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Carving Content at the Joints

Volume 38 (2008) Supplement [vol. 34]
pp. 145-177 | 10.1353/cjp.2011.0038

In lieu of an abstract, here is a brief excerpt of the content:

# 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 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.

# 2. Sense

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, 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 raises questions, the main ones being (a) what is...

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