Revisiting Interpolation in Relevant Logics

TL;DR

The paper argues against the widespread belief that relevant logics lack interpolation by showing that the maximal VSP extension ${\mathcal{C}}$ of ${\mathcal{R}}$ enjoys Maehara interpolation (MIP) for deducibility, linking relevance (via VSP) to strong interpolation properties. It establishes this result through algebraic semantics: ${\mathcal{C}}$ forms a congruence-distributive variety with CEP, and, via extensibility of its finitely subdirectly irreducible members, yields AP and thus TIP, culminating in MIP and deductive interpolation. The authors relate MIP to standard interpolation results and discuss how the other maximal VSP extension ${\sf M}$ lacks CEP (hence AP) and leaves the question of deductive interpolation open, highlighting nuanced interactions between relevance notions and interpolation properties. The work clarifies misunderstandings about interpolation in relevant logics and demonstrates the applicability of algebraic tools to foundational questions in logical consequence and relevance.

Abstract

There are exactly two maximal schematic extensions of the relevant logic R with the variable sharing property. We establish that one of them has a strong form of interpolation for deducibility, thereby giving an example of a well-known relevant logic with interpolation.

Figures (2)

• Figure 1: The commutative diagrams illustrating the AP (left) and the TIP (right).
• Figure 2: Labeled Hasse diagram for the relevant algebra ${\bf {C}}$. Here we follow the standard convention that idempotent elements are denoted with filled in nodes, whereas non-idempotent elements are denoted with nodes that are not filled in.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof