A Dictionary of Philosophy, Third Edition
. An AXIOM SYSTEM, in the sense of a set of axioms and rules of inference, is complete in a weak sense if all the truths of the kind it caters for can be derived within the system. It is complete in a strong sense if the addition of any other proposition of the relevant kind as an independent axiom makes the system inconsistent. There are further refinements. In particular, formalizations of the propositional CALCULUS can be complete in both senses; formalizations of the first-order predicate CALCULUS can only be weakly complete.
Also a set of axioms in a formal language is called complete if for every sentence S in the language either S or not-S follows from the axioms. In this sense no explicitly definable set of axioms rich enough for elementary arithmetic is complete (see GÖDEL’S THEOREMS); in fact no such set is complete in any sense mentioned above.
A.H.Basson and D.J.O’Connor, Introduction to Symbolic Logic, 3rd edn, University Tutorial Press, 1959. (See its index.)
E.J.Lemmon, Beginning Logic, Nelson, 1965. (See its index.)
A.N.Prior, Formal Logic, Clarendon, 1955. (See its index.)
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