A Logical Study of Some Common Principles of Inductive Definition and its Implications for Knowledge Representation

TL;DR

This paper provides a formal logical framework for common inductive definitions in science and mathematics, analyzing their properties and implications for knowledge representation.

Contribution

It introduces a uniform logic of definitions, formalizes the induction process, and explores properties like non-determinism, confluence, and the irrelevance of induction order.

Findings

Induction order does not affect the defined set.

The logic formalizes various inductive definitions uniformly.

An inductive construction method avoids reliance on induction order.

Abstract

The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification languages and systems. This paper is a formal study of some of the most common forms of inductive definitions found in scientific text: monotone inductive definition, definition by induction over a well-founded order and iterated inductive definitions. We define a logic of definitions offering a uniform formal syntax to express definitions of the different sorts, and we define its semantics by a faithful formalization of the induction process. Several fundamental properties of definition by induction emerge: the non-determinism of the induction process, the confluence of induction processes, the role of the induction order and its relation to the inductive rules, how the induction order constrains the induction process and, ultimately, that the induction order is irrelevant: the defined set does not depend on the induction order. We propose an inductive construction capable of constructing the defined set without using the induction order. We investigate borderline definitions of the sort that appears in definitional paradoxes.