Fregean directions - Williamson, T (Blackwell Publishing, 1991)
The question 'What is the criterion of identity for directions?' might be construed as asking either 'When do lines have the same direction?' or 'When are directions identical?'. Frege's answer 'When they are parallel' fits the former question, not the latter, for it specifies a relation other than identity between lines, not directions. Jonathan Lowe thinks the latter question more fundamental, and claims that Frege's criterion can be reformulated to answer it: 'When some line with one is parallel to some line with the other'. At p. 147 of Identity and Discrimination I make three points against this view: (a) Lowe's construal does not permit a principled rejection of an intuitively unacceptable answer to the original question, (b) On pain of circularity, Lowe's answer uses conceptual resources ('Of as primitive) beyond those of Frege's, and thus is not a mere reformulation, (c) Lowe's question seems to make the hopeless demand for something more basic (in some sense) than 'x = y Lowe's reply 'One-level versus two-level identity criteria' (ANALYSIS, this issue) does not address (b) or (c). I shall follow him in discussing only (a).