Abstract: A language is second-order, or higher-order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first-order variables. This book presents a formal development of second- and higher-order logic and an extended argument that higher-order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model-theoretic semantics for higher-order languages, and the basic and advanced results in its meta-theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second-order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first-order counterparts. Despite the fact that Quine is the main opponent of second-order logic (arguing that second-order logic is set-theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first-order logic as the de facto standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule-following. The book closes with an examination of several alternatives to second-order logic: first-order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.

Keywords: completeness, foundationalism, foundations, logic, mathematics, model theory, philosophical logic, philosophy of logic, Quine, second-order logic, set, Löwenheim–Skolem, Stewart Shapiro, Wittgenstein