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arXiv · 2004.00989

Lattices of Intermediate Theories via Ruitenburg's Theorem

Gianluca Grilletti·Davide Emilio Quadrellaro
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Abstract

For every univariate formula $χ$ we introduce a lattices of intermediate theories: the lattice of $χ$-logics. The key idea to define chi-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.

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Gianluca Grilletti, Davide Emilio Quadrellaro. 2020-04-02. Lattices of Intermediate Theories via Ruitenburg's Theorem. https://doi.org/10.1007/978-3-030-98479-3_15

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