Vagueness as Closeness
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This paper presents and defends a definition of vagueness, compares it
favourably with alternative definitions, and draws out some consequences of
accepting this definition for the project of offering a substantive theory of
vagueness.  The definition is roughly this: a predicate 'F' is vague just in
case for any objects a and b, if a and b are very close in respects relevant to
the possession of F, then 'Fa' and 'Fb' are very close in respect of truth.  
The definition is extended to cover vagueness of many-place predicates, of
properties and relations, and of objects.  Some of the most important
advantages of the definition are that it captures the intuitions which motivate
the thought that vague predicates are tolerant, without leading to
contradiction, and that it yields a clear understanding of the relationships
between higher-order vagueness, sorites susceptibility, blurred boundaries and
borderline cases.  The most notable consequence of the definition is that the
correct theory of vagueness must countenance degrees of truth.