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(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?

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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.

Mauro ALLEGRANZA
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  • Thanks Mauro for your reply. What about this post, which troubles me a bit: http://philosophy.stackexchange.com/questions/10480/how-to-prove-possibly-p-if-necessarily-p-in-kripke-modal-logic/10483#10483 –  Mar 09 '16 at 14:41
  • @student - as stated in SEP (see link): "The system **K** is too weak to provide an adequate account of necessity. The following axiom [i.e.(M)] is not provable in **K**, but it is clearly desirable. Notice that (M) would be incorrect were □ to be read ‘it ought to be that’, or ‘it was the case that’. So the presence of axiom (M) distinguishes logics for necessity from other logics in the modal family. A basic modal logic **M** results from adding (M) to **K**. (Some authors call this system **T**.)" – Mauro ALLEGRANZA Mar 09 '16 at 14:50
  • So, if I understand correctly, necessity implies possibility in T (and stronger systems like S4 or S5), but not in K. –  Mar 09 '16 at 15:10
  • @student - exactly. – Mauro ALLEGRANZA Mar 09 '16 at 15:13