Modal logic: justification of the rule of necessitation I'm studying some lecture notes about modal logic and I'm now reading a paragraph which goes as follows: 
The rules of inference of system K are modus ponens and the rule of necessitation:
NEC: if A is a theorem, then ◻A is a theorem.
This rule is legitimate in that it preserves truth in a world in any model: if A is true in a world w, it must be true in every world accessible from w, so ◻A is true in w.
I don't quite understand the last line ("This rule is legitimate..."). Do you have any guesses? Maybe the correct phrasing should have been: if A is true in a world w and it is a theorem...

You write "This rule is legitimate in that it preserves truth in a world in any model".
This is not true. Necessitation preserves truth in a model but not truth in a world in a model. It is easy to see that what is true in a world need not be necessarily so in that world, hence adding the box operator may not preserve truth.
