  The Lambek-Grishin calculus is a symmetric extension of the Lambek calculus:
in addition to the residuated family of product, left and right division
operations of Lambek's original calculus, one also considers a family of
coproduct, right and left difference operations, related to the former by an
arrow-reversing duality. Communication between the two families is implemented
in terms of linear distributivity principles. The aim of this paper is to
complement the symmetry between (dual) residuated type-forming operations with
an orthogonal opposition that contrasts residuated and Galois connected
operations. Whereas the (dual) residuated operations are monotone, the Galois
connected operations (and their duals) are antitone. We discuss the algebraic
properties of the (dual) Galois connected operations, and generalize the
(co)product distributivity principles to include the negative operations. We
give a continuation-passing-style translation for the new type-forming
operations, and discuss some linguistic applications.
