An Epimorphism between Fine and Ferguson's Matrices for Angell's AC

TL;DR

This paper demonstrates a surjective homomorphism from Fine's 16-valued matrix to Ferguson's 9-valued matrix for Angell's AC logic, using computational algebra techniques and providing a tableau proof system.

Contribution

It establishes an epimorphism between two finite matrices for Angell's AC and explores the existence of matrices with fewer values, showcasing computational algebra's utility in logic.

Findings

Ferguson's 9-valued matrix is an epimorphic image of Fine's 16-valued matrix.

Candidate 7-valued matrices are ruled out as characteristic of AC.

The MUltlog system effectively aids in investigating finite-valued logics.

Abstract

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. It is shown that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. Some candidate 7-valued matrices are ruled out as characteristic of AC. Whether matrices with fewer than 9 values exist remains an open question. The results were obtained with the help of the MUltlog system for investigating finite-valued logics; the results serve as an example of the usefulness of techniques from computational algebra in logic. A tableau proof system for NC is also provided.