[Paper Review] Knowing the Model

Sergei Artëmov|arXiv (Cornell University)|Oct 17, 2016

Logic, Reasoning, and Knowledge6 references3 citations

TL;DR

This paper challenges the standard use of Kripke models in epistemic logic by proposing a more foundational approach: epistemic situations as sets of maximal consistent worlds (W), where conventional accessibility relations are derived from W. It identifies the 'fully explanatory property' as the necessary and sufficient condition for W to form a Kripke model, revealing this as an overlooked assumption. The work introduces a generalized epistemic model framework that better captures partial knowledge, asymmetric knowledge, and awareness.

ABSTRACT

Epistemic modal logic normally views an epistemic situation as a Kripke model. We consider a more basic approach: to view an epistemic situation as a set W of possible states/worlds -- maximal consistent sets of propositions -- with conventional accessibility relations determined by W. We find that in many epistemic situations, W is not a Kripke model: a necessary and sufficient condition for W to be a Kripke model is the so-called `fully explanatory property' - a propositional form of common knowledge of the model - which has been a hidden (and overlooked) assumption in epistemic modal logic. We sketch a theory that describes epistemic models in their generality. We argue for conceptual and practical value of new models, specifically for representing partial knowledge, asymmetric knowledge, and awareness.

Motivation & Objective

• To re-express epistemic situations not as Kripke models by default, but as sets W of maximal consistent worlds.
• To identify the 'fully explanatory property' as the hidden assumption that makes W a Kripke model.
• To develop a general theory of epistemic models that accommodates partial knowledge, asymmetric knowledge, and awareness.
• To demonstrate conceptual and practical advantages of this generalized framework over standard Kripke models.

Proposed method

• Represent an epistemic situation as a set W of maximal consistent sets of propositions, each representing a possible world.
• Define accessibility relations between worlds in W based on logical relations derived from W, rather than imposing them externally.
• Introduce the 'fully explanatory property'—a propositional condition equivalent to common knowledge of the model structure—as the criterion for W to be a Kripke model.
• Show that without this property, W fails to satisfy Kripke model axioms, revealing a foundational oversight in epistemic logic.
• Construct a generalized epistemic model framework that relaxes the need for the fully explanatory property.
• Use the framework to model scenarios involving partial knowledge, asymmetric knowledge, and awareness, which standard Kripke models cannot represent directly.

Experimental results

Research questions

• RQ1 What conditions must a set W of possible worlds satisfy to be a Kripke model?
• RQ2 Why has the 'fully explanatory property'—a propositional form of common knowledge of the model—been overlooked in epistemic logic?
• RQ3 How can epistemic models be generalized to represent partial knowledge and asymmetric knowledge?
• RQ4 What are the conceptual and practical benefits of moving from Kripke models to a world-set-based foundation?
• RQ5 Can awareness and incomplete knowledge be naturally modeled in this new framework?

Key findings

• The 'fully explanatory property' is both necessary and sufficient for a set W of possible worlds to form a Kripke model.
• This property has been implicitly assumed in epistemic modal logic but never explicitly recognized as a foundational constraint.
• Without the fully explanatory property, standard Kripke models fail to capture the epistemic structure of many real-world situations.
• The proposed framework enables natural representation of partial knowledge and asymmetric knowledge, which standard Kripke models cannot express.
• The generalized model framework provides a more flexible and conceptually accurate foundation for epistemic logic, especially in contexts involving awareness and incomplete information.