A sorites argument is a symptom of the vagueness of the predicate with which it is constructed. A vague predicate admits of at least one dimension of variation (and typically more than one) in its intended range along which we are at a loss when to say the predicate ceases to apply, though we start out confident that it does. It is this feature of them that the sorites arguments exploit. Exactly how is part of the subject of this paper. (...) The majority of philosophers writing on vagueness take it to be a kind of semantic phenomenon. If we are right, they are correct in this assumption, which is surely the default position, but they have not so far provided a satisfactory account of the implications of this or a satisfactory diagnosis of the sorites arguments. Other philosophers have urged more exotic responses, which range from the view that the fault lies not in our language, but in the world, which they propose to be populated with vague objects which our semantics precisely reflects, to the view that the world and language are both perfectly in order, but that the fault lies with our knowledge of the properties of the words we use (epistemicism). In contrast to the exotica to which some philosophers have found themselves driven in an attempt to respond to the sorites puzzles, we undertake a defense of the commonsense view that vague terms are semantically vague. Our strategy is to take fresh look at the phenomenon of vagueness. Rather than attempting to adjudicate between different extant theories, we begin with certain pre-theoretic intuitions about vague terms, and a default position on classical logic. The aim is to see whether (i) a natural story can be told which will explain the vagueness phenomenon and the puzzling nature of soritical arguments, and, in the course of this, to see whether (ii) there arises any compelling pressure to give up the natural stance. We conclude that there is a simple and natural story to be told, and we tell it, and that there is no good reason to abandon our intuitively compelling starting point. The importance of the strategy lies in its dialectical structure. Not all positions on vagueness are on a par. Some are so incredible that even their defenders think of them as positions of last resort, positions to which we must be driven by the power of philosophical argument. We aim to show that there is no pressure to adopt these incredible positions, obviating the need to respond to them directly. If we are right, semantic vagueness is neither surprising, nor threatening. It provides no reason to suppose that the logic of natural languages is not classical or to give up any independently plausible principle of bivalence. Properly understood, it provides us with a satisfying diagnosis of the sorites argumentation. It would be rash to claim to have any completely novel view about a topic so well worked as vagueness. But we believe that the subject, though ancient, still retains its power to inform and challenge us. In particular, we will argue that taking seriously the central phenomenon of predicate vagueness—the “boundarylessness” of vague predicates—on the commonsense assumption that vagueness is semantic, leads ineluctably to the view that no sentences containing vague expressions (henceforth ‘vague sentences’) are truth-evaluable. This runs counter to much of the literature on vagueness, which commonly assumes that, though some applications of vague predicates to objects fail to be truth-evaluable, in clear positive and negative cases vague sentences are unproblematically true or false. It is clarity on this, and related points, that removes the puzzles associated with vagueness, and helps us to a satisfying diagnosis of why the sorites arguments both seem compelling and yet so obviously a bit of trickery. We give a proof that semantically vague predicates neither apply nor fail-to-apply to anything, and that consequently it is a mistake to diagnose sorites arguments, as is commonly done, by attempting to locate in them a false premise. Sorites arguments are not sound, but not unsound either. We offer an explanation of their appeal, and defend our position against a variety of worries that might arise about it. The plan of the paper is as follows. We first introduce an important distinction in terms of which we characterize what has gone wrong with vague predicates. We characterize what we believe to be our natural starting point in thinking about the phenomenon of vagueness, from which only a powerful argument should move us, and then trace out the consequences of accepting this starting point. We consider the charge that among the consequences of semantic vagueness are that we must give up classical logic and the principle of bivalence, which has figured prominently in arguments for epistemicism. We argue there are no such consequences of our view: neither the view that the logic of natural languages is classical, nor any plausible principle of bivalence, need be given up. Next, we offer a diagnosis of what has gone wrong in sorites arguments on the basis of our account. We then present an argument to show that our account must be accepted on pain of embracing (in one way or another) the epistemic view of “vagueness”, i.e., of denying that there are any semantically vague terms at all. Next, we discuss some worries that may arise about the intelligibility of our linguistic practices if our account is correct. We argue none of these worries should force us from our intuitive starting point. Finally, we cast a quick glance at other forms of semantic incompleteness. (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)

I argue for an account of vagueness according to which the root of vagueness lies not in the type of semantic-value that is best associated with an expression, but in the type of linguistic practice that renders the expression meaningful. I suggest, in particular, that conventions about how to use sentences involving attributions of vague predicates to borderline cases prevail to a lesser degree than conventions about how to use sentences involving attributions of vague predicates to clear cases.

THERE IS OFTEN UNCERTAINTY ABOUT WHETHER SOME PREDICATE APPLIES TO SOME PHYSICAL OBJECT OR STATE. THIS UNCERTAINTY MAY HAVE ANY OF THREE SOURCES - VAGUENESS OF A TERM, INEXACTNESS OF A CONCEPT, OR PRACTICAL DIFFICULTY IN DETERMINING ITS APPLICABILITY. VARIOUS WAYS IN WHICH CONCEPTUAL INEXACTNESS OR PRACTICAL DIFFICULTY MAY PRODUCE UNCERTAINTY ARE DISTINGUISHED. NEITHER TERMINOLOGICAL VAGUENESS, NOR PRACTICAL DIFFICULTY IN DETERMINING THE APPLICABILITY OF A CONCEPT ARE NECESSARY FEATURES OF EVERY LANGUAGE IN EVERY PHYSICAL WORLD, BUT CONCEPTUAL INEXACTNESS IS A (...) NECESSARY FEATURE OF EMPIRICAL CONCEPTS. THESE RESULTS ARE UTILIZED TO DETERMINE THE SOURCES AND NECESSITY OF IMPRECISION IN MEASUREMENT. (shrink)

It might be thought that vagueness precludes the possibility of classical conceptual analysis and, thus, that the classical or definitional view of the nature of complex concepts is incorrect. The present paper argues that classical analysis can be had for concepts expressed by vague language since (1) all of the general theories of vagueness are compatible with the thesis that all complex concepts have classical analyses and also that (2) the meaning of vague expressions can be analyzed by having the (...) degree of vagueness of a given analysandum be mapped onto the vagueness of an analysans. (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.

This book, provides a critical approach to all major logical paradoxes: from ancient to contemporary ones. There are four key aims of the book: 1. Providing systematic and historical survey of different approaches – solutions of the most prominent paradoxes discussed in the logical and philosophical literature. 2. Introducing original solutions of major paradoxes like: Liar paradox, Protagoras paradox, an unexpected examination paradox, stone paradox, crocodile, Newcomb paradox. 3. Explaining the far-reaching significance of paradoxes of vagueness and change for philosophy (...) and ontology. 4. Proposing a novel, well justified and, as it seems, natural classification of paradoxes. (shrink)

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence. Charles Sanders Peirce was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics (...) within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic. (shrink)

Cet article aborde la question générale du développement d'une histoire sociale des mathématiques et de la logique, à partir d'un cas historique. Il vise à rendre compte de certains traits de l'histoire récente de la logique floue. À cette fin, il met en oeuvre une sociologie des pratiques de démonstration et une approche fondée sur l'analyse matérielle du travail logique, notamment de l'activité d'écriture et de lecture. Il esquisse la construction d'une histoire sociale des formes de démonstration.

Wright holds that the following two theses are jointly incoherent: Rules determine correct language use. These rules are discoverable via internal reflection on language use. I argue that incoherence is derivable from alone and examine two types of probabilistic accounts that model a modification of, one in terms of inexact knowledge, the other in terms of viewing semantic rules as reasons for linguistic actions. Both accommodate tolerance by breaking the link between justified assertion and truth, but incoherence threatens their conception (...) of justified assertion. I argue that the rules-as-reasons approach can relocate sharp boundaries to a place where they are not only more tolerable, but to be expected. (shrink)

In 1907 Borel published a remarkable essay on the paradox of the Heap (“Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques”), in which Borel proposes what is likely the first statistical account of vagueness ever written, and where he discusses the practical implications of the sorites paradox, including in economics. Borel’s paper was integrated in his book Le Hasard, published 1914, but has gone mostly unnoticed since its publication. One of the originalities of Borel’s essay (...) is that it puts forward a model of vagueness as imprecision, making particular use of the Gaussian law of measurement errors to model categorization. The aim of our paper is to give a presentation of the historical context of Borel’s essay, to spell out the mathematical details of his model, and to provide a critical assessment of his theory. Three aspects of Borel’s account are particularly discussed: the first concerns the comparison between Borel’s statistical account and posterior degree-theoretic accounts of vagueness. The second concerns the anti-epistemicist flavor of Borel’s approach, whereby the idea of statistical fluctuation is used to undermine the notion of sharp boundary for vague predicates. The third concerns the problematic link between Borel’s model of vagueness as imprecision and the notion of semantic indeterminacy. An English translation of Borel’s original essay is appended to this paper (Erkenntnis, this issue). (shrink)

Humans and organizations have limitations of computational capacity and information management. Such constraints are synonymous with bounded rationality. Therefore, in order to extend the human and organizational boundaries to more advanced models of cognition, this research proposes concepts of cognitive machines in organizations. From a micro point of view, what makes this research distinct is that, beyond people, it includes in the list of participants of the organization the cognitive machines. From a macro point of view, this paper relies on (...) the premise that cognitive machines can improve the cognitive abilities of the organization. From such perspectives, it presents rationale and principles of a class of cognitive machines with capabilities to carry out complex cognitive tasks in organizations. It also introduces analyses of the cognitive machines in organizations through theories of bounded rationality, economic decision-making, and conflict resolution. The analyses indicate that these machines can solve or reduce intra-individual and group dysfunctional conflicts which arise from decision-making processes in the organization, and thus they can improve the degree of organizational cognition. From all these backgrounds, this research outlines implications of cognitive machines for organizations. (shrink)

There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems (...) to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the sorites paradox. (shrink)

In a previous paper the concept of disease was fuzzy-logically analyzed and a sketch was given of a prototype resemblance theory of disease (Sadegh-Zadeh (2000). J. Med. Philos., 25:605–38). This theory is outlined in the present paper. It demonstrates what it means to say that the concept of disease is a nonclassical one and, therefore, not amenable to traditional methods of inquiry. The theory undertakes a reconstruction of disease as a category that in contradistinction to traditional views is not based (...) on a set of common features of its members, that is individual diseases, but on a few best examples of the category, called its prototypes, and a similarity relationship such that a human condition is considered a disease if it resembles a prototype. It enables new approaches to resolving many of the stubborn problems associated with the concept of disease. (shrink)

This paper addresses theoretical problems arising from the vagueness of language terms, and intuitions of the vagueness of the concepts to which they refer. It is argued that the central intuitions of prototype theory are sufficient to account for both typicality phenomena and psychological intuitions about degrees of membership in vaguely defined classes. The first section explains the importance of the relation between degrees of membership and typicality (or goodness of example) in conceptual categorization. The second and third section address (...) arguments advanced by Osherson and Smith (1997), and Kamp and Partee (1995), that the two notions of degree of membership and typicality must relate to fundamentally different aspects of conceptual representations. A version of prototype theory—the Threshold Model—is proposed to counter these arguments and three possible solutions to the problems of logical selfcontradiction and tautology for vague categorizations are outlined. In the final section graded membership is related to the social construction of conceptual boundaries maintained through language use. (shrink)

Vagueness is an extremely common feature of natural language, but does it actually play a positive, efficiency enhancing, role in communication? Adopting a probabilistic interpretation of vague terms, we propose that vagueness might act as a source of randomness when deciding what to assert. In this context we investigate the efficacy of multiple sender channels in which senders choose assertions stochastically according to vague definitions of the relevant words, and a receiver then aggregates the different signals. These vague channels are (...) then compared with Boolean channels in which assertions are selected deterministically based on classical definitions. We show that given a sufficient number of senders, a linear stochastic channel outperforms Boolean channels when performance is measured by the expected squared error between the actual value described by the senders and the receiver’s estimate of it based on the signals they receive. The number of senders required for vague channels to be at least as accurate as Boolean channels is shown to be a decreasing function of the size of the language i.e. the number of description labels available to the senders. Vague channels are then shown to be robust to transmission error provided the error rate is not too large. In addition, we investigate the behaviour of both Boolean and vague channels for a parametrised family of distributions on the input values. Finally, we consider optimal vague channels assuming a fixed number of senders and show that, provided there are more than two senders, a vague channel can be found that outperforms the optimal Boolean channel. In this context, we show that for channels with relatively low numbers of senders S-curve production functions are optimal. (shrink)

My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that (...) Dejnožka and others have proposed as potentially influential in Russell’s intellectual reactions to nonclassical logic or to the philosophical concepts that might contribute to his reactions to nonclassical logics. (shrink)

This essay addresses Jacques Derrida’s theory of metaphor, as it has been handed to literary theory and continental philosophy. Our aim is to reassess the relationship between metaphor and metaphysics, using two distinct critical lenses. We will contrast Derrida’s influential position to an anachronistic author – Giambattista Vico (1668–1744). Vico initiated what is now (retrospectively) called the romantic theory of metaphor, but the details of his theory are missing from current discussions. For this reason, Vico’s view is given closer attention. (...) Two new concepts are introduced: mute metaphor and heroic metaphor. These terms help specify precisely how metaphor can be conceived as a process of consciousness rather than a grammatical rule or grammatical exception. Finally, it is suggested that Vico’s theory is preferable to Derrida’s. Vico’s vision of metaphorical language is more fruitful and less problematic than Derrida’s, both methodologically and terminologically. The romanticism espoused by Vico also suggests itself to new modes of contemporary research concerning the purpose and function of metaphor within language. (shrink)

IntroductionScientific pluralism is generally understood in the backdrop of scientific monism. So is mathematical pluralism. Though there are many culture-dependent mathematical practices, mathematical concepts and theories are generally taken to be culture invariant. We would like to explore in this paper whether mathematical pluralism is admissible or not.Materials and methodsMathematical pluralism may be approached at least from five different perspectives. 1. Foundational: The view would claim that different issues within mathematics need support of different foundations, apparently incompatible with one another. (...) 2. Ontological: The world itself is dappled—the mathematical counterpart of which can be traced to the admission of non-Euclidean spaces and also in simultaneous acceptance of set-theoretic and category theoretic entities in the ontology of mathematics. 3. The third route is epistemological. It follows from the view that the nature of reality is so complex that different aspects of it require alternative modes of representation and explanation sometimes severally, sometimes simultaneously, e.g., classical, constructivist, computer-aided and different finitist mathematics can be used depending on the knowledge situation. 4. The fourth route is semantic. Fuzzy mathematics, we know, gives up bivalence in theory of truth, and ascribes truth to mathematical propositions in degrees. 5. The last one is Programmatic: for some, mathematical pluralism is a stance, a programme. It provides a framework which strikes at the root of cultural hegemony and nourishes a climate of intellectual humility and tolerance. It is not my intention to hold the brief of any of these versions. I just want to present to the readers a more or less complete picture of the scenario, though not an exhaustive one. ConclusionUnlike monism, mathematical pluralism does not rule out any possibility—even the possibility of having a unitary over-arching framework of interpretation remains as an open option. This paper highlights the fact that motivations for upholding pluralism in mathematics have been many and one can be a pluralist just by subscribing to any of the views listed above. (shrink)

A speaker often decides whether or not to saysomething based on his assessment of the impact itwould have on his hearer's beliefs. If he thinks itwould bring them more in line with the truth, he saysit; otherwise he does not. In this paper, I developa model of these judgments, focusing specifically onthose of vague sentences. Under the simplifyingassumption that an utterance only conveys a speaker'sapplicability judgments, I present a Bayesian model ofan utterance's impact on a hearer's beliefs. Fromthis model I (...) derive a model of a speaker's judgment ofwhether or not an utterance would be informative. Iillustrate it with several examples of judgments ofvague and non-vague sentences. For instance, I showthat it models the common judgment that assertingeither ``George is tall'' or ``George is not tall'' wouldbe misleading if George were borderline tall, butasserting ``George is tall and he isn't tall'' would notbe. (shrink)

This multifaceted essay emerges from a host of sources within diverse academic settings. Its central thesis is guided by physicist John A. Wheeler's thoughts on the quantum enigma. Wheeler concludes, following Niels Bohr, that we are co-participants within the universal self-organizing process. This notion merges with concepts from Peirce's process philosophy, Eastern thought, issues of topology, and border theory in cultural studies and social science, while surrounding itself with such key terms as complementarity, interdependence, interrelatedness, vagueness, generality, incompleteness, inconsistency, and (...) mestizaje. Ultimately, a sense of semiosic process pervades in light of combined homogenous and hetergenous tendencies. (shrink)