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Jaakko Hintikka has recently proposed that the distinction between the standard and nonstandard interpretations of higher-order quantifiers be used as a new tool for investigations in the history of the foundations of mathematics.¹ This distinction can be described succinctly as follows: let us take a second-order quantifier involving a one-place class variable X, whose values are classes of individuals of a domain do(M). Those adopting the standard interpretation would claim that the range of this quantifier is the entire power set P(do(M)), i.e. some values of X are arbitrary extensionally possible classes, while those adopting the nonstandard interpretation would consider only some such classes as constituting the range of the quantifier. (The same reasoning applies if X is a predicate variable or for function variables.)

DOI: 10.1007/978-94-015-8478-4_14

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