Timeline for Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 30, 2018 at 17:58 | vote | accept | user65526 | ||
| Mar 30, 2018 at 11:14 | comment | added | user65526 | 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. | |
| Mar 29, 2018 at 20:51 | comment | added | user65526 | 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. | |
| Mar 29, 2018 at 20:36 | comment | added | Bjørn Kjos-Hanssen | 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 $\bigcirc N$. | |
| Mar 29, 2018 at 20:20 | comment | added | user65526 | 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? | |
| Mar 29, 2018 at 19:41 | history | answered | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |