Belief in Simplicial Complexes

TL;DR

This paper introduces a new belief semantics using simplicial complexes that satisfies key axioms and rules, addressing limitations of previous models and exploring alternative frameworks for representing belief.

Contribution

It develops a novel simplicial complex-based semantics for belief satisfying KD45 and related axioms, overcoming prior model limitations and discussing alternative frameworks.

Findings

Belief satisfies KD45 axioms in the new model.

The model ensures each facet has exactly one vertex of every color.

Discussion of alternative simplicial sets framework for belief representation.

Abstract

We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the \textsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom ($K\phi \lthen B\phi$); in addition, we adopt the (standard) assumption that each facet in our simplicial models has exactly one vertex of every color. No existing model of belief in simplicial complexes that we are aware of is able to satisfy all of these conditions without trivializing belief to coincide with knowledge. We also address the common technical assumption of ``properness'' for relational structures made in the simplicial semantics literature, namely, that no two worlds fall into the same knowledge cell for all agents; we argue that there are conceptually sensible belief frames in which this assumption is violated, and use the result of ``A Note on Proper Relational Structures'' to bypass this restriction. We conclude with a discussion of how an alternative ``simplicial sets'' framework could allow us to bypass properness altogether and perhaps provide a more streamlined simplicial framework for representing belief.