Saturday, 26 March 2011

Theory of simple types

In the 1920s, Leon Chwistek[2] and Frank P. Ramsey[3] noticed that, if one is willing to give up the vicious circle principle, the hierarchy of levels of types in the "ramified theory of types" (see the History section for more on this) can be collapsed.

    "That the theory of simple types suffices for avoiding also the epistemological paradoxes is shown by a closer analysis of these. (CF. Ramsey 1926 and Tarski 1935, p. 399.)" Gödel 1944:127 footnote 17.[4]

The resulting restricted logic is called the theory of simple types[5] or, perhaps more commonly, simple type theory[6]. Detailed formulations of simple type theory were published in the late 1920s and early 1930s by R. Carnap, F. Ramsey, W.V.O. Quine, and A. Tarski. In 1940 Alonzo Church (re)formulated it as simply typed lambda calculus.[7] and examined by Gödel in his 1944.

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