Lattices of Intermediate Theories via Ruitenburg's Theorem

For every univariate formula $\chi$ we introduce a lattices of intermediate theories: the lattice of $\chi$-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula $\chi^2$, which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of $\chi$-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail

Lattices of Intermediate Theories via Ruitenburg's Theorem

For every univariate formula chi (i.e., containing at most one atomic proposition) we introduce a lattice of intermediate theories: the lattice of chi-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula chi(2), which can be characterised syntactically using Ruitenburg's theorem. We show that chi-logics form a lattice, dually isomorphic to a special class of varieties of Heyting algebras. This approach allows us to build and describe five distinct lattices-corresponding to the possible fixpoints of univariate formulas-among which the lattice of negative variants of intermediate logics.Peer reviewe

On intermediate inquisitive and dependence logics : An algebraic study

Peer reviewe

Dualities in modal logic

Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics

Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it

Polyatomic Logics and Generalised Blok-Esakia Theory

This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure

Uniform interpolation and coherence

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices

Admissible Bases Via Stable Canonical Rules

We establish the dichotomy property for stable canonical multi-conclusionrules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics