First principles don't work in the space of systems geared towards
extreme generalization such as LLMs. You need to be ready to compare anything with anything and build bridges between many principles. In fact there is a deep link between the progress of structuralism in mathematics culminating with homotopy type theory and its parallel (r)evolution in the humanities with the discovery of manuscripts by the founder of structural linguistics, Ferdinand de Saussure.
Identity is what provides the irreducible basis, in the sense that we cannot enter into the consideration of specific facts that are placed under this identity, and it is this identity that becomes for us the true concrete fact, beyond which there is nothing more.
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For example, for a musical composition, compared to a painting. Where does a musical composition exist? It is the same question as to know where 'aka' exists. In reality, this composition only exists when it is performed; but to consider this performance as its existence is false. Its existence is the identity of the performances.
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For each of the things we have considered as a truth, we have arrived through so many different paths that we confess we do not know which one should be preferred. To properly present the entirety of our propositions, it would be necessary to adopt a fixed and defined starting point. But what we are trying to establish is that it is false to admit in linguistics a single fact as defined in itself. There is, therefore, a necessary absence of any starting point, and if some reader is willing to follow our thoughts carefully from one end to the other of this volume, they will recognize, we are convinced, that it was, so to speak, impossible to follow a very rigorous order. We will allow ourselves to present, up to three or four times in different forms, the same idea to the reader because there really is no starting point more appropriate than another on which to base the demonstration.
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As language offers no substance under any of its manifestations, but only combined or isolated actions of physiological, physical, and mental forces, and as nevertheless all our distinctions, our terminology, and all our ways of speaking are based on this involuntary assumption of a substance, we cannot refuse, first and foremost, to recognize that the most essential task of the theory of language will be to untangle what our primary distinctions are all about.
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There are different types of identity. This is what creates different orders of linguistic facts. Outside of any identity relationship, a linguistic fact does not exist. However, the identity relationship depends on a variable point of view that one decides to adopt; therefore, there is no rudiment of a linguistic fact outside the defined point of view that presides over distinctions.
Source: http://www.revue-texto.net/docannexe/file/116/saussure255_6....
TL;DR: identity is equivalent to equivalence