Talk:History of type theory

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I created this page from the "type theory" page and moved some of the talk items from its talk page to here.

History section[edit]

This has only scratched the surface; it could eventually be spun off into its own article, I suppose. I've hidden the following text, because I don't have the (re)sources (e.g. Quine, Tarski) to go further:

Quine reports that Ramsey 1931 urged the "dropping of the ramification and the axiom of reducibility"; Quine opines that "Russell's failure [was] due to this failure to distinguish between propositional functions as notations and propositional functions as attributes and relations.[Quine's commentary before Russell 1908 in van Heijenoort 1967:151-152. Ramsey 1931 The foundations of mathematics and other logical essays, edited by Richard Bevan Braithwaite (Paul, Trench, Trubner, London; Harcourt, Breace, New York); reprinted 1950 (Humanities Press, New York)]."

This is from Hans Riechenbach (1947, reprinted 1980) Elements of Symbolic Logic, Dover Publications, Inc, New York, ISBN:0-486-24004-5

"Although Russell on another occassion,3 suggested the extension of his theory [type theory + axiom of reducibility - ramified types] to a theory of levels of language, this extension ws actually carried through by Ransey4, to whom we own the distinction of logical and semantical antinomies, and by Tarski5 and Carnap6. The ramified theory of types was then dropped1. An interesting attempt to replace the theory of types by weaker formtion rules was constructed by Quine2, whose system employs certain ideas of von Neumann.
3 In his introduction to L. Wittgenstein, Tractatus Logico-Philosophicus, Harcourt, Brace, New York, 1922, p. 23
4 F.P. Ramsey, The Foundations of Mathematics, Harcourt, Brace, New York, 1931, p. 20
5 A. Tarski, 'Der Wahrheitsbegriff in den Formalisierten Sprachen,' Studia Philosophica, Leopoli, 1935
6 R. Carnap, Logical Syntax of Language, Harcourt, Brace, New York, 1937 (German edition, Vienna, 1934)
1 Cf. B. Russell in his Introduction to the second edition of the Principles of Mathematics, Norton, New York, 1938 [ sic! ]
2 W. Quine, Mathematical Logic, New York, 1940, Norton, § 24 and § 29.

Observe Riechenbach's bizarre out-of-sequence history -- Russell had published his 2nd edition to Principia Mathematica in 1927, not 1938. And he has dropped the ramified theory by 1910-1913 with his axiom of reducibility augmented with the "matrix" notion, which he further augments in the 1927 edition (after being exposed to Wittgenstein, cf his references on page xlv-xlvi). He drops the axiom of reducibility but at the loss of "infinite well ordering" as he states at the very end of his 1927 introduction (page xlv). Bill Wvbailey (talk) 16:36, 21 September 2009 (UTC) — Preceding unsigned comment added by Mdnahas (talkcontribs)