  Many models of linear logic proofs have been proposed; we consider that, to date, the simplest and most intuitive construction is those based on the so-called “relational semantics” or “Kripke-style semantics”, where formulas are interpreted as multisets, one-sided sequents are interpreted as tuples of multisets, and proofs are interpreted as relations over the interpretation of sequents. If one wants to give a purely set-theoretic semantics, without resorting to multisets, it is possible to do it by means of coherence spaces, sets equipped with a special coherence relation, as originally shown by in  Girard 1987.  There are interesting category theoretical models of linear logic, such as the *-autonomous categories  (Barr 1991)  and hypercoherences  (Ehrhard 1993).