Philosophers, linguists and others interested in problems concerning natural language frequently employ tools from logic and model theory. The question arises as to the proper interpretation of the formal methods employed—of the relationship between, on the one hand, the formal languages and their set-theoretic models and, on the other hand, the objects of ultimate interest: natural language and the meanings and truth conditions of its constituent words, phrases and sentences. Two familiar answers to this question are descriptivism and instrumentalism. More (...) recently, a third answer has been proposed: the logic as modelling view. This paper seeks to clarify and assess this view of logic. The conclusion is that we can successfully adopt the modelling perspective on a given piece of logical machinery only if we have to hand some other machinery to which we take the descriptive attitude. Thus, logic as modelling is not a full-fledged alternative to the descriptive view— for it cannot stand alone: it can at best be an addition to the descriptive perspective. The paper first presents the argument in a general, abstract form, before working through a detailed case study. The case examined is the one with respect to which the logic as modelling view has been developed in the greatest detail in the literature: the case of fuzzy model theory as an account of vagueness in natural language. (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)

The technical results presented here on continuity and approximate implication are obviously incomplete. In particular, a syntactic characterization of approximate implication is highly desirable. Nevertheless, I believe the results above do show that the theory has considerable promise for application to the areas mentioned at the top of the paper.Formulation and defense of realist interpretations of science, for example, require approximate truth because we hardly ever have evidence that a particular scientific theory corresponds perfectly with a portion of the real (...) world. Realists need to assert, then, that evidence for a theory is evidence for its approximate truth, not its truth (see [3] and [18]). Approximate truth is, however, a vague notion, and specification of quantity terms and of a sense of approximation are needed to make precise applications of it. Suitability of both vocabulary and sense of approximation depend on the subject matter, and their selection is a partly empirical matter that raises complex issues. In light of the number of common inferences which are not continuous, realists also need to be concerned about indiscriminate use of deductive logic to derive consequences from approximately true theories. These issues will be considered further in a future paper.Approximate truth also has potential application in areas of artificial intelligence that require inference from inaccurate data. In the ‘qualitative’ physical theories of de Kleer and Brown [6], for example, ‘qualitative’ values are derived by partitioning the real numbers into regions. Inferences leading from inside to outside a region must be identified and avoided, and approximate implication and continuity may prove useful in doing this. More generally, growing use of predicate logic as a programming language invites application of the theory of approximate truth as a symbolic substitute for numerical evaluation of computation errors. This too will be the subject of a future paper. (shrink)

The postulation by the “epistemic” theory of vagueness of a cut-off point between heaps and non-heaps has made it seem incredible. Surely, the critics argue, a vague predicate doesn’t divide the universe into a set and its complement. I argue in response that an objection of a similar kind can be leveled against most theories of vagueness. The only two which avoid it are untenable. The objection is less compelling than it initially seems. However, even when this obstacle is removed, (...) the epistemic theory is not yet vindicated. (shrink)

If there are vague numbers, it would be easier to use numbers as semantic values in a treatment of vagueness while avoiding precise cut-off points. When we assign a particular statement a range of values (less than 1 and greater than 0) there is no precise sharp cut-off point that locates the greatest lower bound or the least upper bound of the interval, I should like to say. Is this possible? “Vague Numbers” stands for awareness of the problem. I do (...) not present a serious theory of vague numbers. I sketch some reasons for using a many-value semantics. These reasons refer to my earlier treatments of determinacy and definitions of higher-order borderline cases. I also sketch how definitions of independence use the determinacy operator. The distinction between actually assigned values and values whose assignments are acceptable helps avoid unwanted precise cut-off points. (shrink)

Unger claims that we can block sorites arguments for the conclusion that there are no ordinary things only by invoking some kind of miracle, but no such miracle is needed if we reject the principle that every statement has a truth value. Wheeler's argument for the nonexistence of ordinary things depends on the assumptions that if ordinary things exist, they comprise real kinds, and that if ordinary predicates really apply to things, the predicates refer to real properties. If we accept (...) Wheeler's criteria for the reality of kinds and properties, we have no good reason to accept these assumptions. (shrink)

This book proposes a new solution to the problem of vagueness. There are several different ways of addressing this problem and no clear agreement on which one is correct. The author proposes that it should be understood as the problem of explaining vague predicates in a way that systematizes six intuitions about the phenomenon and satisfies three criteria of adequacy for an ideal theory of vagueness. The third criterion, which is called the “criterion of precisification”, is the most controversial one. (...) It is based on the intuition that a predicate is vague only if it is imprecise. The author considers some different definitions of linguistic imprecision, proposing that a predicate is imprecise if and only if there is no sharp boundary between objects to which its application yields some particular truth-value and objects to which its application does not yield that truth-value. The volume critically reviews the current theories of vagueness and proposes a new one, the Theory of Vagueness as Arbitrariness, which defines a vague predicate as an arbitrary predicate that must be precisified in order to contribute to a sentence that has truth-conditions. The main advantages of this theory over the current alternatives are that it satisfies all three criteria and systematizes the relevant intuitions. (shrink)

A semantics of vagueness should reject the principle that every statement has a truth-value yet retain the classical tautologies. A many-value, non-truth-functional semantics and a semantics of super-valuations each have this result. According to the super-valuation approach, 'if a man with n hairs on his head is bald, then a man with n plus one hairs on his head is also bald' is false because it comes out false no matter how the vague predicate 'is bald' is appropriately made precise. (...) But why should a sentence in which components actually remain imprecise be regarded as actually false just because it would be false if its components were precise? On one of the alternative treatments of quantification allowed by the many-value approach, the sentence in question is assigned an intermediate value closer to 'false' than to 'true'. Despite the elegance of the super-valuation approach, there are reasons to prefer the many-value approach. (shrink)