On Vagueness

Vagueness is a very hot problem in Anglo-American philosophy for the last forty years, and it is also a trans-disciplinary problem cross philosophy, logic, cognitive science and linguistics. It is firstly considered as a linguistic phenomenon, having borderline case as its essential characteristic. Thus ordinary language is a kind of vague language in terms of borderline case, because there is no sharp borderline between the cases which the words and expressions of ordinary language can apply to and the cases they cannot apply to. The most serious philosophical problem generated by vagueness is sorites paradox, which consists of a little by little argument, and its primitive instance can be traced back to the paradox of heap and the paradox of bald man constructed by ancient Greek logician Eubulides. According to Hegel’s dialectics, sorites paradox may be explained as ignoring the law of quantity-quality transformation, but the paradox might not be explained away only by dialectics if it is formulated with modern logic. As vagueness is the root of sorites paradox, it is often taken as a logical problem. Moreover, vagueness leads to problems of realism related to reference, problems of meaning related to linguistic use, problems of theory of truth related to "truth", and metaphysical problem relevant to identity.So far, there are several typical theoretical approaches to the study on vagueness: Represenationalism, Many-Valued Logics, Supervaluationalism, Epistemicism and Contextualism. Many-Valued Logics, Supervaluationalism and Epistemicism are the three standard theories of vagueness. Each theory provides a kind of solution to sorites paradox, and according to these solutions either sorites argument is invalid because it uses invalid inference rule, or sorites argument is unsound because it has false premise(s). Each theory of vagueness more or less involves linguistic, logical, epistemic and metaphysical component. This dissertation explains the development of these theories of vagueness from the perspective of philosophy of logic and philosophy of science, and finally gives its own viewpoints.The introduction provides the background of the problem of vagueness, the philosophical problems related to vagueness, and views of early analytic philosophers on vagueness. However, it only focuses on theories of vagueness rather than concerning about individual philosopher’s views on vagueness, if their views can not be contained into certain theory of vagueness.The second chapter discusses representationalism concerning vagueness. Representationalism results from Russell’s views on vagueness, by which representationalism has two types:Russell’s a priori representationalism and descriptive representationalism. According to Russell, vagueness is the feature of representation, specifically one-many relation between words and the world. Vagueness is a linguistic phenomenon, and there is no metaphysical vagueness which means that things or objects (and the real world) is vague. But vagueness invalidates the law of excluded middle in classical logic. Since vagueness is in-eliminable, we can only restrict classical logic to an ideal language. Descriptive representationalism agrees that vagueness is a linguistic phenomenon and there is no metaphysical vagueness. But it claims that vagueness in ordinary language is eliminable by reducing vague language to a precise language, or we can think of vague language as supervening on a precise language, and language of science is the paradigmatic precise language. By this way, classical logic is defended. The problem with Russell’s representationalism is that he constrains the power of classical logic too much, and classical logic may be of no use to inferences and arguments in ordinary language if we keep classical logic away from ordinary language. In addition, sorites paradox is not resolved. The difficulty with descriptive representationalism is that language of science is not really precise, and it is hard for us to find an exemplar of precise language. In this situation, elimination of vague language might undermine our descriptive power of the world.The third chapter discusses many-valued logics. Many-valued logics might be divided into (a) finite many-valued logics, the number of truth values of which is natural number; and (b) infinite many-valued logics, the number of truth values of which is real number between0and1. Three-valued logics are typical finite many-valued logics, and fuzzy logics are typical infinite many-valued logics. An anomaly to classical logic generated by vagueness is that we cannot assign truth values to vague sentence by principle of bivalence, while many-valued logics supply other truth values for us to assign to vague sentences. The arguments of sorites paradox mainly use classical rule of modus ponens and the law of transitivity of identity, which are invalid in many-valued semantics. However, the difficulties with many-valued logics are that it suffers from problem of higher-order vagueness; and if it is implausible for classical logic to divide truth-values into truth and falsity, it is also implausible for three-valued logics to divide truth-values into three categories and for fuzzy logics to divide truth-values into even more categories; what’s more, many-valued logics change the property of classical "truth"; and there are many intuitive contradiction in fuzzy semantics.The fourth chapter discusses supervaluationism. Supervaluationism claims that there are truth-value gaps in borderline sentences, which means that borderline sentences are neither true nor false. So classical principle of bivalence doesn’t hold, but classical law of excluded middle holds. In addition, according to supervaluationism, the truth of existential generalization doesn’t need to have a true instance, and this feature can help to resolve sorites paradox. Supervaluationism has a twin theory, subvaluationism, which claims that there are truth-value gluts in borderline sentences:that is to say, borderline sentences are both true and false. According to subvaluationism, classical principle of bivalence doesn’t hold, but the law of contradiction needs to be mitegated. Sorites paradox can be solved because modus ponens and the law of transitivity of identity are invalid in subvaluational semantics. But supervaluationism also suffers from problem of higher-order vagueness on one hand, and it changes the property of classical "truth" by substituting supertruth for truth on the other hand. The problems with subvaluationism are that it tolerates contradiction and it has more formalization work to do.Nihilism and Epistemicism are discussed in the fifth chapter. According to nihilism, vague terms have no reference and sorites arguments are accepted. Nihilism even claims that there are no ordinary things. Timothy Williamson’s epistemicism claims that vagueness is an epistemic phenomenon; in other words, vagueness is a kind of ignorance. According to epistemicism, there is a "cut-off point" or "borderline" between things vague terms can apply to (extension) and things they cannot apply to (anti-extension). The principle of bivalence holds and classical logic doesn’t need to be revised. Vague sentences in borderline case are still either true or false, but we don’t know which one it is. Our ignorance of the precise borderline of vague terms’extension and our ignorance of the truth-value of vague sentences in borderline case both can be explained away by our possession of inexact knowledge. Therefore, sorites argument is unsound. But the most serious difficulty with epistemicism is that postulating the unknown cut-off point violates our intuition, making epistemicism admit a kind miracle and mystery.The sixth chapter discusses contextualism concerning vagueness. As a result of discontent with supervaluationalism, contextualism intends to give a justification for epistemicism. Some contextualists support a three-valued semantics, and some maintain to accept classical logic. According to contextualism, vague terms have a characteristic of context-sensitivity, and this feature should be added to the semantics of vagueness. Different contextualists have different understanding to "context". Some contextualists treat context as a set of sentences, while more contextualists view context as a unity including external context like comparative classes and internal context like subject’s psychological states. The extension of vague terms will vary with context to context, and so borderline case can be contained into the extension or antiextension of vague terms. Since the property expressed by vague predicates can change with context (eg, internal context, even external contexts are fixed), it is impossible for us to know where the cut-off point between extensions of vague predicates and their antiextensions consists in. there is another twin contextualism which claims that vague terms express a kind of interest-relative property, and this property can change with context rather than properties vague terms express are changing. The most serious problem with contextualism is that vagueness still remains; even if external contexts and internal contexts are both fixed.In the last two chapters, this dissertation claims that vagueness is an ontological "phenomenon", that is to say. it supports metaphysical vagueness. The strong metaphysical vagueness means that there are vague objects, while weak metaphysical vagueness means that there are vague property, vague relations, and vague state of affairs. Since the real world consists of objects, properties, relations and state of affairs, the world as a whole is vague as well. In the vision of this non-classical metaphysics, vagueness is defined as indiscernibility by default, and based on the research achievements of cognitive science such as neuroscience, cognitive psychology and cognitive linguistics, various forms of sorites paradoxes can be dissolved. Sorites paradox reflects inadequate correspondence between the formal valid inferences/arguments in logical systems and the valid inferences/arguments of actual prototype outside the formal systems. Various logics of vagueness try to more adequately correspond to the valid inference and arguments with our ordinary vague language. But from our points of view, sorites paradoxes reveal that we haven’t sufficiently realize the rational power lying behind the formal/symbolic language of modern logic. Ordinary vague language is like our eyes, while the formal language of modern logic is analogous to microscope and telescope. Microscope and telescope are not helpful to improve our insight of ordinary language, but they can make us see more deeply and further. We should not complain and condemn the formal language of modem logic because what microscope and telescope can see is not quite adequately corresponding to what our naked eyes can see.