The premises that a four foot man is short and that a man one tenth of an inch taller than a short man is also short entail by universal instantiation and "modus ponens" that a seven foot man is short. The negation of the second premise seems to entail there are virtually no borderline cases of short men, While to deny the second premise and its negation conflicts with the principle of bivalence, If not excluded middle. But the paradox can (...) be dissolved without resort to degrees of truth or any non-Classical system of logic. If some true predications can be semantically uncertain in a sense suitable for defining borderline cases, The second premise can be denied without denying the vagueness of "short" or reintroducing a sorites paradox along with higher order borderline cases. (shrink)

It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draughtsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the “point-planet” of astronomy, the “perfect gas” of thermodynamics, or the “pure species” of genetics are equally remote from exact (...) realization. Indeed the unintelligibility at the atomic or sub-atomic level of the notion of a rigidly demarcated boundary shows that such objects not merely are not but could not be encountered. While the mathematician constructs a theory in terms of “perfect” objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true. As Duhem remarks, mathematical deduction is not useful to the physicist if interpreted rigorously. It is necessary to know that its validity is unaltered when the premise and conclusion are only “approximately true.” But the indeterminacy thus introduced, it is necessary to add in criticism, will invalidate the deduction unless the permissible limits of variation are specified. To do so, however, replaces the original mathematical deduction by a more complicated mathematical theory in respect of whose interpretation the same problem arises, and whose exact nature is in any case unknown. (shrink)

A Man whose height is four feet is short; adding one tenthof an inch to a short man's height leaves him short; therefore, a man whose height is four feet and one tenth of an inch is short. Now begin again and argue in the same pattern. A man whose height is four feet and one tenth of an inch is short; adding one tenth of an inch to a short man's height leaves him short; therefore, a man whose height (...) is four feet and two tenths of an inch is short. In this way, it seems, we can reach the absurd result that a man whose height is four feet plus any number of tenths of an inch is short. For, if the first argument is sound, so is the second; and if the second, so is the third; and so on. There appears to be no good reason for stopping at any one point rather than at another; it is hard to see why the chain of arguments should ever be broken. But the conclusion is ridiculous; it is, for example, preposterous to say that a man whose height is seven feet is, nevertheless, short. (shrink)

The title of this monograph needs explanation. It certainly sounds too promising. A more adequate, though more cumbersome one, would read: the logical syntax and semantics of the language of empirical theories. The treatment of this subject in the present monograph needs further qualifications. It focusses on what is characteristic of empirical theories as opposed to others, viz. mathematical ones. Now the difference between these two kinds of theories lies evidently, not in their syntax, but semantics. This is why our (...) main concern here is going to be with the problem of interpretation of empirical theories. The fundamental problem here concerns the distinction between the empirical and the a priori elements inherent in any such theory. Accordingly, our final task must include an explication of concepts such as: meaning postulate, analytic, synthetic, empirically meaningful sentence, and others related to them. (shrink)

Originally published in 1966. This volume analyzes the general structure of scientific theories, their relation to experience and to non-scientific thought. Part One is concerned with the logic underlying empirical discourse before its subjection to the various constraints, imposed by the logico-mathematical framework of scientific theories upon their content. Part Two is devoted to an examination of this framework and, in particular, to showing that the deductive organization of a field of experience is by that very act a modification of (...) empirical discourse and an idealization of its subject matter. Part Three analyzes the concordance between theories and experience and the relevance of science to moral and religious beliefs. (shrink)

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...) fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (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)

Many philosophic arguments concerned with infinite series depend on the mutual inconsistency of statements of the following five forms: (1) something exists which has R to something; (2) R is asymmetric; (3) R is transitive; (4) for any x which has R to something, there is something which has R to x; (5) only finitely many things are related by R. Such arguments are suspect if the two-place relation R in question involves any conceptual vagueness or inexactness. Traditional sorites arguments (...) show that a statement of form (4) can fail to be true even though it has no clear counter-example. Conceptual vagueness allows a finite series not to have any definite first member. I consider the speculative possibilities that there have been only finitely many non-overlapping hours although there has been no first hour and that space and time are only finitely divisible even though there are no smallest spatial or temporal intervals. (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)

The model theoretic concept of truth has thus far been applied mainly to mathematical languages and theories. The paper presents an attempt to apply it to languages of empirical theories. Such an application must do justice to some characteristic features of empirical discourse. The paper outlines the main problems which a model theoretic theory of truth for empirical languages is bound to face and suggests some solutions to those problems.

The model theoretic concept of truth has thus far been applied mainly to mathematical languages and theories. The paper presents an attempt to apply it to languages of empirical theories. Such an application must do justice to some characteristic features of empirical discourse. The paper outlines the main problems which a model theoretic theory of truth for empirical languages is bound to face and suggests some solutions to those problems.

CONTENTS: 1 Introductory Remark; 2 Formalism of Empirical Theories; 3 Semantics of Formalized Languages; 4 Interpretation of Empirical Theories; 5 Interpretation of Observational Terms; 6 Interpretation of Theoretical Terms; 7 Main Types of Meaning Postulates for Theoretical Terms; 8 Some Other Kinds of Meaning Postulates for Theoretical Terms; 9 Main Types of Statements in an Empirical Theory; 10 Towards a More Realistic Account; 11 Concluding Remarks; 12 Bibliographical Note.

Fuzzy logics are systems of logic with infinitely many truth values. Such logics have been claimed to have an extremely wide range of applications in linguistics, computer technology, psychology, etc. In this note, we canvass the known results concerning infinitely many valued logics; make some suggestions for alterations of the known systems in order to accommodate what modern devotees of fuzzy logic claim to desire; and we prove some theorems to the effect that there can be no fuzzy logic which (...) will do what its advocates want. Finally, we suggest ways to accommodate these desires in finitely many valued logics. (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...) audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

Putative resolutions of the sorites paradox in which the major premise is declared false or illegitimate, Including max black's treatment in terms of the alleged illegitimacy of vague attributions to borderline cases, Are rejected on semantical grounds. The resort to a non-Bivalent logic of representational "accuracy" with a continuum of accuracy values is shown to resolve the paradox, And the identification of accuracy values as truth values is defended as compatible with the central insight of the correspondence theory of truth (...) and with the practical legitimacy of most applications of ordinary, Bivalent logic to statements involving vague predicates. (shrink)