This post is related to a question I originally asked on Philosophy Stack Exchange
https://philosophy.stackexchange.com/questions/100055/infinitary-modal-logic
In the modal logic (say, ${\bf K}$) the following theorems hold:
(1) $\Box (p\wedge q)\leftrightarrow(\Box p\wedge\Box q)$
(2) $(\Box p\vee\Box q)\rightarrow\Box (p\vee q)$
(3) $\Diamond (p\vee q)\leftrightarrow (\Diamond p\vee\Diamond q)$
(4) $\Diamond (p\wedge q)\rightarrow (\Diamond p\wedge\Diamond q)$
where $\Box$ and $\Diamond$ are the necessity and possibility operators, respectively.
Do these theorems hold in infinitary modal logic, i.e. do we have
(5) $\Box (\bigwedge_{n\in\omega} P_n)\leftrightarrow (\bigwedge_{n\in\omega}\Box p_n)$
(6) $(\bigvee_{n\in\omega}\Box p_n)\rightarrow(\Box\bigvee_{n\in\omega}p_n)$
(7) $\Diamond (\bigvee_{n\in\omega}p_n)\leftrightarrow(\bigvee_{n\in\omega}\Diamond p_n)$
(8) $\Diamond (\bigwedge_{n\in\omega}p_n)\rightarrow (\bigwedge_{n\in\omega}\Diamond p_n)$?