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Consider Propositional Lax Logic ($PLL$)

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.

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.

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    $\begingroup$ (1) Clearly not, as the logic is contained in the logic axiomatized by classical logic and $\bigcirc A$ for all formulas $A$. $\endgroup$
    – Emil Jeřábek
    Commented Mar 29, 2018 at 16:54
  • $\begingroup$ Sorry, I didn't understand your response. You are saying that we clearly do not get the modal collapse with $PLL$ plus excluded middle without $\neg \bigcirc \bot$, since $\bigcirc A$ holds for all formuals $A$? $\endgroup$
    – user65526
    Commented Mar 29, 2018 at 17:01
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    $\begingroup$ Anyway, the argument from the linked page shows that even without assuming $\neg{\bigcirc}\bot$, you get the schema $\bigcirc A\leftrightarrow A\lor{\bigcirc}\bot$, which is hardly any beter than full modal collapse. $\endgroup$
    – Emil Jeřábek
    Commented Mar 29, 2018 at 17:02
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    $\begingroup$ No, that's not what I wrote. "Contained in" means contained in, not equal. $\endgroup$
    – Emil Jeřábek
    Commented Mar 29, 2018 at 17:05
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    $\begingroup$ Ok, but what do you mean by saying "clearly not"? Clearly not what? And why clearly not? $\endgroup$
    – user65526
    Commented Mar 29, 2018 at 17:08

1 Answer 1

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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.

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  • $\begingroup$ I don't fully understand why you get the biconditional $\top \leftrightarrow \bigcirc N$. If we have $\bigcirc \bot$, we can derive $\bigcirc N$. But then how does $\top$ enter the picture? Also, the part of the proof where you say we either have $\neg \bigcirc \bot$ or $\bigcirc \bot$, does that come merely from the assumption the we have the law of the excluded middle? $\endgroup$
    – user65526
    Commented Mar 29, 2018 at 20:20
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    $\begingroup$ We could say that $\bigcirc$ "collapses" if $\bigcirc N$ is equivalent to something somehow trivial. But yeah, $\top\leftrightarrow\bigcirc N$ is just a long way of saying $\bigcirc N$. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 29, 2018 at 20:36
  • $\begingroup$ It's just occurred to me: one way of avoiding the modal collapse is to abandon the inference rule you call (@) in your answer. What do you think about that strategy? Then you could perhaps have the law of the excluded middle but not $\neg \bigcirc \bot$, and the logic wouldn't be forced to be Intuitionistic. But perhaps the inference rule (@) was chosen for well motivated reasons. $\endgroup$
    – user65526
    Commented Mar 29, 2018 at 20:51
  • $\begingroup$ Furthermore if we propose different inference rules, the matter doesn't improve: if we adopt (@_1) $$\frac{M\supset N}{M\supset\bigcirc N}\tag{@_1}$$, we can derive $\top \leftrightarrow \bigcirc N$, and if we add (@_2)$$\frac{M\supset N}{\bigcirc M\supset N}\tag{@_2}$$, we can derive $\top \leftrightarrow N$, by the same argument given above. $\endgroup$
    – user65526
    Commented Mar 30, 2018 at 11:14

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