$□p$, $□q$ therefore, $□(p→q)$
according to the K system of modal logic, the argument is invalid. I tried proving it using a truth tree, but all the branches unfortunately close, I don't know how to proceed.
$□p$, $□q$ therefore, $□(p→q)$
according to the K system of modal logic, the argument is invalid. I tried proving it using a truth tree, but all the branches unfortunately close, I don't know how to proceed.
Your claim that $\Box(p \rightarrow q)$ is not a logical consequence of $\{\Box p, \Box q\}$ in $K$ is ambiguous. It can be read as saying that $\Box(p \rightarrow q)$ is a not local consequence of $\{\Box p, \Box q\}$ and as saying that the formula is not a global consequence of the formula set. For present purposes, local consequence can be defined as preservation of satisfaction at every point from every $K$-model; global consequence means preservation of validity on every $K$-frame.
However, your claim is wrong under both of its interpretations. To see that the formula $\Box(p \rightarrow q)$ is a local consequence, note that it is a local consequence of $\Box q$ and that local consequence is monotonic. Analogous reasoning shows that the formula is a global consequence.