This reprint of Kleene's 1967 text provides an elementary but thorough treatment of mathematical logic of the first order. Kleene (formerly: mathematics, University of Wisconsin, Madison) covers a variety of methods and techniques including model theory (truth tables), Hilbert-type proof theory, and proof theory through derived rules. Subsequent chapters cover topics formal number theory, Godel's completeness theorem, Genzen's theorem, Skolem's paradox, and nonstandard models of arithmetic.