Negative Existentials and Non-denoting Terms - Logic and Its Applications - page 185

according to the standard existence and uniqueness constraints, ∃ x ( ϕ x ⁄ ∀ y( ϕ y ↔ y =
x )). This analysis yields a formula rather than a singular term, and to make a further
assertion about 'the ϕ ', requires an appropriate augmentation of the base formula.
Hence 'The ϕ is Ψ' is formalized as ∃ x ( ϕ x ⁄ ∀ y( ϕ y ↔ y = x ) ⁄ Ψ x ). If there is no
object in the domain of discourse satisfying both the existence and uniqueness con-
straints, then 'the ϕ ' is a vacuous description and the corresponding formula above
will be false, as will any further formula attempting to assert something about 'the ϕ '.
There is no present King of France, and if we let Kx symbolize the property in ques-
tion, then ∃ x ( Kx ⁄ ∀ y( Ky ↔ y = x )) is rendered false by the falsity of the first con-
junct. Consequently 'The present King of France is just' and 'The present King of
France is not just' both turn out false (on both narrow and wide readings of negation),
and now uniform falsity, rather than lack of truth value, propagates through the ac-
count.
But, contra both Frege and Russell, there is an intuitive sense in which we might
want to make true assertions using non-denoting terms, such as those involving basic
logical properties like self identity: 'The present King of France is identical to the
present King of France', or statements using fictional names that affirm details of the
literary context, like 'Sherlock Holmes was a brilliant detective'. It is also convenient
to retain the logical form of a genuine singular term for both proper names and defi-
nite descriptions. But this won't work in classical logic for expressions that don't refer.
If t is a singular term standing, say, for 'Plato's beard', then the negative existential
mentioned above, viz ., ¬ ∃ x ( x = t ) is a contradiction in classical first-order logic with
identity, since it's a basic requirement of the model theory that t be assigned some
object in the domain. This highlights a crucial asymmetry in the classical approach,
where general terms are allowed to have empty extensions while singular terms are
not.

1.2

Free Logic

As Lambert [3] perspicuously observes, the branch of non-classical logic known as
Free logic is largely motivated in response to this asymmetry. The traditional logic of
general terms supposed that the inference from ∀ y( ϕ y →Ψy) to ∃ y ( ϕ y ⁄ Ψy) was
valid, because the terms ϕ and Ψ were thought to have existential import . But this
imposes an unwanted restriction on the range of applicability of formal reasoning, and
on the modern and broader approach no such import is presupposed. The general
terms ϕ and Ψ are allowed to be true of nothing, and hence the inference is invali-
dated. For example, since there are no unicorns, the actual world is a model of the
sentence 'Every unicorn is an aardvark', formalized as ∀ y( Uy → A y), while it is false
that ∃ y ( Uy ⁄ A y), so the actual world serves as a counterexample to the inference. On
the modern approach, an additional premise of the form ∃ y ( Uy ) is required to restore
existential import and yield the valid (but unsound) piece of reasoning: ∀ y( Uy → Ay ),
∃ y ( Uy ) ∴ ∃ y ( Uy ⁄ Ay ).
However, classical first-order logic with identity retains a somewhat curious ex-
ception to the need for an additional premise. If the (potentially complex) 1-place
predicate expression ϕ y is replaced with the complex 1-place predicate y = t , then the
original inference pattern ∀ y ( y = t → Ψ y ) ∴ ∃ y ( y = t ⁄ Ψ y ) goes through on its own.