First, the paradox as stated in the graphic is incorrect. The barber must explicitly be made a man.

More to the point, the paradox can probably (I don’t have the skills) be mapped to “This statement is false.” It’s well-formed, but it is semantically gibberish. It just looks like it says something.

I’m pretty sure Quine wrote a whole thing about this paradox in The Ways of Paraodox. He called them “falsidical” as I recall. (It’s been at least forty years since I looked at it.)

Self-referential paradoxes are used in proofs of Godel’s incompleteness theorem. Again, I’m remembering vaguely, this time from Godel, Escher, Bach. I believe Hoftadter’s demonstration uses a process he called “Quining,” which involved appending an utterance in quotation marks to itself to form a sentence:

“‘Has four letters’ has four letters” is false.

“‘Has more than four letters’ has more than four letters” is true.

How that advances the ball is beyond me, but it should not be beyond the author of an article on the subject. Apparently, Quine resolves all the paradoxes that do not resolve themselves.

The barber paradox, like many self-referential “paradoxes,” is about language. Anyone who can say the phrase “four-sided triangle” can create a paradox by positing a problem that involves one. I submit that the male barber who shaves all the men and only the men who do not shave themselves is no more real than a four-sided triangle, a non-thing by definition, not of the concept denoted by the term “barber” but of the concept denoted by the phrase “all the men and only the men who do not shave themselves.”

BTW, the computer nerd in me sees this problem through the lens of logical and physical devices. The barber is a logical device and a physical device. When he shaves “himself,” the logical barber is shaving a physical person (let’s call him “John”) who does not shave himself but sits patiently while “the barber” shaves him. If we restrict the word “himself” to refer only to a logical device or a physical device acting reflexively on itself as such, the logical barber does not shave himself (because “he” is not a “man”) and John does not shave himself, because he is not “the barber.” So, the barber shaves John. Problem solved…