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Single Idea 9720

[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness ]

Full Idea

If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'.

Gist of Idea

A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity

Source

Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7)

Book Ref

Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.2

The 14 ideas with the same theme [all the truths of a system are formally deducible]:

9544 A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised [Hughes/Cresswell]
19065 Soundness and completeness proofs test the theory of meaning, rather than the logic theory [Dummett]
9720 A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton]
10763 Completeness and compactness together give axiomatizability [Tharp]
10834 Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos]
10069 A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
10598 A theory is 'negation complete' if it proves all sentences or their negation [Smith,P]
10597 'Complete' applies both to whole logics, and to theories within them [Smith,P]
13628 We can live well without completeness in logic [Shapiro]
13698 In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider]
10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
10161 If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman]
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
10761 Completeness can always be achieved by cunning model-design [Rossberg]