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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?

April 6, 2011

Response from Peter Smith on April 8, 2011

What does it mean to say that the logical form of "Every man is mortal" is "Every x is such that, if x is a man, then x is mortal"? The content of this claim is, in fact, quite obscure!

However what is true is that if we are going to translate the English "Every man is mortal" into a a standard single-sorted first-order language of the kind beloved by logicians, the best we can do is along the lines of (For all x)(Fx -> Gx). And, as you say, in so doing, we don't respect the existential commitment which arguably accrues to the "Every" proposition.

OK, but that's one of the prices we pay for trying to shoehorn our everyday claims involving many sorted quantifiers (as in "Every man", "no woman", "some horse", "any natural number") into an artificial language where (in any application) all the quantifiers run over a single common domain. That's a price typically worth paying in order to get other benefits (ease of logical manipulation, etc.). But if, in some context, we don't want to pay the price, then so be it. We could instead regiment the English claims into a logical language with sorted quantifiers (and then we can set things up so that the existential commitment is preserved, if we want it). You pays your money and you makes your choice.

It's the same with the assumption of non-empty domains (and no empty names, etc). It's convenient. A "free" logic with empty names and even empty domains can readily be constructed but has its price in complications. Again, your pays your money and you makes your choice.

(Of course, quite apart from any issue about translating into this logical language or that, there is the question whether "Every" propositions in English do always carry existential commitment. We might wonder: does Newton's law that every particle subject to no net force moves in a straight line with constant velocity imply that there are particles which are subject to no net force?)




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