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Aside from its brevity, this article has a number of mistakes and confusions.
(1) Plato's beard is not an argument.
The article describes Plato's beard as an argument. But as dubbed by Quine it is a doctrine; so why not describe it as such? (It's right there in the quote!)
(2) The doctrine is unclear from the quote given.
The Quine quote is only a short paragraph taken from a more extended discussion in 'On What There Is'. The result leaves the reader significantly in the dark as to what exactly Quine is talking about. The passage just before the one quoted here is:
'Suppose now that two philosophers, McX and I, differ over ontology. Suppose McX maintains there is something which I maintain there is not. Suppose McX maintains there is something which I maintain there is not. McX can, quite consistently with his own point of view, describe our difference of opinion by saying that I refuse to recognize certain entities. I should protest of course that he is wrong in his formulation of our disagreement, for I maintain that there are no entities, of the kind which he alleges, for me to recognize; but my finding him wrong in his formulation of our disagreement is unimportant, for I am committed to considering him wrong in his ontology anyway. When I try to formulate our difference of opinion, on the other hand, I seem to be in a predicament. I cannot admit that there are some things which McX countenances and I do not, for in admitting that there are such things I should be contradicting my own rejection of them.
'It would appear, if this reasoning were sound, that in any ontological dispute the proponent of the negative side suffers the disadvantage of not being able to admit that his opponent disagrees with him.'
(3) Quine, Russell, etc. did not "favor the argument".
The article states, "The argument has been favored by prominent philosophers including Bertrand Russell, A. J. Ayer and C. J. F. Williams." The opposite is true: the doctrine, just outlined in the quote above, is rejected by Quine, Russell, Ayer and Williams. That is, they all deny that 'Nonbeing must in some sense be', since (as Russell showed) claims of non-existence can be made without even seeming to have to presume the existence of the disputed object(s).
(4) The subsequent sentence is confused.
"Declaring that not p (¬p) can't exist, one may be forced to abandon truisms such as negation and modus tollens." I can't even make sense of what this is trying to say. First, the doctrine defined by Quine is about the existence of objects, not propositions (like 'not p'). Second, neither negation nor modus tollens are truisms: negation is a sentential connective (and so neither true nor false); modus tollens is a law of inference (and so sound/unsound, rather than true/false). Third, no clue is offered as to why "one may be forced to abandon" these things if one were to declare that not-p can't exist (whatever it might mean for a proposition not to exist). Certainly, no argument is to be found in the discussions by Russell, Quine, Ayer or Williams.
(5) Popper's quote is not the "inverse" of Quine's claim and has questionable relevance.
I don't know what the author has in mind by "inverse", but Popper's claim here pertains to reductionism in science. He borrows Quine's phrase in a playful way, but his discussion is not germane to the topic. — Preceding unsigned comment added by 81.110.180.111 (talk) 15:37, 14 September 2018 (UTC)