## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 13, Number 5 (2007), 917-930.

### Extending a first order predicate calculus with partially defined iota terms

Albert Hoogewijs and Geert Vernaeve

#### Abstract

Partial functions and "undefinedness" have been around in mathematics for a long time, without causing any trouble. It was only when mathematics and computer science met in projects on ``automatization'' of formal reasoning that some problems came up [Hoogewijs 1987]. Where humans are able to avoid the application of a partial function on an argument outside ``the domain'' of the function, formalizing the rules for this activity seems to be less trivial. In [Farmer 1996] Farmer states that there does not exist a consensus on how partial functions should be mechanized and the developer of a mechanized mathematics system must choose among many different possible ways of representing and reasoning about partial functions. We want to add one more possibility by introducing ``partially defined iota terms'' of the form $\iota x_\psi(\varphi)$ which represents the unique $x$ satisfying $\varphi$ whenever the condition $\psi$ is fulfilled. We present an extension of a two-valued first order sequent calculus for predicate logic with identity [Hermes 1973], where we are able to reason correctly about partially defined iota terms.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 5 (2007), 917-930.

**Dates**

First available in Project Euclid: 1 February 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1170347814

**Digital Object Identifier**

doi:10.36045/bbms/1170347814

**Mathematical Reviews number (MathSciNet)**

MR2293217

**Zentralblatt MATH identifier**

1120.03009

**Subjects**

Primary: 03B35: Mechanization of proofs and logical operations [See also 68T15]

**Keywords**

Mechanisation of proofs sequent calculus partial functions undefinedness

#### Citation

Vernaeve, Geert; Hoogewijs, Albert. Extending a first order predicate calculus with partially defined iota terms. Bull. Belg. Math. Soc. Simon Stevin 13 (2007), no. 5, 917--930. doi:10.36045/bbms/1170347814. https://projecteuclid.org/euclid.bbms/1170347814