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In-Depth Information
The expression ' =
t
' has existential import in the traditional sense, while in general
the expressions
ϕ
y
and Ψ
y
do not. This traditional residue derives from the asymme-
trical fact that singular terms are required to denote while general terms can be empty.
∃
y
(
y
=
t
) is a truth of classical logic for every singular term
t
in the language, and
hence does not need to be introduced as an extra premise. This can itself be viewed as
an undue restriction on the range of applicability of formal reasoning, since it is not
possible to carry out intuitively plausible inferences concerning objects that do not or
might not exist in the actual world. And in the same manner as above, the natural
strategy is to devise a logic free of existence assumptions with respect to its terms,
both
singular
and general (Lambert [4]).
In Free logic, the quantifiers are interpreted in the normal way, as ranging over
some domain of discourse
D
, normally construed as the set of 'existent objects'. But
the singular terms may denote objects outside of
D
, or fail to denote altogether. This
de-coupling of singular reference from the range of the quantifiers undermines two
fundamental inference patterns of classical logic, namely Universal Instantiation (UI)
and Existential Generalization (EG). According to UI,
∀
y
ϕ
y
∴ ϕ
t
is a valid infe-
rence. But it fails in Free logic because the quantifier ∀y only ranges over objects
e
D
has
property
ϕ, it does not follow that
t
does.
And according to EG,
ϕ
t
∴ ∃
y
ϕ
y
is a
valid inference. But similarly this fails in Free logic
because, e.g.,
t
may denote a
nonexistent object not in the range of
∃
y
, thus allowing for the possibility of true pre-
mise and false conclusion.
Analogous to the foregoing transition from traditional to modern logic in the case
of general terms, now that singular terms are also free of existence presuppositions, an
additional premise
is required to restore validity. Existential import with respect to
singular terms is expressed via an existence predicate for individuals (in violation of
Kantian notions), normally using Russell's '
E!
' notation. With the use of identity, the
existence predicate can be defined as
E!
(
t
) :=
def
∃
y
(
y
=
t
). In the case of both UI and
EG,
E!
(
t
) is the suppressed premise required to yield an inference pattern valid in the
context of Free logic. Hence UI
Free
has the form
∀
y
ϕ
y
,
E!
(
t
)
∴
ϕ
t
, and EG
Free
has the
form
ϕ
t
,
E!
(
t
)
∴
∃
y
ϕ
y. It is now possible to directly articulate the fact that Pegasus
does not exist with the formula ¬
E!
(
t
), letting
t
denote the mythical flying horse. And
while it's true that neither Plato's beard nor Pegasus exist, it's nonetheless
false
that
∃
x
¬
E!
(
x
).
∈
D
, whereas '
t
' may not refer to any such
e
. So from the fact that every
e
∈
1.3
Definite Descriptions Revisited
As noted earlier, Russell's 1905 theory of definite descriptions analysed expression
such as 'the
ϕ
' in terms of a formula rather than a singular term. However, it is often
convenient to be able to render such expressions as genuine terms, and have a uniform
treatment of simple terms such as individual constants or proper names, along with
complex singular terms such as definite descriptions and function terms. In
Principia
Mathematica
, Russell [5] introduced his variable-binding, term-forming 'iota' operator
to do just that. If it's provable that the existence and uniqueness conditions are satis-
fied, then a Russellian iota operator '
i
' yields a complex singular term as follows: if
⊢
∃
x
(
ϕ
x
⁄
∀
y
(
ϕ
y
↔
y
=
x
)) then
i
x
ϕ
x
, read as 'the
x
such that
ϕ
x
', or simply 'the
ϕ
' is
