ABSTRACT: This paper argues that the so-called paradoxes of higher-order vagueness are the result of a confusion between higher-order vagueness and the distribution of the objects of a Sorites series into extensionally non-overlapping non-empty classes.

Supervaluational accounts of vagueness have come under assault from Timothy Williamson for failing to provide either a sufficiently classical logic or a disquotational notion of truth, and from Crispin Wright and others for incorporating a notion of higher-order vagueness, via the determinacy operator, which leads to contradiction when combined with intuitively appealing ‘gap principles’. We argue that these criticisms of supervaluation theory depend on giving supertruth an unnecessarily central role in that theory as the sole notion of truth, rather than (...) as one mode of truth. Allowing for the co-existence of supertruth and local truth, we define a notion of local entailment in supervaluation theory, and show that the resulting logic is fully classical and allows for the truth of the gap principles. Finally, we argue that both supertruth and local truth are disquotational, when disquotational principles are properly understood. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

The use of vague language in law has important implications for legal theory. Legal philosophers have occasionally grappled with those implications, but they have not come to grips with the characteristic phenomenon of vagueness: the sorites paradox. I discuss the paradox, and claim that it poses problems for some legal theorists (David Lyons, Hans Kelsen, and, especially, Ronald Dworkin). I propose that a good account of vagueness will have three consequences for legal theory: (i) Theories that deny that vagueness in (...) formulations of the law leads to discretion in adjudication (including Dworkin's) cannot accommodate “higher-order” vagueness, (ii) A legal theory should accept that the law is partly indeterminate when it can be stated in vague language, (iii) However, the traditional formulation of the indeterminacy claim, that a vague statement is “neither true nor false” in a borderline case, is misconceived and should be abandoned. (shrink)

In this book, I present the results of an investigation which began with an extended stay at Oxford's Balliol College during the first half of 1995. My visit to Oxford was made possible by a grant from the Spanish Ministerio de Educaci6n y Ciencia. My sincere thanks go to Joseph Raz who served as my supervisor in Oxford. For several points of the present study, conversations with Timothy Endicott in Oxford were also of great help. The book is part of (...) a larger project of investigation, directed by Albert Calsami glia, which is a joint effort of a group of legal philosophers from the Universitat Pom peu Fabra and the Universitat de Girona, and which also receives financial support from the Spanish Ministerio de Educaci6n y Ciencia. An earlier version of the manuscript was presented in June 1996 to the selection committee for a tenured professorship in Legal Philosophy at the Universitat de Girona. The members of the committee were Francisco Laporta, Albert Calsamiglia, Gregorio Peces-Barba, Camilo J. Cela Conde, and Francesca Puigpelat. I am grateful to all of them for their comments, which have been extremely useful in preparing the final ver sion of this book. (shrink)

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These (...) explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox. (shrink)

Logicians interested in naive theories of truth or set have proposed logical frameworks in which classical operational rules are retained but structural rules are restricted. One increasingly popular way to do this is by restricting transitivity of entailment. This paper discusses a series of logics in this tradition, in which the transitivity restrictions are effected by a determinacy constraint on assumptions occurring in both the major and minor premises of certain rules. Semantics and proof theory for 3-valued, continuum-valued and surreal-valued (...) semantics are given and the proof theory for the systems outlined. The framework is robust in the sense that no conditional, defined or primitive, which sustains the contraction principles underlying Curry paradoxes can be expressed. Classical recapture is smoothly achievable in the system which however is expressively limited and not semantically closed. The conclusion considers the issue of how to extend the system to capture full naive set theory. (shrink)

What sort of logic do we get if we adopt a supervaluational semantics for vagueness? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recasted. There are several ways of doing that within a supervaluational framework, the main alternative being between “global” construals (e.g., an argument is valid iff it preserves truth-under-all-precisifications) and “local” construals (an argument is valid iff, under all precisifications, it preserves truth). The former alternative is by (...) far more popular, but I argue in favor of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion. (shrink)

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order (...) vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes. (shrink)

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order (...) vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe’s response to combat both paradoxes. (shrink)

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order (...) vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes. (shrink)

The supporter of vague objects has been long challenged by the following ‘Argument from Identity’: 1) if there are vague objects, then there is ontically indeterminate identity; 2) there is no ontically indeterminate identity; therefore, 3) there are no vague objects. Some supporters of vague objects have argued that 1) is false. Noonan (Analysis 68: 174–176, 2008) grants that 1) does not hold in general, but claims that ontically indeterminate identity is indeed implied by the assumption that there are vague (...) objects of a certain special kind (i.e. vague objects*). One can therefore formulate a ‘New Argument from Identity’: 1′) if there are vague objects*, then there is ontically indeterminate identity; 2) there is no ontically indeterminate identity; therefore, 3′) there are no vague objects*. Noonan’s strategy is to argue that premiss 1′) is inescapable, and, as a consequence, that Evans’s alleged defence of 2) is a real challenge for any supporter of vague objects. I object that a supporter of vague objects who grants the validity of Evans’s argument allegedly in favour of 2) should reject premiss 1′). The threat of the New Argument from Identity is thus avoided. (shrink)

This paper is an attempt to show that the subvaluation theory isnot a good theory of vagueness. It begins with a short review of supervaluation and subvaluation theories and procedes to evaluate the subvaluation theory. Subvaluationism shares all the main short-comings of supervaluationism.Moreover, the solution to the sorites paradox proposed by subvaluationists isnot satisfactory. There is another solution which subvaluationists could availthemselves of, but it destroys the whole motivation for using a paraconsistentlogic and is not different from the one offered (...) by supervaluationism. (shrink)

In this paper I develop a semantic theory of vagueness that is immune to worries regarding the use of precise mathematical tools. I call this view semantic quietism. This view has the advantage of being clearly compatible with the phenomenon of vagueness. The cost is that it cannot capture every robust semantic fact.

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order (...) vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes. (shrink)

I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is.

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity are in play (...) (Williamson-style "luminosity" or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys axiom 4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone axiom 4 (by adopting what elsewhere we call columnar higher-order vagueness), and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject axiom 4. Third, we rebut a number of arguments that have been produced by opponents of axiom 4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)

The extensions of vague predicates like ‘is bald’, ‘is tall’, and ‘is a heap’ apparently lack sharp boundaries, and this makes such predicates susceptible to soritical reasoning, i.e. reasoning that leads to some version of the notorious sorites paradox. This essay is concerned with a certain kind of theory of vagueness, according to which the symptoms and puzzles of vagueness should be accounted for in terms of a particular species of context sensitivity exhibited by vague expressions. The basic idea is (...) that the extensions of vague predicates vary with certain contextual factors, and that this fact can explain why they appear to lack sharp boundaries. This kind of view is referred to as contextualism about vagueness. A detailed characterisation of contextualism about vagueness is given in chapter two and three. In chapter two, a generic version of contextualism about vagueness is developed, and some alternative forms of context sensitivity are introduced. In chapter three, the specific contextual factors appealed to by different contextualists are discussed. In chapter four, different contextualist diagnoses of the sorites paradox are considered, and found to be problematic in various ways. It is argued that contrary to what some of its proponents have claimed, contextualism about vagueness is not superior to other comparable theories of vagueness when it comes to explaining the appeal of soritical reasoning. In chapter five, a certain version of the sorites paradox, known as the forced march sorites, is discussed. It is argued that “data” about how speakers would behave in the forced march cannot lend any firm support to contextualism about vagueness. In chapter six, some problems concerning the instability of the contextual factors are considered. One problem is that contextualist diagnoses of the sorites which locate a fallacy of equivocation in the reasoning seem to render non-soritical reasoning fallacious as well. A model for treating this problem is suggested, but on closer consideration, it turns out to be problematic. Moreover, this model is of no help in solving the more general problem that even if classical logic remains valid for vague language on some contextualist views, the instability of the extensions of vague predicates makes it difficult to know when a certain piece of reasoning instantiates a valid argument form. Other difficulties arise with respect to speech reports and belief contents. Chapter seven concludes with a summary and some methodological remarks. (shrink)

It is generally supposed that borderline cases account for the tolerance of vague terms, yet cannot themselves be sharply bounded, leading to infinite levels of higher order vagueness. This higher order vagueness subverts any formal effort to make language precise. However, it is possible to show that tolerance must diminish at higher orders. The attempt to derive it from indiscriminability founders on a simple empirical test, and we learn instead that there is no limit to how small higher order tolerance (...) may become. That means there is no limit to how precisely we may draw the boundaries of borderline cases, thus forestalling any requirement for higher order vagueness. (shrink)

Most words in natural language are vague, that is to say, they lack sharp boundaries and, hence, they have (actual or potential) borderline cases, where the word in question neither definitely applies nor definitely fails to apply. Vagueness gives rise to paradoxes, the best known of which is the sorites (concerned with how many grains of sand are needed to make a heap). Besides offering a solution to such paradoxes, a theory of vagueness should systematically describe how the truth conditions (...) of sentences with vague terms are determined; and it should also define the right logical principles for reasoning with such sentences. This article offers an introduction to the main theories of vagueness and to the problems they have to face. (shrink)