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.