[FOM] Modal logics of contingency

[Joao Marcos]

Consider the unary modal operators L, for necessity, and M, for
possibility, and define the following operators:

C(A) = M(A) & M(~A)
D(A) = L(A) v L(~A)

So, C is the modal operator for contingency, and D is its dual.  Among
other things, it is immediate to check, for instance, that C(A) is
equivalent to C(~A) and to ~D(A), and that D(A) is equivalent to D(~A)
and to ~C(A).  It is equally obvious that D respects the G-rule, but
does not validate the K-axiom.

This much to fix some intended meaning for the above operators.  Now,
start from a modal logic which has only C (and D) as primitive
operators, alongside with the classical ones.  Call any such a logic a
*modal logic of contingency*.  Question:

-o- Were such logics already investigated?  References?

Obviously, given the above properties, there are no formulas depending
only on C-formulas and D-formulas which allow us to define L and M in
terms of them.  Question:

-o- Is there any modal logic of contingency in which L or M can be
introduced by some other kind of definition?

If we restrict our attention to canonical modal frames, a direct proof
of com-pleteness for such modal logics of contingency may present some
gearbox diffi-culties.  (In particular, notice from the above that they
will not in general constitute examples of normal modal logics.)
Question:

-o- Does anyone have an idea of how this proof could be done?  Or had we
better change the underlying semantical structures?

Comments and references welcome!


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