An "abhorrence of contradiction" is necessary for him to even begin to express his having eschewed his former "parochialism". To have said, "abhorrence of contradiction" he has demonstrated an abhorrence of contradiction -- in the very act of making his declaration.
The Principle of Non-Contradiction (PNC) means that every word in the sentence "The line is straight" has a specific meaning. 'The' does not mean 'any', 'all', or 'no'. 'Line' does not mean 'dandelion' or 'doughnut'. 'Is' does not mean 'is not'. 'Straight' does not mean 'white', or anything else. Each word has a definite meaning. In order to have a definite meaning, a word must not only mean something, it must also not mean something. 'Line' means 'line', but it also does not mean 'not-line' — or 'dog', 'sunrise', or 'monkey'.
If 'line' were to mean everything, it would mean nothing; and no one, including him, would have the foggiest idea what he means when he says the word 'line'. PNC means that each word, to have a meaning, must also not mean something. And so; anyone who argues against an "abhorrence of contradiction" must use PNC for that statement to even mean anything, thus undercutting his own argument.
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"There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed."
—George Boole, An Investigation of the Laws of Thought
I don't see that every violation of the principle of noncontradiction trivially leaves definitions meaningless. Suggesting that a principle shouldn't be taken to hold in general doesn't mean that you have to violate the principle at every opportunity, just that there is some place it is (in whatever sense) correct to violate the principle. Of course, under traditional logics, accepting a contradiction leaves everything provable - but we're dealing with nontraditional logics that may have other ways of dealing with that.
As someone not versed in nontraditional logics, can you give an entry-level statement or thought experiment that violates the principle of non-contradiction without leaving everything provable?
I'm also not particularly versed in nontraditional logics. I'm just leery of saying a mathematical structure can't be interesting or has no applications, when mathematicians seem to disagree.
You're being needlessly difficult to no good end. I am not dropping the law of non-contradiction in my comments here, where I speak about logics that drop non-contradiction.
And the point you're trying to make here ignores what I said. It is certainly the case that if you permit every violation of non-contradiction, whether or not it violates any other rule, you wind up with nonsense. In no way does that establish that nonsense follows from any violation of non-contradiction even in the face of other rules. I don't have deep enough knowledge to say one way or the other, but some mathematicians seem to think there are things of interest there and your superficial rejection does nothing to prove them wrong.
You were perceived as difficult because rather than actually address the content of what I said, you tried to make the same point in the same way you had earlier - a way which was undermined by what I had said.
That's just being difficult. The real world does work that way. Take poltical labels for example, liberal, conservative, et al have totally different meanings for different people. Contradictions do exist, just externalizing them in a formal system is lazy.
The Principle of Non-Contradiction (PNC) means that every word in the sentence "The line is straight" has a specific meaning. 'The' does not mean 'any', 'all', or 'no'. 'Line' does not mean 'dandelion' or 'doughnut'. 'Is' does not mean 'is not'. 'Straight' does not mean 'white', or anything else. Each word has a definite meaning. In order to have a definite meaning, a word must not only mean something, it must also not mean something. 'Line' means 'line', but it also does not mean 'not-line' — or 'dog', 'sunrise', or 'monkey'.
If 'line' were to mean everything, it would mean nothing; and no one, including him, would have the foggiest idea what he means when he says the word 'line'. PNC means that each word, to have a meaning, must also not mean something. And so; anyone who argues against an "abhorrence of contradiction" must use PNC for that statement to even mean anything, thus undercutting his own argument.
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"There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed."
—George Boole, An Investigation of the Laws of Thought