Polyatomic Logics and Generalised Blok-Esakia Theory with Applications to Orthologic and KTB
Rodrigo Nicolau Almeida

Abstract:
This thesis presents a study of translations, special “hybrid” logical systems developed on the basis of these translations, and general Blok-Esakia theory. This is done on two levels: the development of a theoretical framework for analysing such questions, as well as an analysis of the special case of orthologic.
On the first front, inspired by the Kolmogorov Double Negation Translation, and DNA-Logics, we develop a general notion of “Polyatomic Logics”, which can be studied for a wide class of translations. These logics serve as “hybrids”, between the two logical systems being translated, which can assist in applications, as well as in the study of the interrelations between the translated systems. We develop the basic theory of Polyatomic Logics, and prove algebraic completeness, and Birkhoff-style definability theorems of such systems.
We then develop a theory connecting such logics with “generalised modal companions” – abstracting from the classic Blok-Esakia isomorphism, and taken to mean a strong and property-preserving connection between the extensions of two logical systems. This is contrasted with the famous Gödel-McKinsey-Tarski situation, where we show that many of the motivating results of that theory can be recovered for a class of translations we call “sober translations”. Our main contribution in this respect is the introduction of the notion of a “Polyatomic Blok-Esakia isomorphism”, which is shown to hold for any sober translation, and which provides a new natural correspondence between logical systems.
As a case study, we provide an analysis of the logic of ortholattices, and the Goldblatt translation of Orthologic into KTB modal logic. Our results show that many natural invariance conditons, including the Polyatomic Blok-Esakia introduced, fail for this setting. We undertake a study of the reasons for this failure, and analyse whether restricted versions of it might hold. With this goal in mind, we introduce a new duality between a subcategory of the category of ortholattices, and a subcategory of the category of orthospaces. This representation is shown to have desirable category-theoretic properties, which we use to identify appropriate expansions of orthologic and KTB. With these tools, we prove the existence of a Polyatomic Blok-Esakia isomorphism between “Orthoimplicative Logic” and “Sober KTB”.