Establish whether the proposed system is legitimately "linear"

Ascertain whether the proposed "very linear logic" qualifies as a linear logic in the proof-theoretic sense by constructing a proof system that justifies treating multiplication and related operations as linear connectives.

Background

The authors develop a semantics with multiple families of connectives and argue for a linear reading of certain operations. However, they acknowledge not having a proof system that would justify the linear status of these connectives, creating uncertainty about whether the logic should be called "linear."

They suggest it is likely that such a justification could be made, referencing developments in Lawvere-style logarithmic logics, but leave the question unresolved pending a suitable proof theory.