First-order quantifier logic is complete; higher-order quantifier logic is not. A few formally minded philosophers of logic — such as Quine and some of his followers — appear to believe that this is sufficient grounds for concluding that the only legitimate quantifier logic is first-order, not higher-order. However, most leading formally minded philosophers of logic over the past hundred years — Frege, Russell, Church, Carnap, Henkin, Montague, Kaplan — believe that the higher-order approach is a natural generalization of the first-order approach and therefore that quantifier logic is properly identified with higher-order quantifier logic. This chapter departs from this majority opinion. It discusses the underlying philosophical differences between the two approaches to quantifier logic, and attempts to illustrate the greater naturalness and generality of the first-order approach.
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