Remarks on a Carnapian extension of S5 (Q2702742)

\({\mathbf C}\) is a modal logic on a language consisting in infinitely elementary formula letters, the usual truth functions and the modal operators \(\square\) and \(\diamondsuit\); formulas are evaluated with respect to `interpretations', sets of elementary letters, with \(\square A\) holding at \(I\) just in case \(A\) holds at every interpretation \(J\), and dually for \(\diamondsuit A\). \({\mathbf C}\) is the set of all valid formulas. This sounds a lot like S5 but it is not, for \({\mathbf C}\) is not closed under substitution for elementary letters, and there are valid formulas not contained in any standard modal logic, e.g., \(\diamondsuit p\), for any elementary letter \(p\). \({\mathbf C}\) is, though, an extension of S5 in its vocabulary. It is `Carnapian' inasmuch as its interpretations correspond to Carnap's `state descriptions'. This paper presents a historical survey of investigations of \({\mathbf C}\) and closely related logics. It also explores some of the fundamental properties of \({\mathbf C}\) and its semantics, especially their relations to S5, and the consequence relation for the logic, leading to a deduction theorem. In addition, connections are drawn to epistemic logic, especially the theory of stable sets, and nonmonotonic logics.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00030].