Finite Axiomatizability of Transitive Logics of Finite Depth and of Finite Weak Width

TL;DR

This paper investigates the conditions under which transitive logics with finite depth and weak width are finitely axiomatizable, providing both positive and negative results based on frame properties.

Contribution

It establishes finite axiomatizability for certain transitive logics characterized by rooted frames with bounded antichains, and shows non-finite axiomatizability for others with larger depth and width.

Findings

Finite axiomatizability for logics with weak width 1 and bounded antichains.

Non-finite-axiomatizability for logics with depth ≥ 3 and width ≥ 2.

Characterization of logics based on properties of rooted transitive frames.

Abstract

This paper presents a study of the finite axiomatizability of transitive logics of finite depth and finite weak width. We prove the finite axiomatizability of each transitive logic of finite depth and of weak width $1$ that is characterized by rooted transitive frames in which all antichains contain at most $n$ irreflexive points. As a negative result, we show that there are non-finitely-axiomatizable transitive logics of depth $n$ and of weak width $k$ for each $n\geqslant3$ and $k\geqslant2$.