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ness have been advanced in attempts to accommodate vagueness within systematic semantic descriptions of natural languages. Most modify bivalence. Many replace the dichotomy between truth and falsity by a continuum of degrees of truth.
Degrees of truth are often associated with many- valued or fuzzy logics. Such logics treat the degree of truth of a complex statement as determined by the degrees of truth of its components; this generalizes the bivalent notion of truth-functionality. 'A and B' is usually assigned the lower of the degrees of truth of 'A' and 'B,' 'A or B' the higher. The associated diag- nosis of sorites paradoxes is that their major premises are less than perfectly true; the degree of truth of n grains made a heap gradually decreases as n decreases. However, generalized truth-functionality has implaus- ible consequences. If Tweedledum and Tweedledee are exactly the same height, it seems completely false to say that Tweedledum is tall and Tweedledee is not tall; but if it is half true that Tweedledum is tall, generalized truth-functionality makes it half true to say that Twee- dledum is tall and Tweedledee is not tall. Another prob- lem is higher-order vagueness; the assignment of degrees of truth to vague statements seems to be as arbitrary as the assignment of bivalent truth-values.
Rejection of bivalence does not require acceptance of generalized truth-functionality. An alternative is the theory of supervaluations. On this view, a vague language permits a range of bivalent interpretations, called 'sharpenings.' A statement is true if all sharp- enings make it true, false if all sharpenings make it false, and neither true nor false otherwise (this assign- ment is the supervaluation; degrees of truth may also be accommodated, if a statement's degree of truth is measured by the proportion of sharpenings that make it true). The failure of truth-functionality is a conse- quence of this view. As for sorites paradoxes, their major premises are false, for every sharpening has a cut-off point somewhere. Thus For some n, n grains made a heap and n —1 did not is true. However, for no n is n grains made a heap and n —1 did not true, for different sharpenings have cut-off points in different places. This result is sometimes found puzzling. Ano- ther problem is that supervaluations do not satisfy a
plausible constraint derived from Tarski's work on truth, that 'n grains made a heap' should be true if and only if n grains made a heap. This might be an acceptable price for a precise concept of truth, but the occurrence of higher-order vagueness means that supervaluationist truth is not precise.
A quite different approach is the epistemic theory of vagueness. On this view, vague terms have cut- off points whose location is unknown. Bivalence is preserved, and vagueness is a kind of ignorance. For somen,ngrainsmadeaheapandn—1didnot,but speakers cannot tell which number n is. The semantic boundaries of 'heap' are not settled by nature or explicit convention; to trace them one would need exact knowledge of both the use of 'heap' and the general principles by which use determines semantic boundaries. Even if one had such knowledge of use, one would still lack it of the general principles, accord- ing to the epistemic theorist. Controversial issues about speakers' semantic knowledge are raised by the claim that they understand vague terms without being able to trace the semantic boundaries which those terms have.
See also: Deviant Logics; Formal Semantics. Bibliography
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