入手文献: Illative Combinatory Logic as Neo-Fregean Logicism

いずれも Illative Combinatory Logic 関係です。

  • Richard B. White  “A Demonstrably Consistent Type-Free Extension of the Logic BCK,” in: Mathematica Japonica, vol. 32, no. 1, 1987
  • Yuichi Komori  “Illative Combinatory Logic Based on BCK–Logic,” in: Mathematica Japonica, vol. 34, no. 4, 1989
  • Uwe Petersen  “Logic Without Contraction as Based on Inclusion and Unrestricted Abstraction,” in: Studia Logica, vol. 64, no. 3, 2000


なぜ、Fregeに興味を持っている私が Illative Combinatory Logic 関係の論文を集めているのかというと、私がFregeの Logicism の行方に関心があると言えば、おわかりになると思います。
Fregeの Logicism の行方に関して、近年 Illative Combinatory Logic に注目する研究がいくつか現れてきていると思いますが、‘Illative Logic’という言葉を使って、Fregeの Logicism の行方を解説している論文を一つ挙げますならば、以下があります。

  • 津留竜馬  「概念記法は何故矛盾したのか」、『思想』、岩波書店、no. 954、2003年10月号、154ページ。


また、‘Illative Combinatory Logic’という言葉を使って、それが‘Neo-Fregean’な試みだ、と直裁に述べている文言が次に見られます*1。Gothic は引用者によります。

During the years 1903 to 1905, Russell pursued a version of the foundations of logic that was strongly influenced by Frege's abortive attempt at solving Russell's paradox. The theory was type-free, based on the notion of function, and founded on the idea that the paradoxes could be solved by placing direct restrictions on the comprehension principle.
The working out of the zig-zag theory required a kind of balancing act, in which Russell tried to preserve enough of the comprehension scheme to allow the derivation of ordinary mathematics, while avoiding contradictions. In the end, his efforts failed. In this talk, I survey some of this work, and compare it to modern systems such as illative combinatory logic, which can also be considered as attempts at neo-Fregean foundations for mathematics.


また勉強させていただきます。

*1:Alasdair Urquhart, “Russell's zig-zag theory 1903-05,” Abstract, One Hundred Years of Russell's Paradox: International Conference in Logic and Philosophy, University of Munich, Germany, Sat. June 2 - Tue. June 5, 2001. ちなみに Godehard Link ed., One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, De Gruyter Series in Logic and its Applications, vol. 6, 2004 には、このUrquhartさんの話は載っていません。