism makes two fundamental
claims. First, vagueness is weakly
logically incoherent without being strongly logically incoherent. Second, vagueness is viable, legitimate, and
indeed essential in human language and thought; its weak logical incoherence is
benign rather than malevolent. 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 in a certain way, can and should be affirmed and embraced, not
nihilisticaly repudiated.[14]
If
vagueness is really boundarylessness, as it certainly appears to be, then,
since boundarylessness involves disciplined weak logical incoherence, an
adequate treatment of vagueness will have to be some version of
transvaluationism. Moreover, since the
weak logical incoherence involved is generic, transvaluationism itself is a fairly
generic approach, potentially open to further development and articulation in a
variety of different ways. Numerous
details about the logic and semantics of vagueness remain open within the generic
conception, and might get handled differently in different versions.[15] But regardless of how the details go, any
account of vagueness that seriously comes to grips with boundarylessness must
be a version of transvaluationism—whether its proponents acknowledge this fact
or not. Much of the recent literature on
so-called “higher-order vagueness”—i.e., vagueness of categories like truth,
falsity, and the category “neither true nor false”—is best viewed as falling
within the transvaluationist framework.
In effect, specific proposals concerning higher-order vagueness amount
to suggested strategies for implementing the dominance-without-defeat of
collectivistic semantic standards over individualistic ones.[16]
The
weak generic logical incoherence that any such proposal must take on board, at
least implicitly, will inevitably reveal itself when one considers what the
advocate of the particular proposal will be forced to say when confronted with
what I call a “forced march” through a sorites sequence. Consider, for instance, a sorites sequence
for baldness: B(0), B(1), ..., B(1017). A forced march through this sequence is a
series of questions, with respect to each successive statement, “Is it
true?” Each of the questions is
perfectly meaningful. And for no two
successive questions could it be correct to give different answers; for, that
difference would mark a determinate semantic transition, contrary to the nature
of vagueness. So the only thing to do,
when confronted with the prospect of forced-march querying, is to refuse
steadfastly to play that question-and-answer game. Instead of taking the forced march, adopt a
Zen attitude: be tranquilly silent in the face of those persistent queries, in
the knowledge that no complete set of answers is semantically correct.[17] This is the right thing to do, because
it reflects the dominance of collectivistic-pole semantic requirements over
individualistic-pole requirements. But
although this refusal to take the forced march is entirely appropriate as a
tactic for avoiding commitment to any logically contradictory statements, it
would be self-deception to think that such an avoidance tactic somehow
eliminates the weak logical incoherence of vagueness. 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 each still demands the same answer as its
predecessor, even though it is proper and respectable to duck those
individualistic semantic requirements by refusing to take the forced
march. The forced march is essentially
just the sorites paradox itself, with our noses rubbed in it.[18] What it reveals is the weak generic logical
incoherence of vagueness.
6. Conclusion.
To
summarize: Although vagueness is a
legitimate and essential feature of human language and thought, it is weakly
generically logically incoherent; yet it is not strongly generically logically
incoherent, because it exhibits logical discipline involving the dominance,
without defeat, of collectivistic semantic standards over individualistic
ones. This generic approach to
vagueness, which I call transvaluationism, has several important
consequences. First, there cannot be
ontological vagueness. Second, truth,
for vague discourse, is indirect correspondence between vague language and
non-vague reality. Third, any adequate
account of the logic of vagueness will incorporate the weak generic logical
incoherence of boundarylessness; it will be an implementation of transvaluationism,
rather than an alternative to it.
I
remarked earlier that indirect correpondence is best understood as semantically
correct assertibility, under contextually operative semantic standards that are
not referentially strict. I also said
that we sometimes employ language under limit-case, referentially strict,
direct-correspondence, semantic standards.
These are the standards appropriate for serious ontological
inquiry. When they are in play, so is
classical two-valued logic. Under this
limit-case use of language, sorites reasoning can be correctly employed to
construct reductio ad absurdum arguments against the existence of vague
properties or vague objects—including, of course, not only mountains and
clouds, but also tables, chairs, and (regrettably) persons.[19] Fortunately, however, ordinary uses of vague
language are not governed by limit-case semantic standards, and normally the
contextually operative standards conspire with non-vague reality to render much
of our vague discourse true.[20]
REFERENCES
Fine, K., 1975: “Vagueness,
Truth and Logic,” Synthese (30), 265-300.
Horgan, T. 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.
_____ 1998a: “Actualism,
Quantification, and Contextual Semantics,” Philosophical Papers (12),
503-09.
_____ 1998b: “The
Transvaluationist Conception of Vagueness,” The Monist (81),
316-33. Issue on
vagueness.
_____ in press: “The Benign Logical
Incoherence of Vagueness,” Acta Analytica. Issue containing papers
from the 1998 Bled conference on vagueness.
Keefe,
R. and Smith, P. 1996: Vagueness: A Reader.
Quine, W. V. 1995: From
Stimulus to Science.
Sainsbury, R. M. 1990: Concepts
without Boundaries. Inaugural Lecture, King’s College London. Reprinted in Keefe and
Smith 1996.
Tye, M. 1990: “Vague Objects,”
Mind (99), 535-57.
Tye, M. 1994: “Sorites
Paradoxes and the Semantics of Vagueness,” Philosophical Perspectives
(8): 189-206.
Tye, M. 1995: “Vagueness:
Welcome to the Quicksand,” Southern Journal of Philosophy (33), Spindel
Conference Supplement on Vagueness, 1-22.
Unger, P. 1979a: “I Do Not
Exist,” in G. F. Macdonald (ed.), Perception and Identity.
Unger, P. 1979b: “There Are
No Ordinary Things,” Synthese (41), 117-54.
Williamson,
T. 1994: Vagueness.
Zadeh, L. A. 1975: “Fuzzy
Logic and Approximate Reasoning,” Synthese (30): 407-25.
[1] See Horgan 1994, 1995, 1998b, in press. Other papers of mine discussing vagueness are cited therein.
[2] For more extended overviews, including bibliographical information, see Tye 1995 and the editors’ introductory essay in Keefe and Smith 1996. For a thorough critical overview, including rich bibliographical information, see Williamson 1994.
[3] For articulation and defense of epistemicism, see Williamson 1994 chapters 7 and 8, and the additional sources cited there.
[4] The locus classicus of the supervaluationist theory of vagueness is Fine 1975.
[5] The locus classicus of the fuzzy-logic treatment of vagueness is Zadeh 1975.
[6] A mixed view is possible too, asserting that some vagueness is ontological and some is purely semantic; but I will ignore this hybrid position, for simplicity.
[7] Supervaluationism preserves standard logic’s theorems and inference rules, but not its bivalent semantics. A statement applying a vague predicate to a borderline instance will be neither true nor false, since it will come out true under some permissible interpretations and false under others. With respect to the sorites paradox, the supervaluationist affirms both statement (1) and statement (2) above. But under supervaluationist semantics, statement (2) can no longer be understood as asserting a determinate transition-point between baldness and non-baldness. Rather, (2) is true because it comes out true under all permissible interpretations of ‘bald’.
[8] I call this general approach to truth “contextual semantics.” It is explored and developed in Horgan 1995, 1998a, and in various other papers of mine cited therein (some co-authored with Mark Timmons). The overall defense of contextual semantics in these papers proceeds largely by appeal to metaphysical and epistemological considerations orthogonal to matters of vagueness. (For awhile I called the view “language-game semantics, then “psychologistic semantics.”) 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.
[9] Contextual semantics 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 stake, for instance, typically the standards governing a local mode of discourse become referentially strict--or anyway, closer to referentially strict than they normally are. 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!”.
[10] Fuzzy logic typically treats all real numbers from zero to one as truth values. Consider, for instance, the predicate ‘B(n)’ introduced earlier, and the sequence of statements B(0), B(1), ..., B(1017). Those who address vagueness via fuzzy logic will say that the numerical truth values of these successive statements gradually diminish from one (outright truth) to zero (outright falsity) as we progress through the sequence, and that this is the key to blocking the above paradoxical sorites argument without positing any transition-point between bald and non-bald. For an approach that embraces ontological vagueness but weds it to a logic significantly different from fuzzy logic, see Tye 1990, 1994.
[11] For convenience of exposition, here
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 (i) initially in
this sequence there are heaps (with each predecessor of a heap being a heap);
(ii) later there are non-heaps (with each successor of a non-heap being a
non-heap); and (iii) there is no determinate fact of the matter about the
transition from heaps to non-heaps.
[12] For example, there cannot be a genuine property H (for ‘heaphood’), and a sequence of sand conglomerations each of which has one fewer grain than its predecessor, such that (i) initially in the sequence there are instances of H (with each predecessor of an H instance being an H instance), (ii) eventually there are non-H instances (with each successor of a non-H instance being a non-H instance), and (iii) for each pair of successive piles in the sequence, either both are H instances, or both are non-H instances, or both are neither.
[13] Does this mean that under the correct collective assignment of semantic status to all the statements in a sorites sequence, no two adjacent statements differ in semantic status? No. According to the collectivistic-pole requirements, there is no correct collective assignment of semantic status to all the statements in the sequence.
[14] One reason I call my position transvaluationism is to emphasize that it is not a species of what Williamson 1994 calls nihilism—the view that “vague expressions are empty; any vaguely drawn distinction is subverted” (p. 165). Another reason is to emphasize the need for a “transvaluation of all truth values,” so to speak—i.e., the need to transcend the impossible goal of finding some logically coherent, semantically correct, collective assignment of semantic status to all the statements in a sorites sequence. The proper goal for a semantics of vagueness, rather, is to provide an adequate account of the normative standards governing semantically correct assertoric practice.
[15] Perhaps transvaluationism can even be implemented by standard two-valued logic, employed in a way that respects in practice the logically disciplined weak generic incoherence of vagueness. Concerning our accommodation of vagueness, Quine 1995 remarks, “What I call my desk could be equated indifferently with countless almost coextensive aggregates of molecules, but I refer to it as a unique one of them, and I do not and cannot care which. Our standard logic takes this...in stride, imposing a tacit fiction of unique though unspecifiable reference” (p. 57).
[16] One salient example, discussed in Horgan 1998b section 4, is what I call “iterated supervaluationism.” The core idea of this approach is that the metalanguage for stating supervaluationist semantics is itself vague, and thus it too is subject to a supervaluationist treatment in a meta-meta-language; and so on, all the way up the metalinguistic hierarchy. More specifically, the metalinguistic expression ‘permissible interpretation’ is itself vague, and thus has various permissible interpretations itself in a meta-meta-language; and so on up. I myself am inclined to prefer a different approach to the logic of vagueness, described in Horgan 1994, sections 2 and 3.
[17] The phrase ‘Zen attitude’, suggested by Matjaz Potrc, seems to me to capture well the spirit of my recommended approach—more so than my use, in earlier papers, of sado-masochistic imagery (the dominatrix, cracking her whip) in connection with the theme of dominance without defeat.
[18] As my colleague John Tienson once remarked to me.
[19] Peter Unger was right! See Unger 1979a, 1979b.
[20] I thank Robert Barnard, Matjaz Potrc, John Tienson, and Mark Timmons for helpful comments and discussion.