Sunday, February 21, 2016

Number, concepts and existence

The Maverick Philosopher is agonising about number and existence in this post. It would be simpler if we returned to the original text of Frege which started all this (Die Grundlagen der Arithmetik 1884. Page numbers are to the original edition).

Frege claims with a concept the question is always whether anything, and if so what, falls under it. With a proper name such questions make no sense. (§51, p. 64). He also claims that when you add the definite article to a concept word, it ceases to function as a concept word, although it still so functions with the indefinite article, or in the plural (ibid).

This is part of a section of the Grundlagen where he develops the thesis that number is a property of concepts, not of things. Thus if I say (§46, p. 59) that ‘the King’s carriage is drawn by four horses’, I am ascribing the number 4 to the concept horse that draws the King’s carriage. The number is not a property of the horses, either individually or collectively, but of a concept.

From these two claims, namely that number is a property of concept words, and that proper names are not concept words, it seems to follow that ‘Socrates exists’ makes no sense. If ‘Socrates’ is not a concept word, then it seems no concept corresponds to it, but existence means that some concept is instantiated, so ‘Socrates exists’cannot express existence. This is the difficulty that Bill is grappling with.

But why, from the fact that ‘Socrates’ is not a concept word, does it follow that there is no corresponding concept? Frege has already told us that a concept word ceases to be such when we attach the definite article to it. So while ‘teacher of Plato’ signifies a concept, ‘the teacher of Plato does not. Why can’t the definite noun phrase ‘Socrates’ be the same, except that the definiteness is built into the proper name, rather than a syntactical compound of definite article and concept word. Why can’t ‘Socrates’ be semantically compound? So that it embeds a concept like person identical with Socrates, which with the definite article appended gives us ‘Socrates’?

As I argued in one of the comments, the following three concepts all have a number

C1: {any man at all}
C2: {any man besides Socrates}
C3: {satisfies C1 but not C2}

If the number corresponding to C1 is n, then the number of C2 is n-1. And the number of C3 is of course 1, and if C3 is satisfied, then Socrates exists. Simple.

David Brightly said...

Ed, how will you avoid the regress

Socrates = the person identical with Socrates
= the person identical with (the person identical with Socrates)
= the person identical with (the person identical with (the person identical with Socrates))
= ...
?

Edward Ockham said...

Hi David,

I think it would only be a regress if I were trying to define 'Socrates' as 'the person identical with Socrates'. That would violate the requirement that the defining term must not contain the defined term, or any part of it.

Rather, I am claiming that 'Socrates' is semantically complex in the way that 'the teacher of Plato' is complex, i.e. implicitly embeds a determiner + concept term. So I am saying that 'Socrates' = 'the' + 'S', where 'S' is a concept term. Or if you like, define S as 'Socrates' minus 'the'.

David Brightly said...

Ah, that makes more sense. Isn't that the logician's 'iota' quantifier?

℩x.P(x) = the unique x satisfying P(x)

Socrates = ℩x.*Socrates*(x)

There are lots of competing possibilities for *Socrates* just as

pi = twice the unique smallest positive x satisfying cos(x)=0

is just one of many ways of specifying pi.

Edward Ockham said...

Yes but only one competing possibility in that sense. Consider:

There is a man called 'Socrates'.

This introduces the singular concept of *Socrates*, in that sense. Then

Socrates is a man who lives down the road.

invokes that concept.

David Brightly said...

>> Yes but only one competing possibility in that sense.

Hmmm. It looks a little as if you want to assimilate singular concepts to general concepts. But 'man who lives down the road' isn't unique and 'the man who lives down the road' doesn't quite do it either. You have to add something like 'you know, the one I told you about'. I certainly accept the idea of singular concepts. It makes sense to talk about 'my idea of Socrates'. Asked to elaborate I'd have to say, 'Well, there's a man, a philosopher, goes around asking awkward questions...That's Socrates', even though there might be no such person. What I find surprising is that it seems not to be in the philosophical mainstream. Why does it need arguing for? Is Frege's Object/Concept distinction to blame?

Edward Ockham said...

There is scattered material in the philosophical mainstream, particularly on anaphora/back reference, but the dominant theory is still direct reference.

Why does it need arguing for? Well it’s a weird theory, and takes a while getting your head round. Bill certainly thinks it is weird/bizarre etc.

David Brightly said...

Ah yes. I guess I have the enthusiasm of the convert.