118 research outputs found
Lattices of Intermediate Theories via Ruitenburg's Theorem
For every univariate formula we introduce a lattices of intermediate
theories: the lattice of -logics. The key idea to define chi-logics is to
interpret atomic propositions as fixpoints of the formula , which can
be characterised syntactically using Ruitenburg's theorem. We develop an
algebraic duality between the lattice of -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
An Algebraic Approach to Inquisitive and DNA-Logics
This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA -varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff's classic variety theorems. We also introduce locally finite DNA -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal omega + 1 and give an axiomatisation of these logics via Jankov DNA -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].(1)Peer reviewe
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
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
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