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There are three laws that are essential to Aristotelian logic: the law of identity ‘A = A’; the law of self-contradiction ‘not p and not-p’; the law of the excluded middle ‘p or not-p’.
Gibson claims that of the three laws the law of identity is the more fundamental as it is implied in any proposition and presupposed by the law of non-contradiction. As Gibson notes “… the principle of identity is logically the more fundamental. It is implied in the stating of a proposition and is therefore presupposed in the very enunciation of the principle of non-contradiction.”
As Aristotle noted without an essence, or ‘identity’ ontological, or nominal i.e. definitional, the law of self-contradiction is useless. If the law of self-contradiction is useless then our logic breaks down and becomes useless as an epistemic condition of truth
Thus as O’ Hear notes, logic is not ontologically neutral it implies an ontology. Again as he notes “[l]ogic, indeed is not metaphysically neutral but may well reveal the types of things various forms of thought and argument commit us to.” Putnam similarly claims that logic derives from metaphysics the belief in substances. As he notes, “[w]e get at the very beginning of logic, a metaphysics accompanying it and conditioning it
Putnam claims that if “the metaphysical picture that grew up with and conditioned classical logic is wrong, then some of the tautologies of classical logic may have to be given up
Thus if the anti-essentialists are right then logic become useless since there is no essence for the law of self-contradiction to work. The only way that logic could be an epistemic condition of truth is that it does not break down, and for this to be, there must be an ontological or a nominal definitional one. The problem with this consequence is that the notion of essence reduces to absurdity.
Every one on here debates using Aristotelian logic, but if this logic is not a criteria for truth then their arguments are nonsense ie maths and science