We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that the exhibited extensions of classical and intuitionistic linear logic all enjoy a guarded form of Craig interpolation. We also give continuum-many axiomatic extensions of classical linear logic without the deductive interpolation property.
Fussner et al. (2024) studied this question.
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