Extending a first order predicate calculus with partially defined iota terms

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.