Introduction

One could define vagueness as the existence of

borderline cases

and characterise the philosophical debate on vagueness as being about the nature of these. The prevalent theories of vagueness can be divided into three categories, paralleling three logical interpretations of borderline cases: (i) a borderline case is a case of a truth-value gap; it is neither true nor false; (ii) a borderline case is a case of a truth-value glut; it is both true and false; and (iii) a borderline case is a case where the truth-value is non-classical. The third of these is proposed in the fuzzy logic approach to vagueness. Three-valued approaches have only

$\frac{1}{2}$

as a value in addition to the standard values 1 and 0. These approaches can be interpreted either as allowing for gaps or gluts, depending on how the notion of satisfaction or truth is defined. If a sentence is taken to be true only if its value is 1, it allows for gaps, but if it is taken to be true already if its value is at least

$\frac{1}{2}$

it allows for gluts. The most popular theories advertising gluts and gaps, however, are supervaluationism and subvaluationism, both of which make use of the notion of precisifications, that is, ways of making things precise. Truth-value gaps in supervaluationism are due to the way truth simpliciter, or supertruth, is defined: A proposition is supertrue (superfalse) if it is true (false) at all precisifications. This means that a proposition can be neither true nor false in case there exist two precisifications, one of which make it true and one of which makes it false. Conversely, in subvaluation theory, the same scenario would lead to a truth-value glut. That is, the proposition would be both true and false. This is because subvaluationism defines truth simpliciter as being true at some precisifcation.