You have normal additive (and/with, or/plus) and multiplicative operators (forall/times, nonsuch/par) as well as exponential ones (ofcourse/ever, whynot/sometimes).
The final two are the linear part as they cause the logic to be positional.
The closest CS equivalent are categorial grammars (including combinatorial ones), the difference is that pure linear logic is commutative while these are not in general.
It has several interpretations. One is close to game theory, one is related to ownership, one is related to probability, one is related to interfaces. These are all the same idea.
