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Let's take the Law of Identity and change it to the Law of Identity with Overlapping Categories:

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Would classical logic with the modified Law of Identity be completely different than classical logic. If so, how so? And is it something that has been explored before? I am wondering if there are ontological implications in using this as a basis for all physical models, or it won't have any difference whatsoever. My understanding is that because it is a model it won't have any effect.

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  • Several people already told you that modern classical logic does not have a "law of identity", it is an outdated phrase from Aristotle's times. There is only syntactic convention that identified expressions can be substituted for each other. How your proposed modification is supposed to function in this regard is unclear. Moreover, it has ∈ in it, and so belongs to set theory rather than logic. Is the idea that any of Ci can be substituted for A in all contexts? If so, there will not be much difference, you are just duplicating labels.
    – Conifold
    Sep 30 at 5:21
  • The "law" above only asserts that entity A belongs to a "universe" organized in categories Ci.
    – Mauro ALLEGRANZA
    Sep 30 at 6:45
  • You have apparently been reading either some old and outdated logic text or someone more modern (Ayn Rand?) who has an outmoded theory of logical foundations where you try to start with a single universal principle and build all of logic from it. The problem with such foundational approaches is that they aren't actually simpler, they have to make just as many assumptions as other systems; they just bury those assumptions as manufactured consequence of the one special principle rather than positing them as explicit axioms.
    – David Gudeman
    Sep 30 at 13:36

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