A family of neighborhood contingency logics. (English) Zbl 1444.03052

A contingency logic is a kind of modal logic to study the notion of contingency (or noncontingency) by formalizing it as a unary logical operator. In this paper, as seems to be rather common in the literature, the noncontingency denoted by \(\Delta\) is employed as the primitive and interpreted in terms of the necessity as for any formula \(\varphi\) it is noncontingent that \(\varphi\) iff it is necessary that \(\varphi\) or it is necessary that not \(\varphi\). Thus any formula \(\varphi\) with \(\Delta\) can be translated into a usual modal formula \(\varphi^*\) by the replacement, which then gives rise to the noncontingency logic \(L\Delta\) of given modal logic \(L\) by considering the inverse image of the translation. Since the interpretation also allows us to develop a semantical analysis of \(\Delta\) in a natural way, the semantical characterization of \(L\Delta\) follows from that of \(L\) straightforwardly. For example, \(K\Delta\) is the noncontingency logic characterized by all Kripke frames (under the interpretation of \(\Delta\) given in the obvious way) where \(K\) is the least normal modal logic as usual. However, the axiomatization of \(L\Delta\) does not follow from that of \(L\) thus immediately because it is not possible to characterize the necessity by \(\Delta\) in general. So the axiomatization is one of the problems of (non)contingency logics which have been studied by the author, Cresswell, Humberstone, Kuhn and others, where Humberstone suggested to modify the definition of accessibility relation in the canonical Kripke model construction (due to the so-called Henkin method for the normal modal logic). In fact, the axiomatization of \(K\Delta\) has been established by S. T. Kuhn [Notre Dame J. Formal Logic 36, No. 2, 230–234 (1995; Zbl 0833.03005)] according to a similar canonical Kripke model construction, while the definition of accessibility is slightly different in its appearance from that suggested by I. L. Humberstone [Notre Dame J. Formal Logic 36, No. 2, 214–229 (1995; Zbl 0833.03004)]. Thus the study has been developed mainly in relation to normal modal logics. But Z. Bakhtiari et al. [Lect. Notes Comput. Sci. 10119, 48–63 (2017; Zbl 1485.03063)] proposed the axiomatization problem of noncontingency logic induced by a modal logic weaker than \(K\), in particular, those of \(E\Delta\) and \(M\Delta\) where \(E (M)\) is the least regular (monotone) modal logic. In this paper, the author gives the answer to these two cases as well as to some others weaker than \(K\Delta\). The logics taken up in this paper are first characterized by means of neighbourhood semantics according to four additional conditions on neighbourhood system, which are roughly the complimentation condition characteristic to the noncontingency and the superset condition (i.e., closure under supersets) to build up the normality in conjunction with other two conditions as usual. So, a simple calculation gives rise to 16 combinations, among which the author shows the axiomatization for 10 cases including the above mentioned two. The principal issue is the completeness proof according to the canonical neighbourhood model construction, in which for the cases without the superset condition the construction can be carried out exactly in the similar way as usual for regular modal logic but for the cases with the superset condition such a straight definition does not work and is to be modified by a similar idea as that of Kuhn [loc. cit.] introduced in the Kripke semantics. Then, the author analyzes the definitions of Kuhn [loc. cit.] and Humberstone [loc. cit.] for more details and shows that they are actually the same even in the generalized setting of neighbourhood semantics. Among the 6 cases for which the axiomatization problem has been left open, two couples are shown to be the same (exactly, \(MZ\Delta = RZ\Delta\) and \(EMNZ\Delta = KZ\Delta\)) with simple axiomatizations respectively as a matter of fact, even though they are dealt with as different (cf. Figure 1). There are some gaps in the argument of 3 of Remark 24.


03B45 Modal logic (including the logic of norms)
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