Lattices of Intermediate Theories via Ruitenburg’s Theorem

Abstract

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.

We would like to thank Nick Bezhanishvili for comments and discussions on this work. Also, we would like to thank the two anonymous referees for several useful remarks and suggestions. The first author was supported by the European Research Council (ERC, grant agreement number 680220). The second author was supported by Research Funds of the University of Helsinki.

A Proof of Lemma 7

Proof

(Proof of Lemma 7). As shown in [21, 24], the following is a presentation of all the non-constant univariate intuitiornistic formulas modulo logical equivalence:

$$\begin{aligned} \beta _1 := p \qquad \beta _{n+1} := \alpha _n \vee \beta _n \qquad \alpha _1 := \lnot p \qquad \alpha _{n+1} := \alpha _n \rightarrow \beta _n. \end{aligned}$$

We consider the following two properties for a univariate formula $\phi$:

• 1. $\lnot \lnot \phi \equiv \top$.

• 2. If $\psi$ has property 1, then .

In particular, if $\phi$ has both properties then $\phi ^2 \equiv \top$, that is, the fix-point of $\phi$ is $\top$.

Firstly notice that

$$\begin{aligned} \alpha _5 = ((\lnot \lnot p) \rightarrow p\vee \lnot p) \rightarrow (\lnot p \vee \lnot \lnot p) \qquad \beta _5 = ((\lnot \lnot p) \rightarrow p\vee \lnot p) \rightarrow (\lnot \lnot p \rightarrow p) \end{aligned}$$

have both properties.

Moreover, we can show that, if $\alpha _n$ and $\beta _n$ have both properties, then this holds for $\alpha _{n+1}$ and $\beta _{n+1}$ too.

So, by induction all the formulas $\alpha _n, \beta _n$ with $n\ge 5$ have index at most 2 and fixpoint $\top$. As for the remaining formulas, one can easily show their fix-points are as follows:

This concludes the proof.    $\square$