A quick guided tour to the modal logic S4.2

Abstract

The normal modal system S4.2 is a useful tool in Epistemic Logic: it has been advocated as the 'correct' logic of knowledge by Lenzen (1979, Erkenntnis, 14, 33-56) and Stalnaker (2006, Philos. Stud., 128, 169-199) and it has been declared as the logic of justified true belief (JTB) by Voorbraak (1993, PhD Thesis). Goldblatt (1980, Stud. Logica, 39, 219-236) has proved that S4.2 is the temporal logic of relativistic spacetime, in particular it is the logic of the Diodorean modality when time ismodelled by T[double-struck]4, the four-dimensionalMinkowskian geometry forming the basis of Einstein's theory of special relativity.More recently, J. D. Hamkins and B. Löwe (2008, Trans. Am. Math. Soc., 360, 1793-1817) have employed S4.2 in the metamathematics of Set Theory and in particular in the study of Paul Cohen's method of forcing: S4.2 is exactly the modal logic of forcing, the collection of valid principles emerging when □ϕ is interpreted as 'ϕ is true in every forcing extension'. In this survey paper we provide a guided tour to S4.2 focusing on its epistemic interpretation and its model-theoretic characterizations. We briefly review the results of Lenzen (1979, Erkenntnis, 14, 33-56) and Stalnaker (2006, Philos. Stud., 128, 169-199) and provide an epistemic interpretation of S4.2 models. We survey the frame completeness results for S4.2, contributing also a new simple proof that it is determined by the class of partial pre-orders with a final cluster. En route, we provide a quick glimpse to other semantics for this modal system. Our goal is to provide a useful guide to researchers interested in the basic facts about the modal logic S4.2.