1. The Problem
Here is Frege in Foundations of Arithmetic, § 64:
The judgment 'Line a is parallel to line b', in symbols: ab, can be taken as an identity. If we do this, we obtain the concept of direction, and say: 'The direction of line a is equal to the direction of line b.' Thus we replace the symbol by the more generic symbol =, through removing what is specific in the content of the former and dividing it between a and b. We carve up the content in a way different from the original way, and this yields us a new concept.
Something important is going on in this passage. But at the same time it borders on incoherent. For Frege is saying at least the following:
1. 'dir(a ) = dir(b )' has the same content as 'ab'
2. reflecting on that can lead one to the concept of direction.
Why doesn't (2) contradict (1)? (2) has a neophyte acquiring the concept of direction — and so presumably a grasp of the content of 'dir(a ) = dir(b )' — by reflecting on a certain content-identity. But then it is hard to see how the postulated content-identity can really obtain; Leibniz's Law would seem to forbid it. If one grasps content X at a certain time, and content X = content Y, then one grasps content [End Page 145] Y at that time. The neophyte grasped the content of 'ab' before encountering (1), so if that content is also the content of 'dir(a ) = dir(b ),' she must have grasped the content of 'dir(a ) = dir(b )' before encountering (1) as well.
I know of only one good way of getting around this. The neophyte did grasp the content of 'dir(a ) = dir(b )' before encountering (1); she just failed to know it as the content of an identity-sentence. She doesn't know it as the content of an identity-sentence until she acquires the concept of direction: perhaps knowing it that way is acquiring the concept of direction.
What should content be, for this way around the problem to work? One natural hypothesis is that content is sense; and Frege certainly says things that suggest this. But the suggestion is problematic, if we take Frege at his word that the sense of part of a sentence is part of the sense of the whole.
Remember, the neophyte has to grasp the content of 'dir(I) = (b)' before acquiring the concept of direction. So if content is sense, she must be able to grasp the sense of 'dir(a ) = dir(b)' before acquiring the concept of direction. If she lacks the concept of direction, though, how is she supposed to grasp the sense of direction-terms? And if she does not grasp the sense of direction-terms, how is she supposed to grasp the sense of 'dir(a ) = dir(b)'? The problem is that each of the indicated achievements presupposes the one 'before' it:
Frege's strategy does not appear to work, then, if content is sense. What else could it be?
The downward-facing arrow is compulsory, for the passage clearly states that the concept of direction is acquired by grasping the content of the direction-sentence. The left-to-right arrow is compulsory too, for grasping the sense of 'direction-of ' is appreciating that it expresses the relevant concept. The upward-facing arrow is not forced on us, [End Page 146] though. Why shouldn't grasping the content of a sentence leave one still undecided about its sense?
The obvious way to arrange for that is to make content coarser-grained than sense, though presumably still finer-grained than reference. Then everything hangs together just right:
1. 'dir(a ) = dir(b )' shares something with 'ab,' but
2. the shared something is content, not sense;
3. the shared content can be carved in two ways,
4. corresponding to the sentences' two senses;
5. we start out knowing one carving, then learn the other;
6. the directional carving teaches us the concept of direction.
This has got to be the way to go. But every step...