Generalizations

Abstract

In this chapter we ask to what extent our results transfer to logics weaker than normal. We investigate this question only for our main results and in cases where a transfer is easily possible, and leave more comprehensive investigations to further studies. The weakest kind of a modal logic in which the modal operator represents some operation which applies to a proposition is classical modal logic: the only genuin modal principle which is required in the minimal system of propositional classical logic, C ^a 0, is the equivalence rule (ER): A↔B/□A↔□B (cf. Segerberg 1971, ch. 1; Chellas 1980, part III). Investigations in classical modal predicate logics are rare (cf. Gabbay 1976). The minimal classical predicate logic of type 1, C ^a 1, is obtained from C ^a 0 by adding axioms and rules of nonmodal predicate logic (cf. ch. 2.4.1: ∀13 and ∀R). The Barcan formula does not belong to the minimal classical 1-logic because the assumption of constant domain and rigid designators in neighbourhood semantics (see below) does not imply (aBF) nor its converse. Three sucessive strengthenings of classical modal logics are of particular importance: the monotonic, the regular, and finally the normal ones. Here are the basic definitions.