Counting Coordination Categorially

Certain categorial calculi exhibit a property that is known as count invariance (Van Benthem 1986). In these grammars, it is true that if a proposition Y ⇒ z is derivable, the string to the left of ⇒ and the type to its right share the results of a particular way of counting occurrences of basic types. This count protocol discriminates between positive occurences (heads) and negative occurrences (arguments) of basic types; for each basic type x and for each string of types S it yields an integer representing the occurrences of x in S.