A quick guided tour to the modal logic S4.2.

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 is modelled by T-4, the four-dimensional Minkowskian geometry forming the basis of Einstein's theory of special relativity. More recently, J. D. Hamkins and B. Lowe (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 rectangle phi is interpreted as 'phi 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.