# Necessity in relation to possibility

(1) Does necessity (materially) imply possibility?

(2) Does possibility (materially) imply necessity?

From a logical point of view:

If by material implication (A -> B), we mean (-A or B), then it seems that necessity does not imply possibility. For if A denotes the necessary and B the possible, then -A is impossible, and the disjunction of the impossible with the possible is possible (but not necessary).

On the other hand, if A denotes the possible, then -A is not necessary, and the disjunction of the not necessary with the necessary is necessary. Hence, the possible implies the necessary.

Yet, from a pure philosophical point of view, the opposite implications seem to hold, that is to say, the necessary implies the possible, and the possible does not imply the necessary. How is that (so to speak) possible?

An axiom of Modal Logic (at least: of some ML) is:

(M) □A → A : "whatever is necessary is the case".

Thus, with ~A in place of A and using contraposition:

~~A → ~□~A.

With double negation and the definition of the operator ◊ (‘It is possible that’) in terms of □ (‘It is necessary that’): ◊A := ~□~A, we conclude with:

A → ◊A.

Now we can apply transitivity to (M) and the last formula to get:

(1) □A → ◊A.