Abstract
Epistemic reading of Kripke models relies on a hidden assumption of common knowledge of the model which is too restrictive in epistemic contexts since agents may have different views of the situation. We explore possible worlds models in their full generality without common knowledge assumptions. Our starting point is a collection of possible worlds with accessibility relations “whatever is known in u is true in v.” We call such a structure an observable model since, contrary to the popular belief, it is not generally a Kripke model but rather an “observable section” of some Kripke model. We sketch a theory of observable models and argue that they bring a new conceptual clarity to epistemic modeling. In practical terms, observable models are as manageable as Kripke models and have advantages over the latter in representing (un)awareness and ignorance. Similar analysis applies to intuitionistic models.
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- 1.
In this text we will also be using terms state or world for global states, when convenient.
- 2.
Analyzing the role of knowing the model, normally assumed and not acknowledged in formal epistemology, has been long overdue. The paper that prompted completing this study was [6].
- 3.
State w is constructive: one can check that \(\varGamma \) is a complete set of formulas, i.e., for each F, either \(\varGamma \) proves F or \(\varGamma \) proves \(\lnot F\), and w is the set of formulas derivable from \(\varGamma \), cf. also Sect. 7.
- 4.
It appears that a natural formalization of this condition leads us beyond the current level of propositional modal logic.
- 5.
We regard this as a feature that makes observable models more flexible and realistic.
- 6.
A similar normal form-based proof of completeness can be given for each of A–D, but we have opted for Necessitation-based proof for A and D to underline the fact that both A and D enjoy Necessitation.
- 7.
Here again, we ignore indices i in \(\mathbf {K}_i\) and \(R_i\).
- 8.
We drop brackets in \({ CM}(\{p\})\) and similar cases for better readability.
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Artemov, S. (2020). Observable Models. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham. https://doi.org/10.1007/978-3-030-36755-8_2
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DOI: https://doi.org/10.1007/978-3-030-36755-8_2
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