The SEP article on modal logic states the necessitation rule for system K (after Saul Kripke):
Necessitation Rule: If A is a theorem of K, then so is □A.
The wikipedia article on modal logic uses a more careful formulation:
Many modal logics, known collectively as normal modal logics, include the following rule and axiom:
- N, Necessitation Rule: If p is a theorem (of any system invoking N), then □p is likewise a theorem.
- K, Distribution Axiom: □(p→q) → (□p→□q).
The weakest normal modal logic, named K in honor of Saul Kripke, is simply the propositional calculus augmented by □, the rule N, and the axiom K.
For the SEP formulation, if we add the axiom □A→A to system K in order to get system M, it seems like □(□A→A) wouldn't be a theorem of system M. The wikipedia formulation on the other hand makes it clear that □(□A→A) is a theorem of system M. But when we are using system K or M and have a premise p, we certainly don't want to conclude □p from the necessitation rule.
But how to make it clear whether the necessitation rule should be applicable to axiom schemes concerning the inner structure of proposition? Consider the axiom scheme for transitivity of equality (p=q)∧(q=r)⇒(p=r) in a formal system operating on equations between terms (p=q for terms p and q) as basic propositions. Can we deduce □((p=q)∧(q=r)⇒(p=r)) from the necessitation rule, or do we have to add it explicitly as an additional axiom scheme? In case we can deduce it, what is the difference in terms of semantics between a theorem from an axiom scheme (for which the necessitation rule applies) and a deduction from a premise (for which the necessitation rule doesn't apply)?