Russell's 1903 – 1905 Anticipation of the Lambda Calculus
HISTORY AND PHILOSOPHY OF LOGIC, 24 (2003), 15–37 Russell’s 1903 – 1905 Anticipation of the Lambda Calculus KEVIN C. KLEMENT Philosophy Dept, Univ. of Massachusetts, 352 Bartlett Hall, 130 Hicks Way, Amherst, MA 01003, USA Received 22 July 2002 It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church’s ‘Lambda Calculus’ for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903 and 1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the Lambda Calculus. Russell also anticipated Scho¨ nfinkel’s Combinatory Logic approach of treating multi- argument functions as functions having other functions as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and ‘value-ranges’. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Rus- sell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the Lambda Calculus. 1. Introduction The Lambda Calculus, as we know it today, was initially developed by Alonzo Church in the late 1920s and 1930s (see, e.g. Church 1932, 1940, 1941, Church and Rosser 1936), although it built heavily on work done by others, perhaps most notably, the early works on Combinatory Logic by Moses Scho¨ nfinkel (see, e.g. his 1924) and Haskell Curry (see, e.g.
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