Modern logicians teach us that some of the inferences embodied by the Aristotelian square of opposition (i.e., the A-E-I-O scheme) are not valid. Take the inference from the Universal Affirmative "Every man is mortal" to the Particular Affirmative "Some men are mortal": the logical form of the first proposition is a conditional ("Every x is such that if x is a man, then x is mortal") and we know that a conditional is true whenever its antecedent is false. In other words, the proposition "Every x is such that if x is a man, then x is mortal" is true even if there were no man, so the aforementioned inference is invalid. But if the universal quantifier has not ontological import, why such a logical truth as "Everything is self-idential" implies that there is something self-identical? And, above all, why the classical first order logic needs to posit a non-empty domain?
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