Necessity and contingency (Q1116939)

This paper explores conditions under which necessity, Lp, can be defined in terms of non-contingency, \(\Delta\) p (where \(\Delta\) \(p\equiv Lp\vee L\sim p\) must always be true). For systems containing T a defnition is easy, viz. \(L\alpha\) \(=_{df}\alpha \&\Delta \alpha\), because of the axiom Lp\(\supset p\). Necessity is also easily defined in the verum system \(=K+Lp,\) since \(\vdash_{Ver} Lp\equiv (p\supset p)\). In most other systems not containing T, necessity is not definable in terms of non- contingency. Nevertheless, there is at least one in which it is, namely \(S=K+Lp\equiv (\Delta p\&(p\equiv \Delta \Delta p)).\)