There are many situations in logic, theoretical computer science, and category theory where two binary operations—one thought of as a (tensor) “product”, the other a “sum”—play a key role, such as in distributive categories and in -autonomous categories. (One can regard these as essentially the AND/OR of traditional logic and the TIMES/PAR of (multiplicative) linear logic, respectively.) In the latter example, however, the distributivity one often finds is conspicuously absent: in this paper we study a “linearisation” of distributivity that is present in this context. We show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules), and show how it can be strengthened in two natural ways, one to generate full distributivity, and the other to generate -autonomous categories.