Actuality without possible-worlds: Ivlev-like versions of Crossley and Humberstone’s logics of actuality

Abstract

The notion of actuality is historically dependent on the notion of possible world. This paper contains a novel approach to logics of actuality, viz. we will develop an account of actuality that does not depend on possible worlds. We do so by utilizing the non-deterministic generalization of logic matrices known as non-deterministic matrices (Nmatrices, for short). By following the pioneering ideas of Ivlev, we propose a non-normal version of the well-known modal system of actuality $S5A$, introduced in 1977 by Crossley and Humberstone. Our system, called T45Am, is presented by means of a 6-valued Nmatrix. The 6 elements (truth-values) of this Nmatrix, as well as its multifunctions for the connectives, admit an intuitive interpretation in terms of swap structures. Soundness and completeness of a Hilbert calculus for T45Am is obtained w.r.t. its Nmatrix. One interesting feature of our system is that the actuality operator is hyperintensional, allowing us to analyze questions about actuality under a new formal perspective.