We saw that in the minimal logic NL completeness with respect to the frame semantics for composition and its residuals doesn't impose restrictions on the interpretation of the merge relation R⊗. The same holds for R◊ in the pure residuation logic of ◊,□. From the residuation laws, one easily derives monotonicity of the control operators (A ⊢ B implies ◊A ⊢ ◊B and □A ⊢ □B); their compositions satisfy ◊□A ⊢ A and A ⊢ □◊A. These properties can be put to use in refining lexical type assignment so that selectional dependencies are taken into account.  Compare the effect of an assignment A/B versus A/◊□B. The former will produce an expression of type A in composition both with expressions of type B and ◊□B, the latter only with the more specific of these two, ◊□B. An expression typed as □◊B will resist composition with either A/B or A/◊□B. Bernardi and Szabolcsi (2008) present accounts of licensing of polarity-sensitive expressions and scope construal based on (a generalization of) this modalisation strategy.