Reasoning about constructive concepts

TL;DR

This paper addresses issues with second order quantification in concept reasoning, proposing a constructive interpretation to develop a consistent formal system capable of interpreting set and number theories.

Contribution

It introduces a constructive approach to reasoning about concepts that resolves second order quantification problems and provides a formal system with proven consistency.

Findings

Constructive interpretation resolves second order quantification issues.

The formal system is consistent.

It can interpret set and number theoretic systems.

Abstract

We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation suffices to resolve the difficulty. We are then able to present a formal system for reasoning about concepts. We prove that this system is consistent and we investigate the extent to which it is able to interpret set theoretic and number theoretic systems of a more standard type.