To accommodate vague statements and predicates, I propose an infinite-valued, non-truth-functional interpretation of logic on which the tautologies are exactly the tautologies of classical two-valued logic. iI introduce a determinacy operator, analogous to the necessity operator in alethic modal logic, to allow the definition of first-order and higher-order borderline cases. On the interpretation proposed for determinacy, every statement corresponding to a theorem of modal system T is a logical truth, and I conjecture that every logical truth on the interpretation corresponds (...) to a theorem of T. the interpretation is extended to predicate logic. A borderline case of a predicate 'F’ is neither determinately F nor determinately not-F. Traditional sorites arguments are seen to fall apart early in their gradual stepwise passage from truth to falsity. (shrink)

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (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.

This brief paperback is designed for symbolic/formal logic courses. It features the tree method proof system developed by Jeffrey. The new edition contains many more examples and exercises and is reorganized for greater accessibility.

The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in truth conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.

One of the few points of agreement to be found in mainstream responses to the logical and semantic problems generated by vagueness is the view that if any modification of classical logic and semantics is required at all then it will only be such as to admit underdetermined reference and truth-value gaps. Logics of vagueness including many valued logics, fuzzy logics, and supervaluation logics all provide responses in accord with this view. The thought that an adequate response might require the (...) recognition of cases of overdetermination and truth value gluts has few supporters. This imbalance lacks justification. As it happens, Jaskowski's paraconsistent discussive logic-a logic which admits truth value gluts-can be defended by reflecting on similarities between it and the popular supervaluationist analysis of vagueness already in the philosophical literature. A simple dualisation of supervaluation semantics results in a paraconsistent logic of vagueness based on what has been termed subvaluational semantics. (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)