Principles of Mathematical Logic is the 1950[1] American translation of the 1938 second edition[2] ofDavid Hilbert's andWilhelm Ackermann's classic textGrundzüge der theoretischen Logik,[3] on elementarymathematical logic. The 1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known asfirst-order logic (FOL). Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonical status. FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments ofPeano arithmetic and nearly all treatments ofaxiomatic set theory.
The 1928 edition included a clear statement of theEntscheidungsproblem (decision problem) for FOL, and also asked whether that logic wascomplete (i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules). The former problem was answered in the negative first byAlonzo Church and independently byAlan Turing in 1936. The latter was answered affirmatively byKurt Gödel in 1929.
In its description ofset theory, mention is made ofRussell's paradox and theLiar paradox (page 145). Contemporary notation for logic owes more to this text than it does to the notation ofPrincipia Mathematica, long popular in the English speaking world.
