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Single Idea 10597
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
]
Full Idea
There is an annoying double-use of 'complete': a logic may be semantically complete, but there may be an incomplete theory expressed in it.
Gist of Idea
'Complete' applies both to whole logics, and to theories within them
Source
Peter Smith (Intro to Gödel's Theorems [2007], 03.4)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.25
The
14 ideas
with the same theme
[all the truths of a system are formally deducible]:
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9544
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A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised
[Hughes/Cresswell]
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19065
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Soundness and completeness proofs test the theory of meaning, rather than the logic theory
[Dummett]
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9720
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A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
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10763
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Completeness and compactness together give axiomatizability
[Tharp]
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10834
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Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
[Boolos]
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10069
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A theory is 'negation complete' if one of its sentences or its negation can always be proved
[Smith,P]
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10598
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A theory is 'negation complete' if it proves all sentences or their negation
[Smith,P]
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10597
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'Complete' applies both to whole logics, and to theories within them
[Smith,P]
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13628
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We can live well without completeness in logic
[Shapiro]
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13698
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In a complete logic you can avoid axiomatic proofs, by using models to show consequences
[Sider]
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10127
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A 'complete' theory contains either any sentence or its negation
[George/Velleman]
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10161
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If a sentence holds in every model of a theory, then it is logically derivable from the theory
[Feferman/Feferman]
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13538
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If a theory is complete, only a more powerful language can strengthen it
[Wolf,RS]
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10761
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Completeness can always be achieved by cunning model-design
[Rossberg]
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