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The well known paradoxes that arise from defining terms with a strict binary classifications, such as the Sorites paradox (heap/not heap), show the absurdities resulting from a strict seperation between what is and is not.

Then as all human words have some inherent ambiguity it stands to reason that dichotomies, whenever referring to real constructs and purporting to establish a strict binary, are incorrect.

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    Not all human words have inherent "ambiguity" (the correct term is vagueness). Not all predicates are vague, terms in mathematics and science are specifically designed not to be vague. A number is either prime or not, and one can not be just a little bit pregnant. – Conifold Jan 24 at 1:31
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    @Conifold - Agreed about pure mathematics, but arguably any term describing physical reality that isn't defined as some precise mathematical function of underlying quantum states is at least a little vague. With pregnancy, the fertilization process takes some time once a sperm cell meets an egg cell, and I don't think biologists have any precise criteria that would allow one to say the exact nanosecond when it goes from "unfertilized egg cell in contact with a sperm cell" to "fertilized zygote". Likewise with other important distinctions like living vs. non-living, human vs. non-human etc. – Hypnosifl Jan 24 at 4:29
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    @Conifold Actually, 5 is prime and irreducible in the ring of integers, but 5 = (2+i)(2-i) is reducible (and hence its principle ideal not prime) in the ring of Gaussian integers. Even math requires some (sometimes a lot) context to get rid of ambiguity – Hagen von Eitzen Jan 24 at 10:02
  • @HagenvonEitzen That we tend to reuse terms in different contexts and omit it when it is implied are just pragmatic shortcuts, "abuse of terminology", it has nothing to do with the nature of the predicates involved. The same when we do not make the terms more precise than is practically necessary. Loose talk, or even insufficient knowledge to resolve the terms are not vagueness. Being a heap is vague not because we skip context or lack knowledge about heaps, and being pregnant is not simply because biologists did not bother with nanoseconds. Dichotomies are about how things are, not our talk. – Conifold Jan 25 at 20:44
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From experience with mathematics, not all dichotomies are incorrect. However, the dichotomies I have found which are consistent have a limited universe from which they pull. "All natural numbers are either even or odd" is a valid dichotomy, but it is limited to natural numbers.

When one attempts to extend the domain that such statements are made over to "everything," that is when the dichotomies seem to fall apart. There are many well understood paradoxes that arrive, and many more that are not understood.

Just to turn the whole thing on itself, it is a fascinating dichotomy indeed to partition statements into those that are correct and those that are incorrect.

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  • Absolute generalizations are always wrong. Including that one. Which proves my point. – Hagen von Eitzen Jan 24 at 10:04
  • Thanks for the insight. It seems then to be more sensible to limit the question to words describing real constructs as i had not intended to ask about formal words that only refer to purely formal constructs.. – M. Y. Zuo Jan 24 at 17:14
  • @MichaelZ Yep, we've been doing such narrowing of scope for millennia =) Now the real bugger is whether "everything" is a real construct or a formal one. Most religions would quickly argue that there is a meaningful "everything," and most atheists tend to have their opinion on that as well! And yet that bugger of a word creates so many paradoxes! – Cort Ammon Jan 24 at 17:32

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