Quine's Paradox - Application

Application

Quine suggested an unnatural linguistic resolution to such logical antinomies, inspired by Bertrand Russell's Type theory and Tarski's work. His system would attach levels to a line of problematic expressions such as falsehood and denote. Entire sentences would use a higher hierarchy each of their parts'. The form "'Clause about falsehood₀' yields falsehood₁" will be grammatically correct, and "'Denoting₀ phrase' denotes₀ itself" – wrong.

George Boolos, inspired by his student Michael Ernst, has written that the sentence might be syntactically ambiguous, in using multiple quotation marks whose exact mate marks cannot be determined. He revised traditional quotation into a system where the length of outer pairs of so called q-marks of an expression is determined by the q-marks that appear inside the expression. This accounts not only for ordered quotes-within-quotes but also to, say, strings with an odd number of quotation marks.

In Gödel, Escher, Bach: An Eternal Golden Braid, author Douglas Hofstadter suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in many of the proofs of Gödel's incompleteness theorems.