Blackwell Reference Online

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L ogic A property ascribed to a system of formal logic, an axiomatic system or a theory, generally meaning that all truths of the system or the theory can be derived or proved within the system or theory. A logical system is semantically complete if and only if all of its semantically valid formulae are theorems of the system. It is syntactically complete if an addition of a non-theorem will lead to inconsistency. Syntactical completeness is the stronger sense of completeness. A theory is complete or negation-complete if any of its statements or the negation of that statement is provable within the theory. However, according to Gödel 's theorem , none of the systems of ordinary arithmetic is complete for it must either be inconsistent or contain at least one truth that is not provable within the system itself. This thesis of incompleteness effectively undermines Hilbert 's program of providing mathematical proofs of its own consistency. “The notion of completeness of a logical system has a semantical motivation, consisting roughly in the intention that the system shall have all possible theorems not in conflict with the interpretation.” Church, Introduction to Mathematical Logic ... log in or subscribe to read full text