Journal of Symbolic Logic

A Partial Functions Version of Church's Simple Theory of Types

William M. Farmer

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Church's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called $\mathbf\mathrm{PF}$ in which functions may be partial. The semantics of $\mathbf\mathrm{PF}$, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that $\mathbf\mathrm{PF}$ is complete with respect to its semantics. The reasoning mechanism in $\mathbf\mathrm{PF}$ for partial functions corresponds closely to mathematical practice, and the formulation of $\mathbf\mathrm{PF}$ adheres tightly to the framework of Church's system.

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J. Symbolic Logic, Volume 55, Issue 3 (1990), 1269-1291.

First available in Project Euclid: 6 July 2007

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Farmer, William M. A Partial Functions Version of Church's Simple Theory of Types. J. Symbolic Logic 55 (1990), no. 3, 1269--1291.

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