I want to construct an example of a superintuitionistic logic which is a strict superset of $\operatorname{Int}$ and a strict subset of $\operatorname{Cl}$. I think that such logic can be obtained by adding formulas of the form $\lor_{1 \leq i < j \leq n+1}(p_i \leftrightarrow p_j)$. I can show that such logics are stronger than $\operatorname{Int}$ by constructing Kripke models in which such additional formulas are not satisficed. But are such extensions weaker than $\operatorname{Cl}$ and if yes, how I can prove it?
Thanks!
There are many examples of intermediate logics strictly between IPC and CPC (actually continuum many, by Jankov Theorem). If you are looking for examples, you can check logics such as KP, which is obtained from IPC by adding the axiom scheme $(\neg x\to(y\lor z)) \to ((\neg x\to y) \lor (\neg x \to z))$, or WEM, which is obtained by adding $\neg x \lor \neg\neg x$ or others.
Coming to your example, if I do understand correctly you are defining $\omega$-many extensions $L_n$ of IPC. Fix some $n$, then for some $p_i$ we have that $\bigvee_{i\neq j\leq n} (p_i\leftrightarrow p_j)\in L_n$. Since super intuitionistic logics are closed under uniform substitution, we obtain, by the map $p_j\mapsto \neg p_i$, that $\bigvee_{i\neq j\leq n} (p_i\leftrightarrow \neg p_i )\in L_n$ meaning that $L_n$ is inconsistent and thus $\text{CPC}\subseteq L_n$.