Actuality, Tableaux, and Two-Dimensional Modal Logics

The Logic of Sequence Frames

“…It is very natural, as will be seen below, to define indexed tableau systems for following the style of Melvin Fitting’s prefixed tableaux for modal logics 16 . In fact, generalizations of indexed tableau systems for a variety of two-dimensional modal logics appear in Lampert (2018) as well as in Gilbert (2016).…”

“…The goal of this paper is to generalize two-dimensional modal logics with actuality operators to any finite dimension . To be more precise, the logics investigated here are generalizations of the logic for epistemic two-dimensional semantics studied by Fritz (2014) under the name , which in a certain sense is the same logic as those investigated proof-theoretically in Restall (2012) and Lampert (2018)—although the logics in Lampert (2018) contain first-order quantifiers. The formal language defined here contains several modal operators, one for each dimension, including indexed boxes, , as well as actuality operators, .…”

“…That is, the actual world is held fixed while the counterfactual world varies. Additionally, two-dimensional modal languages are equipped with a diagonal necessity operator, interpreted intuitively as an apriority operator, which is sometimes taken as a primitive operator in the language (Restall, 2012; Fritz, 2014; Lampert, 2018), or defined in terms of other operators (Davies & Humberstone, 1980). Here we notate it as , and its corresponding accessibility relation will be , which in turn relates a pair to any pair such that .…”

“…In Lampert (2018), some of the philosophical ramifications of this general case for two-dimensional semantics were explored. In particular, the move from two to three-dimensions was motivated by the fact that some of the other solutions to the expressive deficit of the first-order two-dimensional language—namely, the ones involving the addition of the operators or —would result in the frames validating or , respectively.…”