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 they all enjoy a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic extensions of each of these logics without the deductive interpolation property.

翻译：我们证明，带有交换性的全Lambek演算存在连续统多个具有演绎插值性质的公理化扩张。进一步地，我们将该结果推广至经典及直觉主义线性逻辑，以及它们的乘法-加法片段。我们展示的上述逻辑均不具备Craig插值性质，但证明它们都满足一种受保护的Craig插值形式。此外，我们还构造了这些逻辑中不存在演绎插值性质的连续统多个公理化扩张。