Fregean concepts

January 25, 2008

I’m currently reading up on Frege’s theory of concepts for one of my courses. There are several things that really puzzle me about the theory though.

The first confusing thing is to do with relations. Frege says that relations are doubly unsaturated. You can fill a diadic relation with an object, and get a monadic relation. For example if I have the relation $x < y$ I may fill it out with four to get $x < 4$, the monadic concept of being less than four, or alternatively I may put four in the other slot and get $4 < x$ the monadic concept of being greater than four. And it clearly made a difference, for the former but not the later is true of 3. But what if we consider $x = y$? What about $4 = x$ and $x = 4$ – are they the same or different? If we think analogously to the last case, we might think they were different. Relations are doubly unsaturated, they have two “object shaped holes”*, in the first case the first hole was filled, in the second the second hole was filled – they denote different concepts because in each case a different hole is covered. But this cannot be the case, since for Frege, concepts are extensional. If they’re true of the same things, they are the same concept.

There is the puzzling section 70 in the Grundlagen where he suggests that relations might actually be monadic concepts taking composite objects as arguments. I assume the composite objects would have to have as much structure as ordered pairs if they were to do the job. But I don’t think that really helps us, for if $x < y$ was really a monadic concept of pairs, the notion of filling in only one of the argument places doesn’t make too much sense. The most natural way to do that would be to compose $<$ with $\langle 4, x \rangle$, the function that pairs an object with 4. But this means treating < as a second level concept.

That brings me onto the second thing I find confusing. In “Function and Concept” Frege tries to compositionally derive the truth values of various formulae from the Begriffsschrift from the concepts/functions each expression denotes. This works fine for the simple formulae he tries like $\forall x x=x$. But what about: $\forall x(f(x) = 4)$? It should be derived compositionally from the following pieces:

1. The second level concept $\forall$ true of a first level concept F(x) iff F gives true for each object.
2. The first level function $f(x)$ which for each object gives the number four.
3. $4$ the number four.
4. $x = y$, the first level diadic relation-concept that gives the true when supplemented with x and y, where x and y are identical.

However, I cannot see any plausible way to piece these concepts together – they just don’t fit. Crucially $f(x) = 4$ is not type correct, since = is a first level relation, it cannot take a first level function for its left argument, only an object. But even if you try combining (1) – (4) in other ways (i.e. not in the obviously compositional fashion), you don’t seem to get the right results.

Quite generally, there are all kinds of typing problems with the way Frege uses concepts. For example, in $\forall x \exists y (x < y)$, $\exists y$ is supposed to take a monadic concept as argument like $\forall x$ does, but instead it gets a diadic relation-concept.

I guess I don’t really know the literature, or Frege’s writings well enough to know whether he addressed this, or if there is a straightforward way to get around the type mismatches. Does anyone have any ideas? Any pointers to the literature would be very welcome :D.

*This is my metaphor, not Frege’s. You should probably that argument with a pinch of salt.