Chapter 9 Modal Language, Syntax, and Semantics - David W. Agler

There are four quantifier rules4 ¬P, 0 3 ¬∨D5 ¬Q, 0 3 ¬∨D10.2. Negated Box and Diamond DecompositionThe first two modal tree decomposition rules concern propositions where a negation operatesupon a modal operator.Negated Diamond Decomposition (¬◊D)¬◊P, i □¬P, iNegated Box Decomposition (¬□D)¬□P, i ◊¬P, iNegated diamond decomposition and negated box decomposition.1 ¬◊(P∧R), 0 P2 ¬□(P→R), 1 P3 □¬(P∧R), 0 1¬◊D4 ◊¬(P→R), 1 2¬□D10.3. Box and Diamond DecompositionThe second two modal tree decomposition rules concern propositions where the main operator isa modal operator.Diamond Decomposition (◊D)◊P, I iR jP, jwhere j is a world that does not previouslyoccur in the branchBox Decomposition (□D)□P, iiR j~P, jwhere w i Rw j occurs somewhere in the branchIn the case of diamond decompositionIn the case of box decomposition18