So summarizing the thread and the links I’ve read it looks like there are two basic strategies to solving this problem. One is the Dialetheist strategy used in para-consistent logics. This strategy rejects the principle of non-contradiction and one of the rules that leads to the principle of explosion (any part of the disjunction syllogism used in explosion). The other is the strategy characterized by the formalist school’s approach and cousin_it’s comment variations of which were given throughout the thread (I consider Yvain’s Tarski sentence approach to be an instance of this strategy). The idea here is that by treating the paradox as a bivalent function we notice that it is a function which recurses infinitely. We then add the perhaps not obvious but certainly intuitive premise that sentences which suggest non-terminating functions are part of the subset of ‘meaningless’ sentences.
The Liar Sentence L is meaningless, so the Liar Argument can’t even get started because its main assumption (that the Liar Sentence exists or is meaningful) is faulty. Natural language is incoherent, and its underlying sensible structure is that of an infinite hierarchy of levels. Because the Liar Sentence would have to reside on more than one level simultaneously if it were a legitimate sentence, it’s not really a legitimate sentence. This way out of the paradox is taken by Russell in his ramified theory of types and, following Tarski, by Quine in his hierarchy of meta-languages. For Russell, the referential phrase “This sentence” in L is the culprit because the phrase is not allowed to refer to the sentence in which the phrase itself occurs. For Quine, instead, the culprit is the phrase “is false” in L because the phrase must be satisfied by sentences in a language lower in the hierarchy and not by the very sentence in which the phrase occurs.
Are what has been proposed here different from the approaches of Quine and Russell? If not which of the two is right? It isn’t obviously a difference that makes no difference; Russell’s approach rules out all self-reference while Quine’s does not.
Now, this method (call it the non-termination approach) certainly seems to dissolve the confusion. And compared to the Dialetheist approach the non-termination approach appears much superior. The former gives up the principle of non-contradiction and a useful rule like disjunction introduction (or some other alteration to deductive logic, there appear to be a lot of alternatives and I haven’t gone through them all).
So the question becomes: what advantages does the paraconsistent logic approach have? Does anyone know of examples of logic that don’t have to sacrifice significant power in order to accommodate dialetheias? It doesn’t seem like it would be worth it.
So summarizing the thread and the links I’ve read it looks like there are two basic strategies to solving this problem. One is the Dialetheist strategy used in para-consistent logics. This strategy rejects the principle of non-contradiction and one of the rules that leads to the principle of explosion (any part of the disjunction syllogism used in explosion). The other is the strategy characterized by the formalist school’s approach and cousin_it’s comment variations of which were given throughout the thread (I consider Yvain’s Tarski sentence approach to be an instance of this strategy). The idea here is that by treating the paradox as a bivalent function we notice that it is a function which recurses infinitely. We then add the perhaps not obvious but certainly intuitive premise that sentences which suggest non-terminating functions are part of the subset of ‘meaningless’ sentences.
From the IEP
Are what has been proposed here different from the approaches of Quine and Russell? If not which of the two is right? It isn’t obviously a difference that makes no difference; Russell’s approach rules out all self-reference while Quine’s does not.
Now, this method (call it the non-termination approach) certainly seems to dissolve the confusion. And compared to the Dialetheist approach the non-termination approach appears much superior. The former gives up the principle of non-contradiction and a useful rule like disjunction introduction (or some other alteration to deductive logic, there appear to be a lot of alternatives and I haven’t gone through them all).
So the question becomes: what advantages does the paraconsistent logic approach have? Does anyone know of examples of logic that don’t have to sacrifice significant power in order to accommodate dialetheias? It doesn’t seem like it would be worth it.