An overview of Generalised Veltman Semantics

TL;DR

This paper provides a comprehensive overview of Generalised Veltman semantics, highlighting its advantages over traditional semantics in interpretability logics, including simplified completeness proofs and enhanced filtration techniques.

Contribution

It summarizes the development, properties, and applications of Generalised Veltman semantics, emphasizing its role in advancing interpretability logic research.

Findings

Generalised Veltman semantics simplifies completeness proofs.

It enables filtration arguments not possible with standard semantics.

The semantics has various beneficial properties for interpretability logic.

Abstract

Interpretability logics are endowed with relational semantics \`a la Kripke: Veltman semantics. For certain applications though, this semantics is not fine-grained enough. Back in 1992, in the research group of de Jongh, the notion of generalised Veltman semantics emerged to obtain certain non-derivability results as was first presented by Verbrugge ([76]). It has turned out that this semantics has various good properties. In particular, in many cases completeness proofs become simpler and the richer semantics will allow for filtration arguments as opposed to regular Veltman semantics. This paper aims to give an overview of results and applications of Generalised Veltman semantics up to the current date.