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Ideas of Leslie H. Tharp, by Text
[American, fl. 1975, Taught at MIT.]
1975
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Which Logic is the Right Logic?
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§0
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p.35
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10762
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In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and'
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§1
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p.36
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10763
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Completeness and compactness together give axiomatizability
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§2
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p.37
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10764
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A complete logic has an effective enumeration of the valid formulas
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§2
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p.37
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10765
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Soundness would seem to be an essential requirement of a proof procedure
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§2
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p.37
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10766
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Logic is either for demonstration, or for characterizing structures
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§2
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p.37
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10767
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Elementary logic is complete, but cannot capture mathematics
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§2
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p.38
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10769
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Second-order logic isn't provable, but will express set-theory and classic problems
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§2
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p.38
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10768
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Effective enumeration might be proved but not specified, so it won't guarantee knowledge
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§2
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p.38
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10770
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If completeness fails there is no algorithm to list the valid formulas
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§2
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p.38
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10772
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Compactness blocks infinite expansion, and admits non-standard models
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§2
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p.38
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10771
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Compactness is important for major theories which have infinitely many axioms
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§2
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p.39
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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)
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§3
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p.39
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10774
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There are at least five unorthodox quantifiers that could be used
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§3
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p.40
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10775
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The axiom of choice now seems acceptable and obvious (if it is meaningful)
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§5
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p.41
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10776
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The main quantifiers extend 'and' and 'or' to infinite domains
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§7
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p.43
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10777
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Skolem mistakenly inferred that Cantor's conceptions were illusory
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