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The Law of Identity is one of the fundamental principles in classical logic, stating that an entity is identical to itself, meaning that A is A, and nothing can be other than itself. Without the Law of Identity, it's challenging to create a coherent and consistent system.

A logical system that entirely discards identity would be a highly unconventional approach and may not have been extensively studied or formalized in the field of logic. But is there anything out there that exist, a logic system that doesn't use the law of identity, and even outright reject it.

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    Before anyone mentions paraconsistent logic: those reject one of the steps in the principle of explosion, and do not deny A = A. This question is more radical (and imo, more interesting).
    – Hokon
    Commented Sep 25, 2023 at 0:48
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    You can simply take the conventional FOL without the default equality, then your formalized classic logic cannot even express law of identity, not to say rely on it. However, it's still a coherent and consistent system per your spec and can still be very expressive about many other everyday properties and relations except perhaps equational logic (algebra). FOL without equality is often employed in the context of 2nd order arithmetic and other higher-order theories of arithmetic, where the equality relations between sets of natural numbers are usually omitted by design.... Commented Sep 25, 2023 at 6:52
  • I would add that questions of transworld identity as well as relative identity are, to some extent, questions about the logic of identity, or can be posed as such questions. Then there are defaults such as "logics without transworld/relative identity," or a peculiar negative combination like "logics without normal or relative identity but with transworld ID," etc. (which sounds unstable, offhand, but might be interesting to analyze?). Commented Sep 25, 2023 at 12:00
  • A system entirely discarding identity would be an unconventional approach to those following your views… not anything absolute. What may not have been extensively studied in the field of logic is limited how? Does there exist out there a logic system that doesn't use the law of identity, and might even outright reject it, reads like a Question for search engines or even good old text-books, not for STack Exchange. Commented Sep 25, 2023 at 20:49
  • This condition is actually fairly common in the logics that back/are models of computation.
    – nomen
    Commented Sep 25, 2023 at 22:27

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There are such things, but they are, as you say, highly unconventional. The only application that I have come across lies within quantum mechanics. The idea is that fundamental particles such as electrons don't have an identity, since they cannot be distinguished even in principle, and so they cannot be uniquely labelled.

There is some discussion of this in the SEP article on Identity and Individuality in Quantum Theory, particularly in section 5, Non-individuality and self-identity. One approach to expressing this in formal logic has been developed by da Costa and Krause in what they call Schrödinger logic, which has restrictions on self-identity.

This example might help to explain it. Let's start by way of contrast with a classical situation. Let's say we have a classical box with a hole in it. We put a marble into the box through the hole. Then we put in another marble, qualitatively indistinguishable from the first. We give the box a good shake and after a while one of the marbles pops out. Then the other pops out. Was the first one out the same as the first one in, or was the first one out the second one in? We don't know because we didn't look inside and track their movements. But each marble has a distinct identity, so there is some definite fact of the matter, we just don't know it.

Now change the example to a quantum box. We put two electrons in, give them the equivalent of a shake, and later they come out. Which one is which? Here, there appears to be no fact of the matter. All we can say is two electrons went in and two came back out. The electrons don't have a unique identity. If we had looked inside the box and measured their movements, that would have been a different experiment.

  • da Costa, Newton; Krause, Décio (1994), "Schrödinger logics", Studia Logica, 53 (4): 533–550.
  • Krause, Décio; da Costa, Newton (1997), "An Intensional Schrödinger Logic", Notre Dame Journal of Formal Logic, 38 (2): 179–194.

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