Just to add to the discussion, from a completely definitional point of view, of course it makes sense to consider the concept of no-knowledge. For example, consider the usual definition of proof of knowledge:
We say that $P$ knows $x$ if there is an efficient extractor $E$ that can interact with $P$ to obtain (extract) $x$.
What do we get if we negate this definition? The result would look something like this
We say that $P$ does not know $x$ if for every efficient algorithm $E$ the probability that $E$ outputs $x$ after interacting with $P$ is negligible.
Notice that nothing prevents such definition to exist, and, in fact, it makes a lot of sense! We can say that you don't know something if no matter who you interact with, you never make use of that "something". Now, a different question is whether or not this definition is achievable in any concrete model, which I believe is one of the main arguments against the idea on this question.
As a side note, it may be worth to take a look at this (already hinted by @Geoffroy Couteau). From the abstract: Loosely speaking, such a proof system [proof of ignorance] allows a prover to generate an instance x according to D along with a proof that she does not know a witness corresponding to x.