Yes, that's true. And there are various logical systems which hint at mutability (apart from linear logic itself). I already mentioned how we can find shared-memory futures in semi-axiomatic sequent calculus. Those futures are mutable, but write-once. This write-once aspect induces a degenerate monotonicity property which can be generalized to arbitrary monotonicity. Mutable variables can exhibit a form of CH as long as writes to them are monotonic in a certain sense, in particular new writes must not refute old reads. For example logical variables in a logic languages are exactly this. Safe, shareable mutable variables which denote evolving proof state during proof search.