Possible-worlds semantics for modal notions conceived as predicates (Q1810820)

This is a paper about treating modal concepts as predicates. It is well known that such a syntactical treatment may lead to inconsistency. The authors consider the language of arithmetic supplemented with a one-place necessity predicate that applies to the numerical codes of formulas. Arithmetic formulas are interpreted by the standard model at possible worlds. If \(n\) is the code of \(A\), then \(\square n\) is true at a world \(w\) iff \(A\) is true at every world accessible from \(w\). The problem addressed in the paper is called the characterization problem: identifying classes of Kripke frames that support possible worlds models (i.e., do not lead to inconsistency) for necessity treated as a predicate. Necessary and sufficient conditions for a positive solution to the characterization problem for a certain class of transitive frames and completeness results for the analogues of the modal systems K and K4 are presented. The authors make a very strong philosophical claim, namely that the familiar operator approach ``fails at its main application in philosophical logic: it does not provide an illuminating analysis for necessity, knowledge, obligation and so on. For it does not allow for the formalisation of the most common philosophical claims such as `All laws of physics are necessary' or `There are true but unknowable sentences' ''.