Many philosophical theories make comparisons between objects, events, states of affairs, worlds, or systems, and many such theories deliver plausible verdicts only if some of the elements they compare are ranked as ‘best.’ When the relevant ordering does not have such ‘best’ or ‘tied for best’ elements the theory wrongly falls silent or gives badly counterintuitive results. This paper develops ordering supervaluationism---a very general technique that allows any such theory to handle these problematic cases. Just as ordinary (...) supervaluation helps us save and generalize ‘uniqueness assuming’ theories, ordering supervaluationism helps us save and generalize ‘limit assuming’ theories. With so many otherwise attractive limit assuming theories, this is a sensible, methodologically conservative approach. (shrink)

This paper discusses Fara's so-called 'Paradox of Higher-Order Vagueness' concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call 'regional validity'. Then I show that the supervaluationist might solve Fara's paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

This paper discusses Fara's so-called 'Paradox of Higher-Order Vagueness' concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call 'regional validity'. Then I show that the supervaluationist might solve Fara's paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

Recently, Wilcox (JGPS 51: 65–87, 2020) argued against the so-called wide interval view and in favor of the principle of indifference as the correct response to unspecific evidence. Embedded in a formal model of the beliefs of an agent, the former presupposes imprecise probabilities and the latter numerically precise degrees of belief. His argument is illustrated by a thought experiment that comes with a fundamental intuition. According to Wilcox, the wide interval view is incompatible with this intuition and, thus, undermined. (...) In contrast, I show that the intuition behind the thought experiment is, in fact, compatible with the wide interval view if it is embedded into a specific conception of imprecise probabilities as model of belief. This conception is an extension of a framework which I call modified supervaluationism (MSV) and which I recently presented elsewhere (Karge 2021, 175–191). To accommodate the thought experiment’s fundamental intuition, it introduces a notion of second-order beliefs. (shrink)

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, (...) supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic _NT_. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. (shrink)

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness . This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist (...) theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion. (shrink)

This piece gives an overview of the supervaluationist theory of vagueness. According to that theory, a sentence is true if and only if it is true on all ways of making it precise. This yields borderline case predications that are neither true nor false, but yet classical logic is preserved almost entirely. The article presents the view and some of its merits and briefly compares it with other theories of vagueness. It raises issues about higher-order vagueness and the definitely operator (...) and about the supervaluationist's accounts of truth and validity. (shrink)

Though it is standardly assumed that supervaluationism applied to vagueness is committed to global validity, Achille Varzi (2007) argues that the supervaluationist should take seriously the idea of adopting local validity instead. Varzi’s motivation for the adoption of local validity is largely based on two objections against the global notion: that it brings some counterexamples to classically valid rules of inference and that it is inconsistent with unrestricted higher-order vagueness. In this discussion I review these objections and point out (...) ways to address them not considered in Varzi’s paper. (shrink)

Supervaluationism is one of the most discussed approaches to the semantics of future tense sentences in a branching time. In this paper, we consider the criticism advanced by Malpass against Supervaluationism. This criticism relies on the fact that supervaluationists must accept as supertrue disjunctions whose disjuncts are not only supertrue—which supervaluationists are ready to acknowledge—but also not satisfiable. In order to show this, Malpass proposes a formula, F F 1, which shows the existence of a satisfiable disjunction with (...) unsatisfiable disjuncts in supervaluationist models. In reply, we show that formula F F 1 cannot be expressed within a model (whether Ockhamist or supervaluationist) because it quantifies on models. It can be correctly characterised only within a meta-model that has the resources to quantify on various models. However, once that is done, F F 1 is, for the advocates of Supervaluationism, no more demanding than other disjunctions because it just generalises at the meta-theoretical level what supervaluationists already acknowledge at the theoretical level. (shrink)

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)

Discussions of higher-order vagueness rarely define what it is for a term to have nth-order vagueness for n>2. This paper provides a rigorous definition in a framework analogous to possible worlds semantics; it is neutral between epistemic and supervaluationist accounts of vagueness. The definition is shown to have various desirable properties. But under natural assumptions it is also shown that 2nd-order vagueness implies vagueness of all orders, and that a conjunction can have 2nd-order vagueness even if its conjuncts do not. (...) Relations between the definition and other proposals are explored; reasons are given for preferring the present proposal. (shrink)

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

The Ecclesiastical History of Orderic Vitalis has been called `the greatest of all medieval chronicles'. Written in Normandy between 1114 and 1141, it is a detailed history of the Norman people and their conquests, full of vivid, often penetrating portraits of the lives and characters of kings and queens, lords and bishops, simple knights, and humble villagers. The chronicle gives a unique, authentic picture of feudal society during a period of rapid change in church and state which saw the emergence (...) of the Anglo-Norman realm, the spread of new forms of monasticism, and the launching of the Crusades. Published in 1968, volume II of Marjorie Chibnall's six-volume edition of the Ecclesiastical History is now available for the first time in paperback. `a superbly edited Latin text and a unique English translation of the work of a major historian' American Historical Review `Mrs Chibnall is giving us... her edition of a classical medieval text which will itself become a classic of medieval scholarship.' Times Literary Supplement. (shrink)

Betrifft die Handschrift Cod. 555 der Burgerbibliothek Bern.

Edited with a facing-page English translation from the Latin text by: Chibnall, Marjorie.

After the three families divided up the state of Jin and the Tian family took over Qi, the political situation in the fourth century B.C.E. appeared even more chaotic. Wei conquered Chu's Luyang and Qin's Xihe, Qin defeated Wei at Shimen , and again at Shaoliang , and Wei moved its capital to Daliang. During the mid-Warring States period, Qin became dominant in the west, Qi in the east, Chu in the south, and Wei in the center. Rapid changes occurred (...) one after another; trickery and conflict were rampant. However, the shi , the intelligentsia, actually managed to expand their spheres of existence and activity accordingly in this state of tension between countries, states, and men. A few major states assembled quite a number of literati; and those men of the villages, who had in earlier times merely waited passively for opportunities to fall into their laps, crisscrossed the various feudal states; literati who previously had no cause to use military force became, one by one, the honored guests of the feudal lords. With the breakdown of the previously simple and obvious order, the gradually dismantling of the common body of the ancestral clan of the past, and the increasing complexities of social relationships after the dissolution of blood ties, social order became a matter of concern. Or, to reword this most practical issue: How could this universally chaotic society go from disorder to order? (shrink)

The BRAHMASUTRA of BADARAYANA represents the fundamental text of exegesis of Vedanta. The intent of Badarayana - the sage that for authority and realization of consciousness has been identified with Vyasa, the Rsi who ordered the texts of the Vedas - is that of providing the right perspective in the interpretation of the most profound and meaningful contents of the Upanisads. This had proven necessary in order to rectify some unilateral aspects propounded by several schools of thought, both orthodox and (...) non-orthodox. The Brahmasutra presents, in their simplicity and incisiveness, the assertions of the Sruti and of the Smrti, showing their concordance in the recognition of the Nirguna Brahman as the ultimate Realty. In his notes, RAPHAEL underlines the fact that Badarayana does not oppose the various philosophical schools, but the Rsi allows us to comprehend that their postulates cannot represent the ultimate Truth as expounded in the Vedas and in the Upanisads. Moreover, at times Raphael focuses on certain aspects of the Advaita Doctrine with references to the Western Tradition, and - making the relevant parallels to the philosophy of Parmenides, Plato, Plotinus, and so on - highlights the unity of the sole universal Tradition at the metaphysical level. (shrink)

In this paper I investigate an alternative to imprecise probabilism. Imprecise probabilism is a popular revision of orthodox Bayesianism: while the orthodox Bayesian claims that a rational agent’s belief-state can be represented by a single credence function, the imprecise probabilist claims instead that a rational agent’s belief-state can be represented by a set of such functions. The alternative that I put forward in this paper is to claim that the expression ‘credence’ is vague, and then apply the theory of (...) class='Hi'>supervaluationism to sentences containing this expression. This gives us a viable alternative to imprecise probabilism, and I end by comparing the two accounts. I show that supervaluationism has a simpler way of handling sentences relating the belief-states of two different people, or of the same person at two different times; that both accounts may have the resources to develop plausible decision theories; and finally that the supervaluationist can accommodate higher-order vagueness in a way that is not available to the imprecise probabilist. (shrink)

In ordinary conversation, we describe all sorts of different things as vague: you can have vague plans, vague ideas and vague aches and pains. In philosophy of language, in contrast, it is parts of language – words, expressions and so on – that are said to be vague. One classic example of a vague term is the word ‘heap’. A single grain clearly does not make a heap, and a million grains does make a heap, but where exactly does the (...) boundary lie? How many grains do you need to make a heap? There seems to be no precise answer to this question, and because the term is imprecise in this way, we call it vague. Vague terms are extremely common in natural language. The term ‘bald’ is vague, because there is no precise number of hairs that mark the boundary between ‘bald’ and ‘not bald’; the term ‘hot’ is vague because there is no precise temperature that something must reach to count as hot – and so on. As we have seen, adjectives can be vague, but so can nouns, adverbs and perhaps all parts of language. To find terms which are precise rather than vague, we need to look to the languages of logic and mathematics. We can use vague terms to construct paradoxes known as sorites paradoxes. From an obviously true premise, such as that a collection of 1 million grains is a heap, together with the claim that ‘heap’ has no sharp boundary, we can derive the absurd conclusion that just 1 grain counts as a heap. Any theory of vagueness must offer some solution to this paradox. Some of the most popular theories of vagueness include supervaluationism, the degree theory of truth and the epistemic theory, and many of the available theories demand a radical rethink of classical accounts of logic and language. (shrink)

A number of Bayesians claim that, if one has no evidence relevant to a proposition P, then one's credence in P should be spread over the interval [0, 1]. Against this, I argue: first, that it is inconsistent with plausible claims about comparative levels of confidence; second, that it precludes inductive learning in certain cases. Two motivations for the view are considered and rejected. A discussion of alternatives leads to the conjecture that there is an in-principle limitation on formal representations (...) of belief: they cannot be both fully accurate and maximally specific. (shrink)

Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (shrink)

Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature (...) of Shapiro's account is that the extensions of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis is that in some cases, a competent speaker of the language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. Shapiro calls this open texture, borrowing the term from Friedrich Waismann.The formal model theory has a similar structure to the supervaluationist approach, employing the notion of a sharpening of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The ultimate goal of the technical aspects of the work is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox.Later chapters deal with what passes for higher-order vagueness - vagueness in the notions of 'determinacy' and 'borderline' - and with vague singular terms, or objects. In each case, the philosophical picture is developed by extending and modifying the original account. This is followed with modifications to the model theory and the central meta-theorems.As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak. But vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. Vagueness is also due to the kinds of beings we are. There is no need to blame the phenomenon on any one of those aspects. (shrink)

Supervaluational theories of vagueness have achieved considerable popularity in the past decades, as seen in eg [5], [12]. This popularity is only natural; supervaluations let us retain much of the power and simplicity of classical logic, while avoiding the commitment to strict bivalence that strikes many as implausible. Like many nonclassical logics, the supervaluationist system SP has a natural dual, the subvaluationist system SB, explored in eg [6], [28].1 As is usual for such dual systems, the classical features of SP (...) (typically viewed as benefits) appear in SB in ‘mirror-image’ form, and the nonclassical features of SP (typically viewed as costs) also appear in SB in ‘mirror-image’ form. Given this circumstance, it can be difficult to decide which of two dual systems is better suited for an approach to vagueness.2 The present paper starts from a consideration of these two approaches— the supervaluational and the subvaluational—and argues that neither of them is well-positioned to give a sensible logic for vague language. §2 presents the systems SP and SB and argues against their usefulness. Even if we suppose that the general picture of vague language they are often taken to embody is accurate, we ought not arrive at systems like SP and SB. Instead, such a picture should lead us to truth-functional systems like strong Kleene logic (K3) or its dual LP. §3 presents these systems, and argues that supervaluationist and subvaluationist understandings of language are better captured there; in particular, that a dialetheic approach to vagueness based on the logic LP is a more sensible approach. §4 goes on to consider the phenomenon of higher-order vagueness within an LP-based approach, and §5 closes with a consideration of the sorites argument itself. (shrink)

Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professionalbasketball players. The main feature of (...) Shapiro's account is that the extensions of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis is that in some cases, a competent speaker ofthe language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. Shapiro calls this open texture, borrowing the term from Friedrich Waismann.The formal model theory has a similar structure to the supervaluationist approach, employing the notion of a sharpening of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The ultimate goal of the technical aspects of the work is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox.Later chapters deal with what passes for higher-order vagueness - vagueness in the notions of 'determinacy' and 'borderline' - and with vague singular terms, or objects. In each case, the philosophical picture is developed by extending and modifying the original account. This is followed with modifications to the model theory and the central meta-theorems.As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak. But vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. Vagueness is also due to the kinds of beings we are. There is no need to blame the phenomenon on any one of those aspects. (shrink)

Intuitively, vagueness involves some sort of indeterminacy: if Plato is a borderline case of baldness, then there is no fact of the matter about whether or not he’s bald—he’s neither bald nor not bald. The leading formal treatments of such indeterminacy—three valued logic, supervaluationism, etc.—either fail to validate the classical theorems, or require that various classically valid inference rules be restricted. Here we show how a fully classical, yet indeterminist account of vagueness can be given within natural semantics, an (...) alternative semantics for classical proof theory. The key features of the account are: there is a single notion of truth—definite truth—and a single notion of validity; sentences can be true, false, or undetermined; all classical theorems and all classical inference rule are valid; the sorites argument is unsound; ‘definitely’ is treated as a meta-language predicate; higher-order vagueness is handled via semantic ascent. (shrink)

This article analyzes approaches to nondeterminacy that suggest that one can make justified choices when primary criteria fail to fully determine a best alternative by introducing a secondary criterion. It is shown that these approaches risk violating Basic Contraction Consistency. Some ways of adjusting two-step models in order to protect against this are addressed, and it is suggested that proponents of two-step models should adopt formal conditions which qualify what counts as a permissible secondary criterion that resemble supervaluationist conditions that (...) qualify what counts as admissible precisifications of vague terms. (shrink)

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to special kinds (...) of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification. (shrink)

The Boolean many-valued approach to vagueness is similar to the infinite-valued approach embraced by fuzzy logic in the respect in which both approaches seek to solve the problems of vagueness by assigning to the relevant sentences many values between falsity and truth, but while the fuzzy-logic approach postulates linearly-ordered values between 0 and 1, the Boolean approach assigns to sentences values in a many-element complete Boolean algebra. On the modal-precisificational approach represented by Kit Fine, if a sentence is indeterminate in (...) truth value in some world, it is taken to be true in one precisified world accessible from that world and false in another. This paper points to a way to unify these two approaches to vagueness by showing that Fine’s version of the modal-precisificational approach can be combined with the Boolean many-valued approach instead of supervaluationism, one of the most popular approaches to vagueness. (shrink)

:The Small Improvement Argument is the leading argument for value incomparability. All vagueness-based accounts of the SIA have hitherto assumed the truth of supervaluationism, but supervaluationism has some well-known problems. This paper explores the implications of epistemicism, a leading rival theory. We argue that if epistemicism is true, then options are comparable in small improvement cases. Moreover, even if SIAs do not exploit vagueness, if epistemicism is true, then options cannot be on a par. The epistemicist account of (...) the SIA has an advantage over leading existing rival accounts of the SIA because it accounts for higher-order hard cases. (shrink)

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. (shrink)

The goal of this book is to defend a supervaluationist theory of vagueness. Keefe begins by laying out a series of desiderata for an adequate theory of vagueness generally: among other things, such a theory will need to solve the sorites paradox, provide a plausible analysis of borderline cases, preserve so-called penumbral connections among borderline predications, accommodate the phenomenon of higher-order vagueness, and comport with as many of our ordinary linguistic intuitions as possible. She then proceeds to evaluate in turn (...) each of the main competitors to a supervaluationist theory—including epistemic, many-valued, degree-theoretic, and pragmatic accounts—in light of these desiderata, and finds each seriously lacking. Having thus cleared the ground, she advances a version of the supervaluationist approach and replies to a number of potential objections. (shrink)

I advocate a two part view concerning vagueness. On one hand I claim that vagueness is logically incoherent; but on the other hand I claim that vagueness is also a benign, beneficial, and indeed essential feature of human language and thought. I will call this view transvaluationism, a name which seems to me appropriate for several reasons. First, the term suggests that we should move beyond the idea that the successive statements in a sorites sequence can be assigned differing truth (...) values in some logically coherent way that fully respects the nature of vagueness -a way that [1] fully eschews any arbitrarily precise semantic transitions. We should transcend this impossible goal by accepting that vagueness harbors logical incoherence. Second, just as Nietzsche held that one can overcome nihilism by embracing what he called the transvaluation of all values, my position affirms vagueness, rather than despairing in the face of the logical absurdity residing at its very core. This affirmation amounts to a transvaluation of truth values, as far as sorites sequences are concerned. Third, the term ’transvaluationism’ has a nice ring to it, especially since one of the principal philosophical approaches to vagueness is called supervaluationism. I will call the first claim of transvaluationism, that vagueness is logically incoherent, the incoherence thesis . I will call the second claim, that vagueness is benign, beneficial, and essential, the legitimacy thesis . The legitimacy thesis, taken by itself, seems overwhelmingly plausible; anyone who denies it assumes a heavy burden of proof. But prima facie, it seems dubious that the legitimacy thesis can be maintained in conjunction with the incoherence thesis. For, there is reason to doubt whether there is any cogent way to embrace the incoherence thesis without thereby becoming mired in what Williamson (1994) calls global nihilism about vagueness -the view that vague terms are empty (i.e., they do not, and cannot, apply to anything). Global nihilism, Williamson argues, has such destructively negative consequences that it does not deserve to be taken seriously -for instance, the consequence that vastly many of our common sense beliefs are false, and the consequence that these beliefs are not even useful (since the constituent terms in ’Common sense beliefs are useful’ are vague and hence this statement turns out, given the [2] incoherence thesis, to be false itself). In short, the idea that one can adopt the incoherence thesis and then somehow transcend nihilism might initially seem hopelessly optimistic; transvaluationism would then be an unattainable, chimerical, goal rather than an intelligible and conceptually stable position concerning vagueness. Given certain widely held philosophical views about how language and thought must map onto the world in order for statements and the beliefs they express to be true -views that fall appropriately under the label ’referential semantics’ -transvaluationism probably is a chimerical goal. (shrink)

Vague expressions are omnipresent in natural language. Their use in legal texts is inevitable. A law phrased in vague terms will often leave it indeterminate whether it applies to a particular case. This places the law at odds with legal values. One of the fundamental pillars of the rule of law is legal certainty. The determinacy of the law enables people to use it as a guide and allows judges make impartial decisions. Vagueness poses a threat to these ideals. In (...) borderline cases, the law seems to be indeterminate and thus incapable of serving its core rule of law value. -/- In the philosophy of language, vagueness has become one of the hottest topics of the past two decades. Linguists and philosophers have investigated what distinguishes ‘soritical’ vagueness from other kinds of linguistic indeterminacy, such as ambiguity, generality, open texture, and family resemblance concepts. There is a vast literature that discusses the logical, semantic, pragmatic, and epistemic aspects of these phenomena. Legal theory has hitherto paid little attention to the differences between the various kinds of linguistic indeterminacy that are grouped under the heading of ‘vagueness’, let alone to the various theories that try to account for these phenomena. -/- Bringing together leading scholars working on the topic of vagueness in philosophy and in law, this book fosters a dialogue between philosophers and legal scholars by examining how philosophers conceive legal ambiguity from their theoretical perspective and how legal theorists make use of philosophical theories of vagueness. -/- The chapters of the book are organized into three parts. The first part addresses the import of different theories of vagueness for the law, referring to a wide range of theories from supervaluationist to contextualist and semantic realist accounts in order to address the question of whether the law can learn from engaging with philosophical discussions of vagueness. The second part of the book examines different vagueness phenomena. The contributions suggest that paying greater attention to these phenomena can make lawyers aware of specific issues and solutions as yet overlooked. The third part deals with the pragmatic aspects of vagueness in law and with the professional, political, and moral issues to which such vagueness gives rise. (shrink)

Supervaluation is a method which has been invented to deal with the reference failure. In his 1975 paper K. Fine suggested that it might be applied to the analysis of the phenomenon of vagueness as well. The paper tries to assess the pros and cons of the supervaluation theory of vagueness. Supervaluation provides us with the means for analysing vagueness without eliminating it from the language, and allows to solve the main paradox connected with vagueness; i.e. the sorities paradox. The (...) preservation of classical logic was thought to be one of its main virtues. The solution to sorities which supervaluationism proposes is a very counterintuitive one, however. Moreover, it seems that it does not preserve classical logic after all. Besides, the theory of supervaluation is not able to handle the higher-order vagueness. Nevertheless it remains one of most attractive semantic theories of vagueness available. In conection with the objections raised against supervaluationism arises the problem concerning the interpretation of the meaning of the supervaluationism's key notion, namely the notion of supertruth. The paper offers one such interpretation. (shrink)

In this dissertation, I argue that vagueness is a metaphysical phenomenon---that properties and objects can be vague---and propose a trivalent theory of vagueness meant to account for the vagueness in the world. In the first half, I argue against the theories that preserve classical logic. These theories include epistemicism, contextualism, and semantic nihilism. My objections to these theories are independent of considerations of the possibility that vagueness is a metaphysical phenomenon. However, I also argue that these theories are not capable (...) of accommodating metaphysical vagueness. As I move into my positive theory, I first argue for the possibility of metaphysical vagueness and respond to objections that charge that the world cannot be vague. One of these objections is Gareth Evans' much-disputed argument that vague identities are impossible. I then describe what I call the logic of states of affairs. The logic of states of affairs has as its atomic elements states of affairs that can obtain, unobtain, or be indeterminate. Finally, I argue that the logic of states of affairs is a better choice for a theory of vagueness than other logics that could accommodate metaphysical vagueness such as supervaluationism and degree theories. Preference should be given to the logic of states of affairs because it provides a better explanation of higher-order vagueness and does a better job of matching our ordinary understandings of logical operators than supervaluationism does and because it provides a more general account of indeterminacy than the account given by degree theories. Advisor: Reina Hayaki. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra (...) set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)