The Predicate Calculus
There are many valid argument forms, however, that cannot be analyzed by truth-functional methods, e.g., the classic syllogism: "All men are mortal. Socrates is a man. Therefore Socrates is mortal." The syllogism and many other more complicated arguments are the subject of the predicate calculus, or quantification theory, which is based on the calculus of classes. The predicate calculus of monadic (one-variable) predicates, also called uniform quantification theory, has been shown to be complete and has a decision procedure, analogous to truth tables for truth-functional analysis, whereby the validity or invalidity of any statement can be determined. The general predicate calculus, or quantification theory, was also shown to be complete by Kurt Gödel, but Alonso Church subsequently proved (1936) that it has no possible decision procedure.
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