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 χlogics 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.

Algebraizable Weak Logics

“…See e.g. [18] for a proof of this fact. In particular there is some n < ω such that ρ n (x) ⊣⊢ IPC χ where χ ∈ {⊤, ¬¬x, x ⊗ ¬x, x, ⊥}.…”