Consider Propositional Lax Logic ($PLL$)


* <https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf>


The Hilbert system of $PLL$ takes as axiom schemata all theorems of (or a complete set of axioms for) the Intuitionistic propositional calculus plus the modal axiom schemata $\bigcirc R, \bigcirc M, \bigcirc S$ below. The inference rules are Modus Ponens and the rule "from $M \supset N$ infer $\bigcirc M \supset \bigcirc N$": 


$$\text{Axiom} \bigcirc R: \hspace{0.5cm}M \supset \bigcirc M$$
$$\text{Axiom} \bigcirc M: \hspace{0.8cm} (\thinspace \bigcirc \bigcirc M\thinspace) \supset \bigcirc M$$
$$\text{Axiom} \bigcirc S: \hspace{0.8cm}(\bigcirc M \land \bigcirc N) \supset \bigcirc(M \land N)$$


A proof of the modal collapse (We can derive both $\bigcirc M \supset M$ and $M \supset \bigcirc M$) of $PLL$ obtained by adding the Excluded Middle (EM) and $\neg \bigcirc false$ was given in this answer: [Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic](https://mathoverflow.net/questions/296440/modal-collapse-upon-addition-of-the-law-of-the-excluded-middle-to-an-intuitionis). 


The modal collapse was claimed in <https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf>:



> 
> "...if we add the axiom of the Excluded Middle (EM) and $\neg \bigcirc false$ which is valid for both $\Diamond$ and $\Box$ to the modal system $\bigcirc R, \bigcirc M, \bigcirc S$ then $\bigcirc$ becomes trivial. We can derive both $\bigcirc M \supset M$ and $M \supset \bigcirc M$
>  In other words there is no classical Kripke semantics for $\bigcirc$." (p.4, para 1 of the above article)
> 
> 
> 


However, I was wondering: 


(1) Can we get the modal collapse if we only assume the law of the excluded middle, and we don't assume $¬◯⊥$?


(2) If we are able to avoid the modal collapse by abandoning $¬◯⊥$, would abandoning $¬◯⊥$ be problematic in other ways?




---


A response was given below which I do not understand:


"(1) Clearly not, as the logic is contained in the logic axiomatized by classical logic and ◯A for all formulas A."


I would also be grateful for any clarifications of this comment.



Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?
Notice that by the inference rule
$$\frac{M\supset N}{\bigcirc M\supset\bigcirc N}\tag{@}$$
we have
$$\frac{\bot\supset N}{\bigcirc \bot\supset\bigcirc N}$$
But $\bot\supset N$ always holds. So either we have $\neg\bigcirc\bot$, the case already covered in the linked answer; or we have $\bigcirc\bot$, in which case all statements of the form $\bigcirc N$ are true. In this case $\bigcirc$ "collapses" in a different way: $\top\leftrightarrow\bigcirc N$ instead of $N\leftrightarrow\bigcirc N$.


Moreover, the case $\top\leftrightarrow \bigcirc N$ does happen, as @EmilJeřábek pointed out, because all the other axioms are consistent with this possibility.


Conclusion:



> 
> either way, adding Law of Excluded Middle to PLL is not a good idea.
> 
> 
> 


