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Single Idea 13698
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
]
Full Idea
You can establish facts of the form Γ|-φ while avoiding the agonies of axiomatic proofs by reasoning directly about models to conclusions about semantic consequence, and then citing completeness.
Gist of Idea
In a complete logic you can avoid axiomatic proofs, by using models to show consequences
Source
Theodore Sider (Logic for Philosophy [2010], 4.5)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.105
A Reaction
You cite completeness by saying that anything which you have shown to be a semantic consequence must therefore be provable (in some way).
The
14 ideas
with the same theme
[all the truths of a system are formally deducible]:
9544
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A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised
[Hughes/Cresswell]
|
19065
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Soundness and completeness proofs test the theory of meaning, rather than the logic theory
[Dummett]
|
9720
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A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
|
10763
|
Completeness and compactness together give axiomatizability
[Tharp]
|
10834
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Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
[Boolos]
|
10069
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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
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In a complete logic you can avoid axiomatic proofs, by using models to show consequences
[Sider]
|
10127
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A 'complete' theory contains either any sentence or its negation
[George/Velleman]
|
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]
|
10761
|
Completeness can always be achieved by cunning model-design
[Rossberg]
|