Please quote or cite page numbers from published version only. Saul Kripke pointed out that whether or not an utterance gives rise to a liar-like paradox cannot always be determined by checking just its form or content. 1 Whether or not Jones’s utterance of ‘Everything Nixon said is true ’ is paradoxical depends in part on what Nixon said. Something similar may be said about the sorites paradox. For example, whether or not the predicate ‘are

If we adopt a supervaluational semantics for vagueness, what sort of logic results? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recast. There are several ways of doing this within a supervaluational framework, the main alternative being between ‘global ’ construals (e.g. an argument is valid if and only if it preserves truth-under-all-precisifications) and ‘local’ construals (an argument is valid if and only if, under all precisifications, it preserves truth). The former alternative is by far more popular, but I argue in favour 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. Supervaluationism is a mixed bag. It is sometimes described as the ‘standard ’ theory of vagueness, at least in so far as vagueness is construed as a semantic phenomenon, but exactly what that standard theory amounts to is far from clear. In fact, it is pretty clear that there is not just one supervaluational semantics out there—there are lots of such semantics; and although it is true that they all exploit the same insight, their relative differences are by no means immaterial. For one thing, a lot depends on how exactly supervaluations are constructed, that is, on how exactly we come to establish the truth-value of a given statement. (And when I say that a lot depends on this I mean to say that different explanations may give rise to different philosophical worries, or justify different reactions.) Secondly, and equally importantly, a lot depends on how a given supervaluationary machinery is brought into play when it comes to explaining the logic of the language, that is, not the notion of truth, or ‘super-truth’, as it applies to individual statements, but the notion of validity, or ‘super-validity’, as it applies to whole arguments. (I am thinking for instance of how different explanations may bear on the question of whether, or to what extent, vagueness involves a departure from classical logic.) Here I want to focus on this second part of the story. However, since the notion of validity depends on the notion of truth—or so one may argue—I also want to comment briefly on the first.

this paper I present and defend a definition of vagueness, and draw out some consequences of accepting this definition. 1 asks what we should properly expect of a definition of vagueness, and 2 reviews existing definitions. 3 explains a key background notion necessary to an understanding of the definition to be presented here. 4 presents this definition, and 5 explains its advantages. 6 draws out the consequences of accepting this definition for the project of offering a substantive theory of vagueness, and 7 considers and rejects a variation on the definition offered here. 1 What should we demand of a definition of vagueness? I take a definition of a property, object or phenomenon P to be a statement about what it is to be P that is true, useful and fundamental

Vagueness has always been a problem for philosophers. This is true in a number of ways. One obvious way is that the vagueness inherent in much philosophical discourse has always lead to problems in the interpretation and criticism of philosophical arguments. This is a way in which the vagueness of language

What the world needs now is another theory of vagueness. Not because the old theories are useless. Quite the contrary, the old theories provide many of the materials we need to construct the truest theory of vagueness ever seen. The theory shall be similar in motivation to supervaluationism, but more akin to many-valued theories in conceptualisation. What I take from the many-valued theories is the idea that some sentences can be truer than others. But I say very different things to the ordering over sentences this relation generates. I say it is not a linear ordering, so it cannot be represented by the real numbers. I also argue that since there is higher-order vagueness, any mapping between sentences and mathematical objects is bound to be inappropriate. This is no cause for regret; we can say all we want to say by using the comparative truer than without mapping it onto some mathematical objects. From supervaluationism I take the idea that we can keep classical logic without keeping the familiar bivalent semantics for classical logic. But my preservation of classical logic is more comprehensive than is normally permitted by supervaluationism, for I preserve classical inference rules as well as classical sequents. And I do this without relying on the concept of acceptable precisifications as an unexplained explainer. The world does not need another guide to varieties of theories of vagueness, especially since Timothy Williamson (1994) and Rosanna Keefe (2000) have already provided quite good guides. I assume throughout familiarity with popular theories of vagueness.

We study self-referential sentences of the type related to the Liar paradox. In particular, we consider the problem of assigning consistent fuzzy truth values to collections of self-referential sentences. We show that the problem can be reduced to the solution of a system of nonlinear equations. Furthermore, we prove that, under mild conditions, such a system always has a solution (i.e. a consistent truth value assignment) and that, for a particular implementation of logical “and”, “or” and “negation”, the “mid-point ” solution is always consistent. Next we turn to computational issues and present several truth-value assignment algorithms; we argue that these algorithms can be understood as generalized sequential reasoning. In an Appendix we present a large number of examples of self-referential collections (including the Liar and the Strengthened Liar), we formulate the corresponding truth value equations and solve them analytically and / or numerically.

In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and consider reasons for subscribing to the principle of uniform solution. 1 Introducing the Principle of Uniform Solution It would be very odd to give different responses to two paradoxes depending on minor, seemingly-irrelevant details of their presentation. For example, it would be unacceptable to deal with the paradox of the heap by invoking a multi-valued logic, ̷L∞, say, and yet, when faced with the paradox of the bald man, invoke a supervaluational logic. Clearly these two paradoxes are of a kind—they are both instances of the sorites paradox. And whether the sorites paradox is couched in terms of heaps and grains of sand, or in terms of baldness and the number of hairs on the head, it is essentially the same problem and therefore must be solved by the same means. More generally, we might suggest that similar paradoxes should be resolved by similar means. This advice is sometimes elevated to the status of a principle, which usually goes by the name of the principle of uniform solution. This principle and its motivation will occupy us for much of the discussion in this paper. In particular, I will defend a rather general form of this principle. I will argue that two paradoxes can be thought to be of the same kind because (at a suitable level of abstraction) they share a similar internal structure, or because of external considerations such as the relationships of the paradoxes in question to other paradoxes in the vicinity, or the way they respond to proposed solutions. I will then use this reading of the principle of uniform solution to make a case for the sorites and the liar paradox being of a kind.

A contradiction consists of a pair of sentences, one of which is the negation of the other. 1 From now on I shall settle on statement, but this decision will be of no consequence. 2 Intuitively, the first type of contradiction arises if we assert and deny the same thing in the same breath, whereas the second type of contradiction arises if we end up denying (perhaps unwittingly) something we have already asserted. We could plausibly generalize these formulations by construing in each case one of the two conjuncts, or statements, not as the negation of the other but as a conjunct or statement that is equivalent to the negation of the other. However, the notion of equivalence calls for a logic, and since Im going to be concerned with the logical status of LNC it will be safer to stick to narrow formulations such as (1) and (2). Somewhat more formally, these can also be put thus: (1') A contradiction is a statement of the form

Abstract. Vagueness is blamed for the Sorites Paradox. But the nature of vagueness is not well understood. Initially characterizing vagueness in terms of Sainsbury’s boundary metaphor, I argue that all semantic theories failing to do justice to this boundary metaphor are false. However, it is unclear what kind of semantic theory of vagueness is immune to this charge. I explore the boundary metaphor and find that vague predicates enjoy a peculiar kind of contextsensitivity. This leads to a distinction between m-boundaries and c-boundaries. Vague predicates cannot draw m-boundaries but can draw c-boundaries. This result, along with appreciating the peculiar sense in which vague predicates are context-sensitive, clears the road for a sketch of a solution to the Sorites Paradox: roughly, the conditions for the correct use of vague predicates presuppose c-boundaries, thereby rendering the Sorites Paradox unsound. Finally, the question of whether there is worldly vagueness in addition to representational vagueness is explored. It is found that puzzles purportedly concerning worldly vagueness can be resolved in just the way puzzles involving representational vagueness can. I. Introduction: Vagueness

conference at the University of Connecticut in the Fall of 2002. I am grateful for the help of these audiences, and especially to Prof. Soames. againstgaps.tex: March 7, 2003 (13:13) Many things are neither true nor false: shoes and ships and sealing wax, to name a few. But these things are neither true nor false because they are not the kinds of things that can be either. There are also some things that are apt for being true or false. Preferences vary on exactly what these things are. Common candidates include utterances, interpreted sentences paired with contexts, and propositions. Can there be something that is apt to be true or false, but fails to be either? This is the question of whether there are substantial truth-value gaps. It has been a persistent idea in the philosophy of language that there are substantial truthvalue gaps. This view was held, at some moments, by Strawson1 and by Frege. 2 More recently, Scott Soames3 has presented an argument in favor of the view, and has applied it to some issues related to semantic paradoxes and to vagueness. In its own right, the question is deeply involved with some of the very basic issues in the philosophy of language: content, assertion, and truth. In this essay, I shall argue that there are no substantial truth-value gaps. There are some