Full Idea
'Strict' logical necessity is true by the laws of logic alone; 'narrow' logical necessity is true by the laws of logic plus definitions of non-logical terms; 'broad' logical necessity is true in every possible world where the laws of logic hold.
Gist of Idea
Logical necessity can be 'strict' (laws), or 'narrow' (laws and definitions), or 'broad' (all logical worlds)
Source
E.J. Lowe (The Possibility of Metaphysics [1998], 1.4)
Book Reference
Lowe,E.J.: 'The Possibility of Metaphysics' [OUP 2001], p.14
A Reaction
Lowe then says the third is close to 'metaphysical' necessity. I am unable to distinguish the third from the first. You can't claim that a logical implication holds in this world, but not in another possible world which has the same rules of implication.