What is the truth value of this predicate:
"This sentence is FALSE"?
A statement with a truth value of FALSE would not be a contradiction--it has an unambiguous value--what do you think "contradiction" means?
It occurs with a conjunction of statements that cannot both be true. It is a fundamental of all rational logic.
It is a fundament of all historical formal logic, and it has limited application to the real world. It works pretty well for gross physical objects and relationships, and most formal maths. It fails conspicuously for sets that commit type violations, such as "the set of all sets", it fails conspicuously in subnuclear physics to explain the 2-slit experiment, and it fails conspicuously to explain many subjective phenomena, such as, for example, my ability to be both happy and not(happy) that my mother has died. It is just one of several mathematical descriptions of how elements in well-formed sets behave. As such, it does not constitute the entire warp and weave of the universe. It is a useful tool for many purposes, it is not a ghost that inhabits every corner of the universe.
"A and not A" is false in all rational set theory, grammars
Chomsky level 0 (or 4, I always forget which direction the grammar heirarchies are stacked in) grammars permit exactly such contradictions, as do my dreams and fancies, as do individual molecules going through the slits of the 2-slit one at a time.
That you have construed ways in which you think a contradiction is at times NOT false shows your misapplication of the fundamentals upon which the theories you think you understand are founded.
Kindly just answer the question: is "This sentence is FALSE" FALSE? If we assume the sentence is FALSE, (as you say, because it is contradictory), than upon evaluation, we find it declares itself to be TRUE, which we must believe, because we declared it to be FALSE. If it is true, it must therefore be FALSE, therefore, it must be TRUE...are you getting the drift here? Contradiction does not necessarily just mean FALSE--it could mean you cannot get a value. You hang up if you try. You are using a loose and inadequate notion of contradiction as if it were ubiquitous. It is not, it is quite easy to have validly formed predicates that are neither TRUE nor FALSE in any formally acceptable sense. That is what most of 20th century formal mathematics was about.
Besides. I don't fall for diversions.
Unless you supply them, I assume.
I guess I do fall for diversions.
I will restate, and I doubt even Chomsky would disagree, that the law of noncontradiction is inviolable. You seem well read enough I'm not sure why type violations are such a mystery to you. They are ill-formed concepts. The set of all sets is an ill-formed concept. "This sentence is false" is also ill-formed. In particular it attempts to prescribe a truth value before its definition is made. Just because English words can be put into grammatically correct patterns doesn't mean it makes sense. There's a number of examples in _Through the Looking Glass_.
At any rate, the so-called type violations, and the many paradoxes, or antinomies, of intuitive set theory show that the set theory is not consistent with its own axioms. Rather than revoking the law of noncontradiction, Cantor, Russell, and others had tried to correct the theory to eliminate the paradoxes. That is, they use the law of noncontradiction as a measure of the validity of a theory. As well they should.
You can rest assured that never, in no place, at no time, no matter how many accolades the advocate has, or how complex the theory, false is not true.
Before I dealt with this bird I didn't realize there was a branch of mathematicians called 'formalists' who have this unusual way of looking at everything. That they can't see the fallacy of their thinking is unusual, but there are many means of dodging logic
it fails conspicuously in subnuclear physics to explain the 2-slit experiment
This is the BIG QUESTION that you can expect to face endlessly. That this doesn't refute the law of identity, only that we haven't found the proper identity, is never entertained. According to this view we can't be certain of anything except that the law of identity is invalid in this case. The contradiction of this view is never examined because contradictions are never valid except when he says they are, never when you say they are.
my ability to be both happy and not(happy) that my mother has died. It is just one of several mathematical descriptions of how elements in well-formed sets behave. As such, it does not constitute the entire warp and weave of the universe. It is a useful tool for many purposes, it is not a ghost that inhabits every corner of the universe.
Here is another one. When you are talking reality, he will talk mathematics. When you talk mathematics, he will shift to ephemeral things like emotions. That way neither of you will ever be on the same 'domain of discourse' no matter how hard you try to land it there.
Chomsky level
Chomsky? Pardon me while I go puke. Just say commie idiot and be done with it.
Kindly just answer the question: is "This sentence is FALSE" FALSE? If we assume the sentence is FALSE, (as you say, because it is contradictory), than upon evaluation, we find it declares itself to be TRUE, which we must believe, because we declared it to be FALSE. If it is true, it must therefore be FALSE, therefore, it must be TRUE...are you getting the drift here? Contradiction does not necessarily just mean FALSE--it could mean you cannot get a value. You hang up if you try. You are using a loose and inadequate notion of contradiction as if it were ubiquitous. It is not, it is quite easy to have validly formed predicates that are neither TRUE nor FALSE in any formally acceptable sense. That is what most of 20th century formal mathematics was about.
I liken it to a forebrain problem like sociopaths. They either get it or they don't. You either see the contradiction (speaks against itself) here or you don't. You cannot make someone see a contradiction. You cannot make a person acknowledge a contradiction. That this didn't invalidate all that logic proved was also brushed aside. That this this conundrum is meaningless was never considered, it becomes the holy grail. Then I asked him if fallacies were invalid. He said yes. The conversation stopped. If a person can commit a fallacy and still think he is thinking, there is no hope.
I wish you luck.