Fracterm Calculus for Partial Meadows

TL;DR

This paper introduces a three-valued logic and calculus for partial meadows, formalizing fields with division, and explores their semantics and semi-computability.

Contribution

It provides axioms for fracterm calculus of partial meadows and interprets the logic within $ot$-enlargements of partial algebras.

Findings

The logic cannot express that division by zero is undefined.

The consequence relation of the logic is semi-computable.

The $ot$-enlargement of a partial meadow is a common meadow.

Abstract

Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also known as McCarthy calculus and has been studied extensively. Axioms for the fracterm calculus of partial meadows are given. The case is made that in this way a rather natural formalisation of fields with division operator is obtained. It is noticed that the logic thus obtained cannot express that division by zero must be undefined. An interpretation of the three-valued sequential logic into $\bot$-enlargements of partial algebras is given, for which it is concluded that the consequence relation of the former logic is semi-computable, and that the $\bot$-enlargement of a partial meadow is a common meadow.