Forgetting complex propositions

TL;DR

This paper develops a formal logical framework using possible-world semantics to model how agents forget complex propositions, extending previous models that only considered atomic propositions, and provides a sound axiomatization for this logic.

Contribution

It introduces a new logical operator to model forgetting complex propositions and their atoms, with a complete axiomatization within action model logic.

Findings

Formalization of forgetting complex propositions using possible-world semantics

Introduction of $[oldsymbol{oldsymbol{ riangledown}} ext{pi}]$ operator for minimal forgetting

Sound and complete axiomatization for the logic of knowledge and forgetting

Abstract

This paper uses possible-world semantics to model the changes that may occur in an agent's knowledge as she loses information. This builds on previous work in which the agent may forget the truth-value of an atomic proposition, to a more general case where she may forget the truth-value of a propositional formula. The generalization poses some challenges, since in order to forget whether a complex proposition $\pi$ is the case, the agent must also lose information about the propositional atoms that appear in it, and there is no unambiguous way to go about this. We resolve this situation by considering expressions of the form $[\boldsymbol{\ddagger} \pi]\varphi$, which quantify over all possible (but minimal) ways of forgetting whether $\pi$. Propositional atoms are modified non-deterministically, although uniformly, in all possible worlds. We then represent this within action model logic in order to give a sound and complete axiomatization for a logic with knowledge and forgetting. Finally, some variants are discussed, such as when an agent forgets $\pi$ (rather than forgets whether $\pi$) and when the modification of atomic facts is done non-uniformly throughout the model.