The world of classical logic is a world of the excluded middle. There is no room
for ‘quite true’ and ‘mainly false’. The predicates that express identity are
discrete and clearly defined. The range of possible kinds of objects for which a
predicate holds—its extension in the sense of Frege’s theory of meaning—is
determined unequivocally.
In the real world, there are questions. Is this a creek or a river? Is Humpty
Dumpty clever? How far away from the summit can you be and still truthfully
say you are on Mount Fuji? Because it cannot handle vagueness, classical logic
has no answers to these questions. Philosophers have been divided as to whether
vagueness is a property of mental representations, of linguistic descriptions, or
of things of the world. Russell famously denied that there are vague things,
insisting that vagueness is a matter of imprecise representation.
Wittgenstein emphasized the role of language. His notion of how logic and the
world are connected still reverberates today. As he saw it, we cannot accept that
the world could disobey the laws of logic because ‘we could not say of an
“unlogical” world how it would look’. The general question revolving around
‘identity’ is this: what does how we speak imply about what there is? Does the
fact that there are vague expressions imply that there are vague objects? For our
topic, this is a crucial question, for many predicates essentially important to our
lives are inherently vague: adult, male, intelligent, dead, alive, healthy, normal,
drunk, and on and on.
In response to the limitations of classical logic with regard to vagueness, other
logics arose, notably fuzzy logic, and in recent years, the ontology of vague
objects has become a new field of inquiry explored, for instance, by philosopher
Michael Morreau. Velocity is a pertinent example. Classical logic is confined to
‘fast’ (= ‘not slow’) and ‘slow’ (= ‘not fast’), while fuzzy logic operates with a
variable velocity and three fuzzy sets, ‘slow’, ‘average’, and ‘fast’ with variable
membership values. Fuzzy logic has a fuzzy notion of truth and allows for truth-
value gaps, that is, propositions that are neither true nor false.
Russell once remarked that ‘no one outside a logic-book ever wishes to say “x is
x” ’. Yet it is hard to part with the principle that things are identical with
themselves and harder still to imagine a world where this principle does not
imply sharp boundaries of all things for which it holds. However, both biology
and physics have moved in the direction of accepting vague material objects and