In a 1983 article published in Crítica, Richard Shusterman examples Wittgenstein’s doctrine indeterminacy in aesthetic value and the work of critical evaluation. At first, this may seem rather remote from the matter of persuasion, but Shusterman rightly notes that the work of a critic (literary, musical, dramatic) is a work of persuasive rhetoric. In fact, Shusterman notes a passage in Wittgenstein’s Lectures and Conversations where he equates the nature of legal argument with the rhetorical procedure of a critic. As a lawyer, I can affirm correctness of Wittgenstein on this point.
Briefly, Shusterman notes three ways in which an aesthetic judgment is indeterminate.
Perhaps under the pressure from natural sciences and their seeming claim to indisputable truth, objective and eternal as a sort of sight of Platonic Forms, non-physical sciences beginning in the 19th century often sought to stylize themselves along “scientific” lines. Theology and literary criticism were taken up as sciences (in the sense of chemistry or physics).
A movement was made to formalize aesthetics in terms of deductive and inductive argument. If you have ever seen Dead Poets Society, the introduction from the beginning of the poetry book on how to graph the quality of a poem – which the new literature teacher has the students tear from their book – is a perfect example.
Wittgenstein critiques such deductive and inductive evaluations of art on the ground that the necessary grounds for evaluation have a degree of flux (Shusterman calls this “radical indeterminacy”, but I think he overshoots the mark; there is however a relative indeterminacy without question).

First, the basic concepts of balance and beauty do not have hard edges. Second, the work of the critic depends upon what sort of question the critic is asking (Wittgenstein’s concept of “game”). Thus, a critic who considers Hamlet psychological or the emotive or the political or metaphysical work will come to different conclusions. (Merrill Tenney’s book on Galatians is an example of how to perform this sort of critical evaluation. I also just realized that I have lent this book to someone and now I can’t find my copy.) Third, evaluations of beauty and art take place in a larger cultural context.
When we come to the contemporary period (granted Shusterman is writing 40 years ago; but the situation is even more extreme at present than then), the cultural context does not provide a universal scheme in which we can make a deductively valid and sound argument: who can say whether the premises are true.
What then does a literary critic do in such an environment? The critic cannot present “a nice knock-down argument for you.”
The critic’s work in this environment is thus to bring you to see the work from a particular point of view:

Validity is success, success in inducing the desired perception of the work, if not the desired critical verdict. He held that ‘aesthetic discussions were like discussion in a court of law’, where the goal and criterion of success is that ‘what you say will appeal to the judge.’ Elsewhere, Wittgenstein suggests that the criterion for adequacy of argument and correctness of explanation is acceptance or satisfaction. ‘The answer in these cases is the one that satisfied you.’ ‘That explanation is the right one which clicks,’ and is accepted by the interlocutor; ‘if he didn’t agree, this wouldn’t be the explanation.’

Richard Shusterman, “Aesthetic Argument and Perceptual Persuasion,” Critica 15, no. 45 (Dec. 1983): 51-74.
There is an important point: a deductive or inductive argument is not persuasive in some manner abstracted from the person who hears the argument. A deductive argument only works if it persuades the hearer in the direction intended by the speaker (it is possible that the deductive argument merely annoys the hearer and you have merely persuaded him to dislike you).
Thus, the work of the critic is to bring you see the artwork from a particular perspective: I seek to have you understand this poem, this painting as I do. When you see it as I see it, my literary criticism has been “worked.”
If anything, legal persuasion falls into this category even more than literary criticism. The lawyer seeks to bring the court to see the world from the perspective of the client.
Wittgenstein also notes that science functions in a similar manner:

Wittgenstein in fact suggests that such persuasion is also present in science. For instance, it underlies our firm and ready acceptance of the theories of Darwin and Freud, even when the grounds for their doctrines were in strictly logical terms of confirmation ‘extremely thin.’ We have been largely persuaded by the attraction of looking at things the way these theories present them.

Subsequent research into things such as confirmation bias only strengthen this insight. The strength of an argument – even a scientific argument – ultimately lies in the fact that it persuades: it causes one to see the world from a particular point of view.
Shusterman goes to offer a correction to those who take Wittgenstein further than is warranted. While literary criticism may be to bring you see a work from a particular perspective (and thus perhaps is the intended – and not always successful — end of all argument), that does not mean that deductive and inductive arguments are illegitimate.
Since arguments exist in the context of “conventions”, the “conventions” may permit or even dictate the use of deductive or inductive aesthetic judgment. Thus, “logical” arguments are not wrong even in the case of aesthetics. They just may not be effective (as determined by the intent of the one making the argument).

Finally, this understanding of persuasion as seeing the world from a particular perspective helps explain why certain ideas will be particularly difficult to change. When we ask someone to give up a certain perspective, they must not only give up the particular idea under review, they also must give up their conclusions on all of the world as seen from that perspective. Certain ideas form the context and basis for a worldview.

It is one thing to change a window on the second floor; it is quite another to tear our the foundation on the entire high-rise.