## Abstract

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functions*in theory.* However, the approach to partial functions traditionally employed by mathematicians is quite adequate*in practice.* This paper shows how the traditional approach to partial functions can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems.

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## Additional information

Supported by the MITRE-Sponsored Research program. This paper is a written version (with references) of an address given at the Partial Functions and Programming: Foundational Questions conference held 17 February 1995 at the University ol California at Irvine.

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Farmer, W.M. Reasoning about partial functions with the aid of a computer.
*Erkenntnis* **43, **279–294 (1995). https://doi.org/10.1007/BF01135375

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### Keywords

- Computer Science
- Logical Formalism
- Strong Support
- Traditional Approach
- Type Theory