Review: "Foundations Without Foundationalism, A Case for Second-Order Logic," by S. Shapiro

Abstract

Foundations Without Foundationalism, A Case for Second-Order Logic (Ox-ford University Press, Oxford, 1991), by Stewart Shapiro, is an excellent book, covering of all of the main results in second-order logic and its applications to mathematical theories. Its main theme is that first-order logic does not adequately "codify the descriptive and deductive components of actual mathematical practice", and that first-order languages and semantics are also inadequate models of mathematics" (43). Second-order logic (under its "standard" semantics), Shapiro maintains, "provides better models of important aspects of mathematics, both now and in recent history, than first-order logic does" (v); and in that regard it is second-order, and not only first-order, logic that "has an important role to play in foundational studies" (ibid.). Indeed, the restriction of logic to first-order logic (without Skolem relativism) in such studies is "the main target of this book" (196).