More Square

As I mentioned in my last post we are discussing the traditional vs modern square of opposition in my logic course (which did not go over well btw, most students reacted viscerally to the claim that modern logicians reject the entailment of the I proposition by the A proposition…to the point that one student exclaimed that it was a betrayal of Aristotle!) but at any rate I was reading the entry at the Stanford Encyclopedia on the Traditional Square of Opposition which is written by Terrance Parsons. The article is very interesting and provides a valuable history of the development of the square.

Along the way Parsons develops the very interesting idea that the modern problem with empty terms is not a problem for Aristotle’s original formulation of the square and especially because of the way he formulated the O statement, which is not as the traditional ‘some A is not B’ but rather is ‘Not every A is B’. This, argues Parsons, solves the problem with existential import since ‘not every A is B’ does in fact seem to be true as is required. Parsons blames Boethius for the rewording of the O form,

Aristotle’s work was made available to the Latin west principally via Boethius’s translations and commentaries, written a bit after 500 CE. In his translation of De interpretatione, Boethius preserves Aristotle’s wording of the O form as “Not every man is white.” But when Boethius comments on this text he illustrates Aristotle’s doctrine with the now-famous diagram, and he uses the wording ‘Some man is not just’. So this must have seemed to him to be a natural equivalent in Latin. It looks odd to us in English, but he wasn’t bothered by it.

But isn’t it obvious that ‘not every unicorn is an animal’ is truth-functionally equivalent to the traditional ‘some unicorn is not an animal’? That is to say, it is clearly the case that ~(x) (Hx –> Mx) is equivalent to Ex (Hx & ~ Mx) [by quantifier exchange, the definition of ‘–>’ and DeMorgan’s law]. So…Boethius was right, wasn’t he? And not just because it is a natural translation in Latin, but because the two statements are logically equivalent….right?


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