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### Single Idea 10069

#### [catalogued under 5. Theory of Logic / K. Features of Logics / 4. Completeness]

Full Idea

Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability.

Gist of Idea

A theory is 'negation complete' if one of its sentences or its negation can always be proved

Source

Peter Smith (Intro to Gödel's Theorems [2007], 01.1)

Book Reference

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.2

A Reaction

The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ.