Probabilistic Epistemic Updates on Algebras

Abstract

The present article contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here, we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL.

Rough concepts

“…for modelling cognitive and social development of individuals [40]), and management science (to analyze market dynamics in terms of the evolution of categories of products and producers [64]). This is presently ongoing work, based on a suitable generalization of the techniques and results on dynamic updates on algebras developed in [55,50,18,19] J c. join-generates F + i.e ∀x ∈ X R (0) [I (1) [R (0) [x ↓↑ ]]] ⊆ R (0) [x ↓↑ ] translation iff ∀x ∈ X R (0) [I (1) [R (0) J c. join-generates…”

Sahlqvist via Translation

“…Let A, B, e : A → B and r : B X → A X be as above. 10 Then the co-GMT translation σ satisfies conditions Extending the GMT and co-GMT translations to bi-intuitionistic logic. In the present subsection we consider the extensions of the GMT and co-GMT translations to bi-intuitionistic logic (aka Heyting-Brouwer logic according to the terminology of Rauszer [56] who introduced this logic in the same paper, see also [55,57]).…”

“…The paper [66] considers a modal expansion of bi-intuitionistic logic with box and diamond operators where the extended GMT translation is used to establish a Blok-Esakia result and to transfer properties such as completeness, finite model property and decidability. 10 The assumption that e is a homomorphism of the lattice reducts of A and B is needed for the inductive steps relative to ⊥, , ∧, ∨ in the proof this proposition, while condition (5.2) is needed for the step relative to > . The language of bi-intuitionistic logic is given by…”

“…The same modus operandi, which achieves generalization through algebras via duality, has been fruitfully employed by some of the authors also for very different purposes, such as the definition of the non-classical counterpart of a given logical framework (cf. [42,44,10]).…”