Second-Order Logic

Abstract

This expository article focuses on the fundamental differences between first-order logic and second-order logic. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between first-order logic and second-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as involving radical repudiation of part of our scientific tradition. But even if genuine logic comes to be regarded as excluding second-order reasoning, which is a real possibility, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may some day be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach an interdisciplinary audience, but every effort has been made to limit the inevitable sacrifice of rigor.