Lattices of Intermediate Theories via Ruitenburg's Theorem

Abstract

For every univariate formula χ we introduce a lattices of intermediate theories: the lattice of χ-logics. The key idea to define χ-logics is to interpret atomic propositions as fixpoints of the formula χ 2 , which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of χ-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.

An Algebraic Approach to Inquisitive and -Logics

“…This allows to introduce the -variant of an intermediate logic L as L = {ϕ ∈ L P : ϕ[ n (p)/p] ∈ L} and to generalize our study of DNA-logics to arbitrary -variants. We refer the reader to the upcoming [24].…”