Keefe, Rosanna. Theories of vagueness / Rosanna Keefe. p. cm. Includes bibliographical references and index. ISBN 0 4 (hardback). 1. Vagueness. Rosanna Keefe, Theories of Vagueness, Cambridge University Press, philosophers, I suspect, are partial to supervaluational theories of vagueness. Most expressions in natural language are vague. But what is the best semantic treatment of terms like 'heap', 'red' and 'child'? And what is the logic of.
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related controversial issues in the theory of vagueness are introduced. Either issue 60To be clear, Keefe  herself subscribes to a standard version of . Theories of VaguenessRosanna KeefeCambridge University Press Theories of Theories of Vagueness Rosanna Keefe Cambridge University Press Theories. timely book Rosanna Keefe explores the questions of what we should want from an account of vagueness and how we should assess rival theories.
There have, however, been some attempts at this type of solution to the sorites paradox using, for example, more elaborate notions of the context of a subject's judgement see e. Raffman But many vague predicates are multi-dimensional: several different dimensions of variation are involved in determining their applicability.
The three central features of vague predicates are shared by multidimensional ones. Next consider whether multi-dimensional predicates may lack sharp boundaries. In the one-dimensional case, F has a sharp boundary or sharp boundaries if possible candidates for it can be ordered with a point or points marking the boundary of F's extension, so that everything that falls on one side of the point or between the points is F and nothing else is F.
For a multi-dimensional predicate, there may be no uniquely appropriate ordering of possible candidates on which to place putative boundary-marking points. Rather, for a sharply bounded two-dimensional predicate the candidates would be more perspicuously set out in a two-dimensional space in which a boundary could be drawn, where the two-dimensional region enclosed by the boundary contains all and only instances of the predicate.
With a vague two-dimensional predicate no such sharp boundary can be drawn. Similarly, for a sharply bounded predicate with a clear-cut set of n dimensions, the boundary would enclose an n-dimensional region containing all of its instances; and vague predicates will lack such a sharp 11 Theories of vagueness boundary.
But this, I claim, is distinctive of the vagueness of such predicates: they have no sharp boundary, but nor do they have a fuzzy boundary in the sense of a rough boundary-area of a representative space. Finally, multi-dimensional vague predicates are susceptible to sorites paradoxes. Next, I shall argue that comparatives as well as monadic predicates can be vague. This will be the case if there is a determinate ordering of candidates for F-ness allowing ties.
It may seem that 4 Could there be a single, determinate way of balancing the various dimensions of a multi-dimensional predicate that does yield a unique ordering? But, in fact, there could be borderline instances of the comparative due to indeterminacy over exactly what should count as the instant of someone's birth and so whether it is before or after the birth of someone else.
Similarly, though there is a single dimension of height, people cannot always be exactly placed on it and assigned an exact height.
For what exactly should count as the top of one's head? Can comparatives also lack sharp boundaries? Talk of boundaries, whether sharp or fuzzy, is much less natural for comparatives than for monadic predicates. But we might envisage precise comparatives for which we could systematically set out ordered pairs of things, hx, yi and draw a sharp boundary around those for which it is true that x is F-er than y.
Another possible sense in which comparatives may lack sharp boundaries is the following. First, there can be other vague dyadic relational expressions. And, just as comparatives can be vague, particularly when related to a multi-dimensional positive, so can superlatives.
A theory of vagueness should have the resources to accommodate all the different types of vague expression. And, for example, we should reject an account of vagueness that was obliged to deny the above illustrated features of certain comparatives in order to construct its own account of vague monadic predicates.
The typical focus on monadic predicates need not be mistaken, however. Perhaps, as Fine suggests, all vagueness is reducible to predicate vagueness , p. Alternatively, vagueness might manifest itself in different ways in different kinds of expression, and this could require taking those different expression-types in turn and having different criteria of vagueness for comparatives and monadic predicates. Another possibility is to treat complete sentences as the primary bearers of vagueness, perhaps in their possession of a non-classical truth-value.
Provided one can still make sense of a typical attribution of vagueness to some element of a sentence in the uncontroversial cases, I suggest that this strategy is an appealing one. Surely not: thoughts and beliefs are among the mental items which share the central characteristics of vagueness; other controversial cases include perceptions.
What about the world itself: could the world be vague as well as our descriptions of it? Can there be vague objects? Or vague properties the ontic correlates of predicates? So it may seem that Ben Nevis has fuzzy boundaries, and so, given the common view that a vague object is an object with fuzzy, spatio-temporal boundaries, that it is a vague object. See e. Parsons , Tye and Zemach for arguments that there are vague objects.
But there are, of course, other contending descriptions of the situation here. It would then be at the level of our representations of the world that vagueness came in. This would be a mistake if a theory of linguistic vagueness had to rely on ontic vagueness. But that would be surprising since it seems at least possible to have vague language in a non-vague world. Similarly, in a precise world we would still use vague singular terms, perhaps to pick out various large collections of precise fundamental particulars e.
So it seems that language could still be vague if the concrete world were precise. The second task is that of addressing the sorites paradox. Borderline case predications are either true or false after all, though we do not and cannot know which. And a pragmatic account of vagueness also seeks to avoid challenging classical logic and semantics, but this time by accounting for vagueness in terms of pragmatic relations between speakers and their language: see chapter 6.
They also do not seem to bear on the question whether there can be vague sets, which might also be counted as a form of ontic vagueness. Tye, for example, believes that there are vague sets and maintains that they are crucial to his own theory of the linguistic phenomena see Tye This generates a number of non-classical options. Note that a borderline case of the predicate F is equally a borderline case of not-F: it is unclear whether or not the candidate is F.
This symmetry prevents us from simply counting a borderline F as not-F. But there are several ways of respecting this symmetry. Some take the line that a predication in a borderline case is both true and false: there is a truth-value glut.
A more popular position is to admit truth-value gaps: borderline predications are neither true nor false.
One elegant development is supervaluationism. In this way, the supervaluationist adopts a non-classical semantics while aiming to minimise divergence from classical logic. A theory of this type will be defended in chapters 7 and 8.
Alternatively, degree theories countenance degrees of truth, introducing a whole spectrum of truth-values from 0 to 1, with complete falsity as degree 0 and complete truth as degree 1. So far the sketched positions at least agree that there is some positive 17 Theories of vagueness account to be given of the logic and semantics of vagueness. Other writers have taken a more pessimistic line.
In particular, Russell claims that logic assumes precision, and since natural language is not precise it cannot be in the province of logic at all , pp. And arguments involving vague predicates are clearly not all on a par. And, similarly, there are other ways of arguing with vague predicates that should certainly be rejected. Some account is needed of inferences that are acceptable and others that fail, and to search for systematic principles capturing this is to seek elements of a logic of vague language.
So, I take the pessimism of the no-logic approach to be a very last resort, and in this book I concentrate on more positive approaches. They may be true or false, or have no truth value at all in particular, being neither true nor false , or be both true and false, or have a nonclassical value from some range of values.
For example, 2C1 if x1 is tall, so is x2 2C2 if x2 is tall, so is x3 and so on. As well as needing to solve the paradox, we must assess that general form of argument because it is used both in philosophical arguments outside the discussion of vagueness e. We can: a deny the validity of the argument, refusing to grant that the conclusion follows from the given premises; or b question the strict truth of the general inductive premise 2 or of at least one of the conditionals 2Ci ; or c accept the validity of the argument and the truth of its inductive premise or of all the conditional premises but contest the supposed truth of premise 1 or the supposed falsity of the conclusion 3 ; or d grant that there are compelling reasons both to take the 7 As a further example of the former, consider Kirk pp.
He presents his argument as using mathematical induction but does not ask whether its employment of vague predicates casts doubt on that mode of argument. Wright argues that different responses could be required depending on the reasons that support the inductive premise.
More generally, a theory should account for the persuasiveness of the paradox as a paradox and should explain how this is compatible with the fact that we are never, or very rarely, actually led into contradiction. This can be seen most clearly when the argument takes the second form involving a series of conditionals, the 2Ci.
The only rule of inference needed for this argument is modus ponens. I agree on both points and shall not pursue the matter further here. There is, however, a different way of rejecting the validity of the many-conditionals form of the sorites.
It might be suggested that even though each step is acceptable on its own, chaining too many steps does not guarantee the preservation of truth if what counts as preserving truth is itself a vague matter. As Dummett again notes, this is to deny the transitivity of validity, which would be another drastic move, given that chaining inferences is normally taken to be essential to the very enterprise of proof.
The sorites arguments, on his view, cannot be valid because, containing vague expressions, they are just not the kind of thing that can be valid or invalid. This implies the existence of sharp boundaries and the epistemic theorist, who takes this line, will explain why vague predicates appear not to draw sharp boundaries by reference to our ignorance see chapter 3. In a non-classical framework there is a wide variety of ways of developing option b , and it is not clear or uncontroversial which of these entail a commitment to sharp boundaries.
And other non-classical frameworks may allow that 2 is not true, while not accepting that it is false. Degree theorists offer another non-classical version of option b : they can deny that the premises are strictly true while maintaining that they are nearly true. The essence of their account is to hold that the predications Fxi take degrees of truth that encompass a gradually decreasing series from complete truth degree 1 to complete falsity degree 0.
If the sorites argument based on many conditionals is to count as strictly valid, then 21 Theories of vagueness an account of validity is needed that allows a valid argument to have nearly true premises but a false conclusion. Putnam suggests this strategy. But critics have shown that with various reasonable additional assumptions, other versions of sorites arguments still lead to paradox.
In particular, if, as might be expected, you adopt intuitionistic semantics as well as intuitionistic logic, paradoxes recur see Read and Wright And Williamson shows that combining Putnam's approach to vagueness with his epistemological conception of truth still faces paradox.
See also Chambers , who argues that, given Putnam's own view on what would make for vagueness, paradox again emerges. The bulk of the criticisms point to the conclusion that there is no sustainable account of vagueness that emerges from rejecting classical logic in favour of intuitionistic logic.
If we accept these premises and the validity of the argument, it follows that we will never get a heap, no matter how many grains are piled up: so there are no heaps. The thesis, put in linguistic terms, is that all vague predicates lack serious application, i.
Classical logic can be retained in its entirety, but sharp boundaries are avoided by denying that vague predicates succeed in drawing any boundaries, fuzzy or otherwise. There will be no borderline cases: for any vague F, everything is F or everything is not-F, and thus nothing is borderline F. For example, he shows how the nihilist cannot state or argue for his own position on 22 The phenomena of vagueness The response of accepting the conclusion of every sorites paradox cannot be consistently sustained.
Such reversibility is typical; given a sorites series of items, the argument can be run either way through them. Systematic grounds would then be needed to enable us to decide which of a pair of sorites paradoxes is sound e.
Unger is driven to such an extreme position by the strength of the arguments in support of the inductive premises of sorites paradoxes. If our words determined sharp boundaries, Unger claims, our understanding of them would be a miracle of conceptual comprehension , p. A different miracle of conceptual comprehension would be needed then to explain how we can understand that meaning and, in general, how we can use such empty predicates successfully to communicate anything at all.
Sorites paradoxes could then demonstrate the inconsistency of such a set of rules, and this is option d. Responses c and d are not always clearly distinguished. Writers his own terms e. My discussion of methodological matters in chapter 2 will suggest that a swifter rejection of the position is warranted anyway.
But other writers, for example Dummett, explore these conceptual questions. The paradoxes thus reveal the incoherence of the rules governing vague terms: by simply following those rules, speakers could be led to contradict themselves.
This inconsistency means that there can be no coherent logic governing vague language. So option d can be developed in such a way that makes it compatible with option a , though this route to the denial of validity is very different from Russell's. Being outside the scope of logic need not make for incoherence.
The acceptance of such pervasive inconsistency is highly undesirable and such pessimism is premature; and it is even by Dummett's own lights a pessimistic response to the paradox, adopted as a last resort rather than as a positive treatment of the paradox that stands as competitor to any other promising alternatives.
Communication using vague language is overwhelmingly successful and we are never in practice driven to incoherence a point stressed by Wright, e. And even when shown the sorites paradox, we are rarely inclined to revise our initial judgement of the last member of the series.
It looks unlikely that the success and coherence in our practice is owed to our grasp of inconsistent rules. A defence of some version of option a or b would provide an attractive way of 10 See also Rolf , Horgan , advocates a different type of the inconsistency view. Such higherorder paradoxes must also be addressed. There are also related metalinguistic paradoxes which threaten any theory of vagueness that introduces extra categories for borderline cases assuming they can thereby classify every predication of a given vague predicate in some way or other.
Horgan instructs us to take, in turn, successive pairs of a sorites series x1 and x2, x2 and x3 etc. This emphasises how theorists need to avoid solutions to the original sorites which are still committed to sharp boundaries between semantic categories.
Unlike most the solutions I have been outlining in a to d , these treatments are not situated in the context of a theory of vagueness more generally. Some, I suggest, may be better seen as tackling a somewhat different issue. But we may hope to express it without that device. Among other non-solutions are discussions which give some remedy through which we can avoid actually being driven to paradox as if it wasn't already clear how this could be done. For example, Shapiro distinguishes serial processes from parallel ones and attributes the paradox to the use of a serial process that assigns values to predications on the basis of the assignment to the previous member of the ordered sorites series.
Such a procedure is wrong because it yields absurd results, Shapiro argues, but he gives no indication of why it is plausible nonetheless and reliable in other contexts , or what the consequences are regarding sharp boundaries.
Moreover, in treating something like the inductive premise of the sorites as an instruction for applying the predicate given certain other members of its extension, Shapiro appears to ignore the fact that it can be treated as a plausible generalisation about the members of the series.
On this typical interpretation the paradox persists in abstraction from contexts of running through the sorites sequence via some chosen procedure. We may hold that no sentence can be true without being determinately true. For how can a be F without being determinately F? Dp and p will then be true in exactly the same situations. But the operator is not thereby redundant: for example, :Dp will be true in a borderline case, when :p is indeterminate.
When there is some deviation from classical logic and semantics, the fact that p and Dp coincide in the way described does not guarantee that they are equivalent in the embedded contexts generated by negating them.
According to the epistemic view, which allows no deviation from classical logic, p can be true without Dp being true, namely when p is borderline and not known to be true. For the D operator must, on that account, be an epistemic operator. The degree theorist can say that Dp is true if p is true to degree 1 and is false if p is true to any lesser degree. A supervaluationist, on the other hand, will say that Dp is true just in case p is true on all ways of making it precise and is false otherwise so if p is borderline, p itself will be neither true nor false, but Dp will be false.
Note that it is not only on the supervaluationist scheme that the comparison with modal logics is appropriate. Williamson, for example, explains its applicability within an epistemic view of vagueness see especially his And, Wright also maintains, we need to use the D operator to say what it is for a predicate to lack sharp boundaries. Consider a series of objects xi forming a suitable sorites series for F e.
Wright proposes , p. This latter condition gives rise to paradox; but lacking sharp boundaries in the sense of W does not lead straight to paradox. Suppose someone were to take Wright's claim about the importance of D to show that a theory of vagueness should proceed by introducing a primitive D operator and focusing on its logic and semantics. They would, I argue, be pursuing the wrong approach. In particular, replacing statements naturally used to express our intuitions about borderline cases and the lack of sharp boundaries with different but similar statements involving the D operator does not provide an excuse to ignore the very questions that are, and should be, at the centre of the debate, namely ones about the original intuitions.
We need still to ask how we should classify the original premise itself. If we say that premise is true, as it seems to be, the paradox remains untouched.
But can we be content to call it false and hence accept that there is a last patch of a sorites series that is red and an adjacent pair in the series of which one is F and the other is not-F? Wright suggests that the inductive premises of some but not all sorites paradoxes may be of indeterminate status , p. But how are we to understand this claim? But what should we say when we do not attach that operator?
As editors, Keefe and Smith have made two wise decisions. They have selected and arranged just the right readings in just the right way, and, because of the scope and complexity of issues involving the philosophical treatment of vagueness, they have provided a clear and extensive introductory essay.
This introductory essay should be read both initially and then, in sections, along with the subsequent essays. This first includes brief accounts of the phenomenon of vagueness from classical sources such as Diogenese Laertius, Galen, and Cicero, as well as pres papers by Russell, Black, Hempel, and Mehlberg.
In the s there was, as the editors point out, an explosion of interest in vagueness resulting in a number of detailed theories. The third group of essays is made up of more recent papers, most of which build upon, extend, or criticize ideas first broached in the s. As I have already indicated, this selection, and its more or less chronolog- ical ordering, is just right.
A brief note on the main modern theories is in order. The epistemic theory holds that vague terms are not vague because they signify vague properties but because we just happen to be ignorant about the bound- aries of such properties. In a few pages of his The Reach of Science, Mehlberg inaugu- rated the supervaluation theory later developed by van Fraassen and formulated as an account of vagueness by Fine.
Vagueness is thus accounted for by means of a nonclassical semantics for the language. Recently, philosophers who are less sanguine about truth-value gaps have opted for theories that admit either a third value between truth and falsity or an infinite number of truth-values to account for bor- derline cases.
Unlike the supervaluationist, these philosophers have chosen to preserve a standard semantic theory while turning to a non- standard either three-valued or fuzzy logic to account for vagueness.
The best of these are represented by the papers by Wright, Dummett, and Sainsbury included here.