A few days ago I stumbled upon [this question][1] on MS. The question is: Does the lattice of intermediate logics have an atom, *i.e*. an element that is strictly stronger than IPC but not strictly stronger than another logic that is itself strictly stronger than IPC? The question has not been answered on MS and I didn't find an answer in any other source too.

Thinking about the question myself I had the idea to find a systematic way to weaken the sentences an intermediate logic includes in addition to IPC but I couldn't find any way to do so repeatedly in a way that doesn't result in either intuitionistic tautologies or additional sentences intuitionistically equivalent to the sentences I began with. I would appreciate any answer or any proposal about possible ways to go forward with solving the problem.


  [1]: https://math.stackexchange.com/questions/4598211/is-it-known-whether-there-are-atoms-in-the-complete-lattice-of-intermediate-logi