A similar strategy can be followed within typelogical grammar (Moortgat 1996, Kurtonina 1995).  The type language is extended with a pair of unary operators which we will write ◊,□; formulas now are built out of atomic formulas p with the unary and binary type-forming operations.  As with the binary vocabulary, we start with a minimal logic for ◊,□; facilities for structural control can then be added in the form of extra postulates. The truth conditions below characterize the control operators ◊ and □ as inverse duals with respect to a binary accessibility relation R◊. This interpretation turns them into a residuated pair, just like composition and the left and right division operations.