Journal of Symbolic Logic 55 (3):1269-1291 (1990)

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 PF in which functions may be partial. The semantics of 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 PF is complete with respect to its semantics. The reasoning mechanism in PF for partial functions corresponds closely to mathematical practice, and the formulation of PF adheres tightly to the framework of Church's system
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DOI 10.2307/2274487
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References found in this work BETA

On Denoting.Bertrand Russell - 1905 - Mind 14 (56):479-493.
Singular Terms, Truth-Value Gaps, and Free Logic.Bas C. van Fraassen - 1966 - Journal of Philosophy 63 (17):481-495.
On Second-Order Logic.George S. Boolos - 1975 - Journal of Philosophy 72 (16):509-527.
Sense Without Denotation.Timothy Smiley - 1959 - Analysis 20 (6):125 - 135.
Existential Import Revisited.Karel Lambert - 1963 - Notre Dame Journal of Formal Logic 4 (4):288-292.

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The Seven Virtues of Simple Type Theory.William M. Farmer - 2008 - Journal of Applied Logic 6 (3):267-286.
Higher-Order Multi-Valued Resolution.Michael Kohlhase - 1999 - Journal of Applied Non-Classical Logics 9 (4):455-477.

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