 It is possible to provide a canonical treatment of additive connectives, even with first-order quantification  (Heijltjes et al. 2018).  Proof nets for formulas containing both multiplicative and additive connectives have various technical presentations, none of which appears canonical and satisfactory. Their treatment in proof-net-like proof systems is currently a topic of active research. In particular, see (Hughes and van Glabbeek 2005) and  (Hughes and Heijltjes 2016). 