Infinitary Expressibility of Necessity in Terms of Contingency

This paper consists of two parts. In the first part, which is rather techni-cal, we present axiomatizations of versions of some well-known modal logics in the language with non-contingency operator as the sole modal primitive symbol. The second part, having a certain philosophical flavour, is devoted to the old question of possibility to define the necessity operator in terms of the contingency operator. In general, a notion α is said to be definable in terms of a notion β if there exists an expression A = A(β) containing β such that A is equal (or equivalent) to α. From this point of view, necessity is not definable in terms of contingency (cf. [2, 3]). However, this understanding is rather confined, by the author’s opinion, and the more appropriate one is to say that A behaves like α, or A subjects the same laws as α. This approach proves to be successful to give a new, affirmative, answer to the above question. I. The non-contingency operator B is defined in terms of the necessity operator by putting BA: = A ∨¬A. This induces a translation o