I've started studying A New Introduction to Modal Logic by Hughes and Cresswell, and have just gotten to the definition of the necessity operator $\text{L}$. In the book (page 15), they mention that $\text{L}(p\supset q)$ should not be confused with $p\supset \text{L}q$.
However, if you decide to take $\text{L}p$ to mean that $p$ is true, then wouldn't the meanings of $\text{L}(p\supset q)$ and $p\supset \text{L}q$ coincide? The only way I can imagine them being different is if necessarily $p$ is taken instead to mean that $p$ ought to be.
But even then it sounds like making a statement about $\text{M}p$ rather than $\text{L}p$. The book is pretty wordy so some of the explanations are tough to read, any help understanding this would be greatly appreciated!
A link ( by the author of Possible Worlds) :
https://www.sfu.ca/~swartz/modal_fallacy.htm#cannot
Let's admit that being a pianist logically implies being a musician.
It is true that : necesssarily ( John is a pianist $\implies$ John is a musician).
The modal operator applies neither to the anttecedent nor to the consequent, but to the whole conditional. So to say, it applies to the arrow, not to the propositions that are related by the implication relation.
IF John is a pianist, THEN John is necessarily a musician.
Now we are dealing with a false statement: the fact that John is a pianist does not imply that there is no possible world/ situation/ scenario in which John is not a musician.
Even though John is a musician ( due to the fact he is a pianist) he could perfectly not have practiced such an activity.