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The wikipedia page on the laws of thought mentions the The three traditional laws: The Law of the Excluded Middle, Law of Non-contradiction and The Law of Identity. These laws are accepted and used by the vast majority of mathematicians, but not by all.

Intuitionists reject the Law of the Excluded Middle, as it opens the possibility for non-constructive proofs. As such, they may prefer to use some form of intuitionistic logic, where the law is not a tautology.

Dialethics reject the Law of Non-contradiction, on the philosophical standpoint that some statements are contradictory by nature. As such, they may prefer to use some form of paraconsistent logic, where the law is not a tautology.

What about the Law of Identity?

This question is two fold:

  1. Is there a philosophical school that rejects the law of identity as a law of thought?
  2. Is there system of logic in which the law of identity is not a tautology?
Shaun
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Luiz Martins
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    I cannot imagine a possible application of not accepting the law of identity. I consider it part of the definition of equality: if your notion of equality doesn't satisfy $x = x$ then it doesn't deserve the name! Of course there is much to say about equality: http://people.math.harvard.edu/~mazur/preprints/when_is_one.pdf – Qiaochu Yuan Jan 13 '21 at 04:04

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General Semantics as founded by Alfred Korzbiski rejects the principle of identity, as there can't exist a description of empirical facts which is ever one-to-one (according to that school/discipline). As they say, the map is not the territory, but if useful bears a similar structure. I think some of St. Leśniewski's work also rejected the principle of identity, and instead had a concept where statement forms were only 'equiform' instead of 'identical', and if I recall correctly, I think Lukasiewicz only used the term 'equiform' in some of his later work to avoid any sort of question like "how can 'Cpp' in this position be the same as 'Cpp' in this other position, especially considered as material objects?"

Schrodinger logics (see bottom of the link) reject the principle of identity. According to that wikipedia article, and this paper, there has existed motivation for that rejection originating in the study of quantum mechanics.

Doug Spoonwood
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