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By "intermediate logic" I mean a (non-trivial) propositional logic at least as strong as intuitionistic logic whose set of theorems is closed under modus ponens and closed under substitution of propositional letters with formulas. The complete lattice of intermediate logics is ordered by logical strength.

By "atom in the complete lattice of intermediate logics" I mean an element of that lattice which is strictly stronger than intuitionistic logic but is not strictly stronger than another logic which is itself strictly stronger than intuitionistic logic.

RichardBerry
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  • How did this question arise? – Shaun Dec 13 '22 at 22:46
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    @Shaun I'm interested in a Kripke semantics for intuitionistic logic, but with most of the models thrown out, so that the ones that are left have transitive models of ZFC with urelements (where the domain is some inaccessible rank) for worlds, and accessibility corresponds to the inclusion relation between these structures. I know some things about consequence defined over these models, but I wanted a systematic way to narrow things down, so I figured that it would be nice if I could find counterexamples to theorems of some atom(s) in the lattice of intermediate logics. – RichardBerry Dec 13 '22 at 23:15
  • There are no atoms, see https://mathoverflow.net/a/467648 . – Emil Jeřábek Mar 24 '24 at 15:29

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As mentioned in the comments, this question has been answered by Emil Jeřábek on MathOverflow quoted below:

There are no atoms.

Assume for contradiction that $L$ is an atom. Since $L$ strictly contains IPC, there is a finite rooted Kripke frame $F$ that does not validate $L$, thus $L$ proves the Jankov–De Jongh frame formula for $F$, let me denote it $\beta^\sharp(F)$. Let $G$ be a finite rooted frame that properly contains $F$ as a generated subframe. Then $L\supseteq\mathsf{IPC}+\beta^\sharp(F)\supsetneq\mathsf{IPC}+\beta^\sharp(G)\supsetneq\mathsf{IPC}$. (The indicated inclusions are strict because $F$ is a model of $\mathsf{IPC}+\beta^\sharp(G)$ that does not validate $\beta^\sharp(F)$, and $G$ is a model of $\mathsf{IPC}$ that does not validate $\beta^\sharp(G)$.)

FD_bfa
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