Full Idea
Logical necessity is a genus with two species. For classical logic the truth-related notion of validity and the proof-related notion of demonstrability, coincide - but they are distinct concept. In some logics they come apart, in intension and extension.
Gist of Idea
Logical necessity has two sides - validity and demonstrability - which coincide in classical logic
Source
John P. Burgess (Philosophical Logic [2009], 3.3)
Book Reference
Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.46
A Reaction
They coincide in classical logic because it is sound and complete. This strikes me as the correct approach to logical necessity, tying it to the actual nature of logic, rather than some handwavy notion of just 'true in all possible worlds'.
Related Idea
Idea 15418 Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency [Burgess]