A Logic of Contingency with a Propositional Constant

The paper aims at showing that the problem of defining necessity in terms of contingency in weak normal systems may receive an answer in a contingency system K∆τ which is K∆ extended with an axiom for a propositional constant τ. It is proved by semantic tools that the fragment of K∆τ containing two necessity operators  and O is a system of a bimodal logic KD2 with deontic properties. §1. A well-known result proved by M.J.Cresswell in [4] states that the modal operators for necessity  and for con-contingency ∆ turn out to be interdefinable in every system S such that KT ⊆ S thanks to the two equivalences 1) ∆A ≡ A v ¬A and 2) A ≡ ∆A & A but they are not interdefinable, in the mentioned way or in any other possible way, in weaker normal systems such as, for instance, the deontic system KD. In Pizzi [15] it is shown that if contingency systems are extended with propositional quantifiers the definability problem receives a different answer. For instance, let K∆ be a system axiomatized in the following way (where ∇A = Df ¬∆A):