A Completeness Proof for A Regular Predicate Logic with Undefined Truth Value

TL;DR

This paper introduces a sound and complete proof system for an extension of Kleene's ternary logic that handles undefined values in predicates, with applications in computer science and reasoning about partial functions.

Contribution

It extends Kleene's logic with a recursive notion of 'is defined' for functions, providing a complete proof system using Henkin construction.

Findings

The proof system is sound and complete for the extended logic.

Many ternary logics can be reduced to this framework.

The approach models reasoning with partial functions in computer science.

Abstract

We provide a sound and complete proof system for an extension of Kleene's ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of "is defined" is extended to terms and formulas via a straightforward recursive algorithm. The "is defined" formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its "is defined" formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to explain, precisely in terms of logic, typical informal methods of reasoning in such applications.