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Is there any exception to the rule of non-contradiction? In physics, mathematics or philosophy? Is there any system where the law of non-contradiction doesn't apply for a good reason? I can't think of any. I am wondering if quantum superposition is an exception to this rule.

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    See paraconsistent logic. Whether this is motivated by a "good reason" is of course subjective, which is why this is a comment rather than an answer. Dec 6, 2020 at 20:06
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    Sentences that violate the non-contradiction law are called dialetheias, their existence is controversial, see SEP, Dialetheism. But one does not need quantum theory for (alleged) examples: the Liar sentence "I am false" is one, "50 grains are a heap", or something else with a vague predicate like "is a heap", is another.
    – Conifold
    Dec 6, 2020 at 20:32
  • Hegel's dialectical logic and its many offshoots assert the "identity of identity and difference." To put it crudely, defining two "opposites" will necessitate a "position" from which both are seen and thus unified as one "opposition." Hegel begins his Logic by examining the most fundamental oppositions of "being" and "nothing," showing that this mutual definition is unstable, the meaning of each turns into the other, thus generating "becoming." All "contradictions" decay over time. Dialectics is well outside of standard physics, but widely influential and useful. Dec 6, 2020 at 23:31
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    See the tertralemma of the en.m.wikipedia.org/wiki/Catu%E1%B9%A3ko%E1%B9%ADi
    – CriglCragl
    Dec 7, 2020 at 1:33

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There are no real-world examples, because real things don't have opposites, so long as we don't treat the lack of a property as itself a property.

The principle of non-contradiction is mostly a linguistic tool, to point out the absurdity in saying "a non-married married person," for example. Quantum physics also does not contradict this, because superposition, for example, means that every quantum state can be represented as the sum of other quantum states. At worst, this means that two different states can be true at the same time - which is not a contradiction, but rather an ability for particles to have more than one property in areas we normally think of as having only one property.

For example, if a particle is in position A and also in position B, this doesn't mean it is in the position A and also the position not-A, because "not" is a logical function, not the negation of a property; there is no position that is the logical opposite of position A. You can think of this through the analogy of a zebra, which has the property of being white, and also being black, but "black" and "white" are not logical opposites, so there is no contradiction.

The example you are likely thinking of, Schrodinger's Cat, was not intended as a demonstration of quantum physics, but rather the absurdity of the Copenhagen Interpretation.

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    "There are no real-world examples,' I love you and I hate you. I love my job and I hate my job. Everybody wants to go to heaven but nobody wants to die. I want to party, I want to study. We should save the planet by reducing industrialization, we should help de-industrialized societies to industrialize so they won't starve to death. Real life is FULL of contradictions and mutually contradictory imperatives and desires. Haven't you noticed?
    – user4894
    Dec 6, 2020 at 22:23
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    @user4894 None of these are contradictions, they are at best idiosyncrasies. Love is not the opposite of hate, or even the absence of hate. Partying and studying are probably mutually exclusive, but a desire to party and a desire to study are not opposites. Imagine a parent who will buy their child one item, and the child demands two. There is no logical contradiction in the child's desires, but only a choice to be made. Conflicted goals and feelings are not logical contradictions.
    – CodeReaper
    Dec 6, 2020 at 23:11
  • @user4894: I think the point is contradictions are with definitions, and between them, and things in the real world can never fully coincide with definitions, generalisations only ever hold so well. Nancy Cartwright gets at this in 'How The Laws Of Physics Lie'. A contradiction tells you about your system of abstractions representing the world, not about the world itself.
    – CriglCragl
    Dec 7, 2020 at 1:39
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    @CriglCragl OP specifically said "there are no real life examples," thereby bringing the topic up for refutation. The slightest familiarity with adult life shows that the world is full of contradictions and mutually inconsistent imperatives. It's hardly fair for the OP to say there are no real life contradictions, and when I spool out a list of them to have you and the OP said, "Oh I didn't actually mean real life examples." Fact is there ARE many contradictions in real life, an endless procession of them. It's only in formal logic that we can outlaw them.
    – user4894
    Dec 7, 2020 at 20:57
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    @user4894 I said something entirely different than "Oh I didn't actually mean real life examples." Real life has no contradictions, because contradictions are logical opposites. Because of that, there is no such thing as not-love or not-hate. I very specifically meant contradiction in the strictest logical sense, because everyone knows "hate" and "love" seem mutually exclusive, but aren't. The point was that this has literally no bearing on the principle of non-contradiction, because they aren't logical opposites.
    – CodeReaper
    Dec 7, 2020 at 23:19
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Lukasiewicz three valued logic is an example of a logical system in which the classical laws of non-contradiction and the excluded middle are explicitly rejected as general laws applicable to all propositions. Contrary statements 'p' and 'not p' are not necessarily contradictory.

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