以下は単なるメモです。
Aczel explicitly says that the horizontal “is responsible for Russell's paradox” (1989[sic] [“The Structure of the Formal Language of Frege's Grundgesetze”], 17) and “gives an internal definition of the collection of truths” (ibid.). But think again: Since −ξ is definable by ξ = (ξ = ξ), ξ = ζ is likewise undefinable within the system [of Grundgesetze], as indeed stated by Aczel. Why then is it the horizontal that is responsible for Russell's paradox, and not the equality sign? *1
On the way, Frege mentions almost in passing, a fact expressible as an equation which could be taken as a definition of the horizontal, namely
for all a: −a =(a ≈(a ≈ a))
so −is in fact genuinely, logically redundant in the system [of Grundgesetze]! *2
*1:Christian Thiel, “On the Structure of Frege's System of Logic,” in Matthias Schirn, ed., Frege: Importance and Legacy, Walter de Gruyter, Perspektiven der analytischen Philosophie = Perspectives in Analytical Philosophy, Bd. 13, 1996, pp. 264-65.
*2:Peter Simons, “The Horizontal,” in Matthias Schirn, ed., Frege: Importance and Legacy, Walter de Gruyter, Perspektiven der analytischen Philosophie = Perspectives in Analytical Philosophy, Bd. 13, 1996, p. 290. 引用文中の‘ ≈ ’は、現代における‘ = ’に相当する。‘ = ’が文を形成する記号であるのに対し、‘ ≈ ’は、Frege にとって名前を形成する記号であり、この違いを考慮して Simons さんは異なる記号を使い分けている。
