Ideas from 'Which Logic is the Right Logic?' by Leslie H. Tharp [1975], by Theme Structure
[found in 'Philosophy of Logic: an anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21868-8]].
green numbers give full details |
back to texts
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expand these ideas
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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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. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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10766
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Logic is either for demonstration, or for characterizing structures
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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10767
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Elementary logic is complete, but cannot capture mathematics
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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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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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
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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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5. Theory of Logic / G. Quantification / 2. Domain of Quantification
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10776
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The main quantifiers extend 'and' and 'or' to infinite domains
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5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
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10774
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There are at least five unorthodox quantifiers that could be used
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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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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10777
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Skolem mistakenly inferred that Cantor's conceptions were illusory
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5. Theory of Logic / K. Features of Logics / 3. Soundness
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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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5. Theory of Logic / K. Features of Logics / 4. Completeness
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10763
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Completeness and compactness together give axiomatizability
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5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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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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5. Theory of Logic / K. Features of Logics / 6. Compactness
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10772
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Compactness blocks infinite expansion, and admits non-standard models
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10771
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Compactness is important for major theories which have infinitely many axioms
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5. Theory of Logic / K. Features of Logics / 8. Enumerability
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10764
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A complete logic has an effective enumeration of the valid formulas
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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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