## Journal of Symbolic Logic

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

William M. Farmer

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743419

Mathematical Reviews number (MathSciNet)
MR1071328

Zentralblatt MATH identifier
0722.03007

JSTOR