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Single Idea 10764
[filed under theme 5. Theory of Logic / K. Features of Logics / 8. Enumerability
]
Full Idea
A complete logic has an effective enumeration of the valid formulas.
Gist of Idea
A complete logic has an effective enumeration of the valid formulas
Source
Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
Book Ref
'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.37
The
16 ideas
from 'Which Logic is the Right Logic?'
10762
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In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and'
[Tharp]
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10763
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Completeness and compactness together give axiomatizability
[Tharp]
|
10770
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If completeness fails there is no algorithm to list the valid formulas
[Tharp]
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10771
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Compactness is important for major theories which have infinitely many axioms
[Tharp]
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10772
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Compactness blocks infinite expansion, and admits non-standard models
[Tharp]
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10764
|
A complete logic has an effective enumeration of the valid formulas
[Tharp]
|
10768
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Effective enumeration might be proved but not specified, so it won't guarantee knowledge
[Tharp]
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10765
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Soundness would seem to be an essential requirement of a proof procedure
[Tharp]
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10773
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The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals)
[Tharp]
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10766
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Logic is either for demonstration, or for characterizing structures
[Tharp]
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10767
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Elementary logic is complete, but cannot capture mathematics
[Tharp]
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10769
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Second-order logic isn't provable, but will express set-theory and classic problems
[Tharp]
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10775
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The axiom of choice now seems acceptable and obvious (if it is meaningful)
[Tharp]
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10774
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There are at least five unorthodox quantifiers that could be used
[Tharp]
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10776
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The main quantifiers extend 'and' and 'or' to infinite domains
[Tharp]
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10777
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Skolem mistakenly inferred that Cantor's conceptions were illusory
[Tharp]
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