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## The almost necessity operator

We introduce a modal operator of weak necessity, inspired by the canonical model construction for the non-contingency logic developed by Humberstone and Kuhn in 1995. This operator, when applied to a proposition, means that all consequences of a given proposition are non-contingent. We show that, although the weak necessity has many properties inherent to normal modal operators, unfortunately, it is not a normal itself: not all principles of the minimal normal modal logic are valid for it. We study the expressive power of the modal language with weak necessity: for some formulas of this language, equivalent first-order or second-order conditions are found; every class of frames definable in this language is shown to contain the class of functional frames and, dually, there is only one maximal logic of weak necessity. The problem of finding an axiomatization (finitary and infinitary) of the logic of weak necessity is left open