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defined (contextually) as
that
unique
x
. The definite description '
i
x
ϕ
x
' can then be
used as a legitimate complex singular term for making assertions such as
∃
y
(
y
=
i
x
ϕ
x
), Ψ(
i
x
ϕ
x
), and the seemingly innocuous
ϕ
(
i
x
ϕ
x
). In the special case of definite
descriptions, a 1-place predicate
ϕ
x
is used to define a 0-place function, i.e. a singular
term. In the general case, an
n
-ary relation
R
n
(
x
1
,...,
x
n
‒
1
,
y
) can be used to define an (
n
‒
1)-ary
total
function
f
n
‒
1
, if
R
n
satisfies the corresponding existence and uniqueness
constraints
∀
x
1
...
∀
x
n
‒
1
∃
y
∀
z
[R
n
(
x
1
,...
x
n
‒
1
,
y
)
⁄
(
R
n
(
x
1
,...
x
n
‒
1
,
z
) →
z
=
y
)] in which case
f
n
‒
1
(
x
1
,...,
x
n
‒
1
) =
y
and the set of (
n
‒
1)-ary total functions can be viewed as a proper
subset of the set of
n
-place relations.
However, not all functions that we might wish to consider are total, and this can be
due to a failure of either constraint. Furthermore, such failures might not be known to
us at the time the function term is introduced. For example, 0-place definite descrip-
tions are often vacuous, as in 'the greatest prime number', although prior to Euclid's
proof the semantic status of this description was not definitively known. The function
f
(
x
) =
x
‒
1
on the reals is partial, since it is not defined in the case of
x
= 0, and
the description 'the
x
such that
x
2
= 2' fails the uniqueness constraint. Nonethe-
less it is often
expedient
to perform logical and mathematical manipulations
involving partial functions, and thus in the general case Russell's constraints
seem unduly restrictive. For example, on Russell's account, it is a logical truth
that
∃
y
(
y
=
i
x
ϕ
x
). However, it might be useful to be able to introduce the term
i
x
ϕ
x
without
first proving that the existence condition is satisfied,
a la
Free logic, and then
employ the term to articulate the discovery that ¬
E!
(
i
x
ϕ
x
), if it's later found that no
such object exists.
In the context of providing a foundation for mathematics, Frege sought to avoid the
truth value gaps mentioned above that result from descriptions that fail to denote, and
his solution was to assign a 'dummy value' from the realm of existents. This is akin to
the current strategy in computer science of assigning an 'error object' in such cases
(see Gumb [6]). The (generic) Free logic approach is to dispense with existence as-
sumptions for such terms and use the existence predicate to preserve valid patterns of
inference. This is also the intuitive strategy adopted by
Troelstra and van Dalen [7]
with their E-logics in the context of constructive mathematics. Within Free logic there
are various choices regarding descriptions that fail to denote. Making all atomic for-
mulas containing empty descriptions
false
yields a 'negative' free description theory
equivalent to Russell. In contrast, making all identities between empty descriptions
true
yields a 'positive' description theory analogous to Frege's solution above, al-
though instead of taking the 'dummy value' from the realm of existents, it is now more
natural to use a nonextistent object,
as per
the semantics outlined below. So called
'neutral' Free description theories constitute yet a third option, where bivalence is
sacrificed and statements involving empty terms lack a truth value, as on Frege's
strictly compositional approach.
1.4
Inner and Outer Domains
From the point of view of classical semantics, there are two distinct ways in which
singular terms can fail to denote. First, a term can be genuinely empty, in the sense
that it maps to nothing at all, in which case the semantical interpretation function on
