Tuesday, 24 July 2018

Gottlob Frege: Numbers as Properties of Concepts

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.


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?

No comments:

Post a Comment