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it appears reasonable to prefer the affirmative. That means that it is more reasonable
than not to suppose that there are vague objects, vague sets, including relations, and
vague states of affairs. To give three corresponding examples, (i) a
frog
is a vague
animal, i.e., an object with indeterminate spatio-temporal boundaries, because it is
impossible to determine when it emerges from a tadpole. There is no abrupt end
of being a tadpole and no abrupt start of being a frog. The transition is continu-
ous. Similarly, (ii) the class of
bald
human beings has no sharp boundaries. It has
a penumbral region of genuine borderline cases that imperceptibly vanishes into the
set of non-bald people. Finally, (iii) there are also vague
states of affairs
, e.g., vague
events. For, a state of affairs amounts to the belonging of an object to a class. For
example, the state of affairs that
Picasso is bald
entails Picasso's membership in the
class of bald people. If the class to which an object belongs, is a vague set, such as
bald
, and the object resides in its penumbra, the state of affairs turns out to be some-
thing indefinite. To elucidate, let us introduce an operator, symbolized by “
Δ
”and
read “definitely”. For instance, if
α
is a statement,
Δα
means “definitely
α
”. Thus,
“
(Picasso ist a Spanish artist)” says: Definitely, Picasso is a Spanish artist. With
the aid of this operator,
Δ
Δ
, we can define the vagueness of classes in the following
way:
2
Definition 1.
A class C is vague if and only if
∃
x
¬
Δ
(
x
∈
C
)
∧¬
Δ
(
x
∈
C
)
.
For example, the individual Picasso shows that the class of bald people is vague
because:
¬
Δ
(
Picasso is bald
)
∧¬
Δ
(
Picasso is not bald
)
.
That means, it is indefinite whether Picasso is bald and it is indefinite as well
whether he is not bald. Let
α
be any statement, the following sentence:
¬
Δ
(
α
)
∧¬
Δ
(
¬
α
)
says that we neither know whether
α
is true nor know whether
¬
α
is true. This is
equivalent to the following statement:
¬
Δ
(
α
)
∨
Δ
¬
(
α
)
.
From this we can conclude that:
¬
Δ
(
α
∨¬
α
)
.
And that means that in the following disjunction contained in it:
α
∨¬
α
neither
has a truth value. But this sentence is exactly the
Principle of Exluded Middle
of classical logic, and as such, a tautology. That is, it
α
nor its negation
¬
α
2
The definiteness operator Δ as well as the approach to vagueness using it I owe to Timothy
Williamson ([10], 695 ff.).
