It should be called paradoxology, definable as mapping a set with its content into itself, as done with the well-defined word. One example is "All Cretans are Liars" We all lie on suitable occasions but rarely ALL the time. In Bertrand Russell's "Theory of Classes", are several examples. "There is a barber who does not save himself." So who shaves the barber? This is a self consistent, strict definition of a word but common sense allows that individual real world barbers need not obey this frame; so this hinges on that definition being taken as valid. An inverse of this binary A cannot be not-A, comes as a country's main or national librarian compiles a bibliography of all local libraries' bibliographies, some of which contain themselves and others do not, depending on the definition applied, making this a {(A.~A}.(A)} where the dot = 'and', and ~ = not; not to make things more complex by having a book in a library that is not listed as a book.
The current fad for paradoxes has produced hundreds of examples which then shows that logic if applied too strictly leads to paradox, where the culprit hides in different ways. An incomplete list is: "Allais Paradox, Aristotle's Wheel Paradox, Arrow's Paradox, Banach-Tarski Paradox, Barber Paradox, Bernoulli's Paradox, Berry Paradox, Bertrand's Paradox, Buchowski Paradox, Burali-Forti Paradox, Cantor's Paradox, Catalogue Paradox, Coastline Paradox, Coin Paradox, Elevator Paradox, Epimenides Paradox, Eubulides Paradox, Grelling's Paradox, Hausdorff Paradox, Hempel's Paradox, Heterological Paradox, Hypergame, Leonardo's Paradox, Liar's Paradox, Logical Paradox, Potato Paradox, Pseudoparadox, Richard's Paradox, Russell's Paradox, Saint Petersburg Paradox, Siegel's Paradox, Simpson's Paradox, Skolem Paradox, Smarandache Paradox, Socrates' Paradox, Sorites Paradox, Thompson Lamp Paradox, Unexpected Hanging Paradox, Zeeman's Paradox, Zeno's Paradoxes.
Internet has a million items listed for the reader to practice on. Further Cantor's infinite Set Theory allows A and ~A to happily co-exist otherwise they would all lead into an infinite regression. Godel's proof shows that we cannot exceed or prove our assumptions or beliefs from the inferences we make with such assumptions. This makes paradox out as readily derived from the use of a binary logic. For a silly example If one writes "this is a blackboard" on a blackboard, which is the *real* one? OR, again "it was that that that the master printer approved of" making one 'that' into an object. Moreover hos should one punctuate [or should that be: Grammatify?] that sentence? Apropos of which Alvin Toffler renamed us from a consumer into a prosumer who both consumes and produces whence that is then an oxymoron, antinomy or anomaly as synonyms for paradox. All of this shows that we cannot confine language to the only logical.
