Firstly,
let’s think in terms of predication – that most basic of logical
procedures and an important element of nearly all traditional
ontology.
Take
this Frege-like statement:
Rihanna
is one and Radiohead is three.
In
terms of ontology, Gottlob Frege didn't think that numbers are
properties of objects (i.e., objects like Rihanna and Radiohead).
That is, we can't predicate “unity” or “oneness” of Rihanna
in the way we can predicate “sexiness” of her. In terms of its
grammatical form, that's mainly because the sentence above is a conjunctive identity statement. That is:
Rihanna
= 1 & Radiohead = 3
According
to extensional logic, it
follows from this that if “Rihanna is one”, then “Beyoncé is
one”. And if “Beyoncé is one”, then we can also write “Beyoncé
= 1”, as before. However, according to the extensional principle of
substitutivity, we now have:
Rihanna
and Beyoncé are one.
or
Rihanna
& Beyoncé = 1
That
is, the proper names have the same reference – viz., the number 1.
Frege
(after Kant) argues that this argument is also true of the predicate
“exist” (or “exists”). This too can't be a predicate (or
property) of a concrete object. We can say:
This
man exists.
However,
we "really mean" (or we must mean):
The
concept [man] is instantiated.
In
other words, the concept [man] has at least one instance. (Or,
alternatively, there is at least one instance of the concept [man].)
So if the predicate “exists” can only be applied to concepts (not
to objects), then we can say that "existence" is “a predicate of
predicates”. That is, a predicate of concepts, not of objects. We
can also say that the predicate “exists” is a meta-predicate (or
a meta-concept); which, unlike lower-order predicates, only applies
to concepts. (Just as a meta-truths apply to truths about facts/observations/etc.,
but which aren't themselves about facts/observations/etc. - they're only linguistic
expressions.)
More
importantly for Frege, numbers are predicated of concepts. This is
the case because, for example, the number 5 is the [class of all
five-membered classes]. So, as before, we have a concept [5] which is
applied to other concepts. This means that the number 5 is a
meta-predicate; just like the predicate expression “exists”.
What
about this statement? –
There
are four politicians.
The
Fregean “logical form” of that perfectly grammatical expression
is:
The
concept [politician] is instantiated four times.
That
is, the predicate expression “politician” is used of four objects
– i.e., four politicians. Again, in terms of logic, the concept
[politician] itself is really predicated, not the actual concrete politicians. In consequence, only the concept
[politician] is predicated with the concept or concept-number 4.
Does
it now follow that in 4 isn't really a concept at all: it's a
logical abstract object? Frege himself famously writes:
The
concept [horse] is not a concept.
That
seemingly paradoxical statement can be explained in the sense that in
certain statements (including the one above) the concept
itself is predicated, not the extension of the predicate (or
concept). If that’s the case, it becomes the subject-term of the
statement. Consequently, it must therefore be an object, not a
concept (or a predicate). Hence the prima facie paradoxical
nature of Frege’s statement about the concept [horse].
We
can now say:
The
concept [4] is not a concept.
The
number 4 is (as it were) turned into an object: i.e., a
non-spatiotemporal abstract object. Can we do the same with the
predicate expression “exists”? That is, can we write the
following? –
The
concept [exists] is not a concept.
Is
the “property” existence really an actual object – a
thing of some kind? I don’t think that Frege did think of
existence in the same way as he thought of a number. However,
all the Fregean arguments seem equally applicable to the predicate
expression “exists”; not just to the “four” in “There are four politicians’. And if Frege did think that existence isn't a
genuine object like the number 4, then how did he argue for such a
distinction?
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