ERRETT A. BISHOP The notion of a subset A. of a set A is also of interest. To construct an element of Ao, one must first construct an element of A , and then prove that the element so constructed satisfies certain additional conditions, charac- teristic of the particular subset A Two elements of A are equal if they 0 ' 0 are equal as elements of A . Contrary to classical usage, the scope of the equality relation never extends beyond a particular set. Thus it does not make sense to speak of ele- ments of different sets as being equal, unless possibly those different sets are both subsets of the same set. This is because for the constructivist equality is a convention, whose scope is always a given set all this is conceptually quite distinct from the classical concept of equality as identity. You see now why the constructivist is not forced to resort to the artifice of equivalence classes! After this long digression, consider again the quantifiers. Let A(x) be a mathematical assertion depending on a parameter x ranging over a set S. To prove V xA(x), I ' we must give a method which to each element x of S associates a proof of A(x). Thus the meaning of the universal quantifier "V " is essentially the same as it is classically. We expect the existential quantifier "3." on the other hand, to have a new meaning. It is not clear to the constructivist what the classicist means when he says "there exists. l1 Moreover, the existential quantifier is just a glorified version of "or, I ' and we know that a reinterpretation of this connec - tive was necessary. Let the variable x range over the set S. Then to prove l1 3 xA(x)" we must construct an element x of S, according to the principles 0 laid down in the definition of S , and then prove the statement "A(xo). " Again, certain classical uses of the quantifiers fail constructively. For example, it is not correct to say that "not V xA(x) implies 3 x not A(x). " On the other hand, the implication "not 3 xA(x) implies V x not A(x)I1 is constructively valid. I hope all this accords with your common sense, as it does with mine. Perhaps you see an objection to these developments -- that they appear to violate constructivist principle (D) above. By accommodating our terminol- ogy to the mathematics of finite beings, have we not replaced the classical system, that does not permit the systematic development of constructive mean- ing, by a system that does not permit the systematic development of classical meaning? In my opinion the exact opposite is true -- the constructive
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