Does the Gricean distinction between natural and non-natural meaning exhaustively account for all instances of communication?

The Gricean distinction between natural meaning and non-natural meaning has generally been taken to apply to communication in general. However, there is some doubts that the distinction exhaustively accounts for all instances of communication. Notably, some animal communication seems to be voluntary, though not implying double-barrelled intentions, i.e., falling neither under natural nor under non-natural meaning. Another worry is how the audience can distinguish between that kind of 1st order voluntary communication and non-natural meaning. The paper shows that the Gricean distinction is not exhaustive and that the second intention characteristic of non-natural meaning is presupposed on the basis of the cost of interpretation in linguistic communication given its semantic underdetermination.