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]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice now seems acceptable and obvious (if it is meaningful)
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is either for demonstration, or for characterizing structures
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Elementary logic is complete, but cannot capture mathematics
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic isn't provable, but will express set-theory and classic problems
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and'
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
The main quantifiers extend 'and' and 'or' to infinite domains
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
There are at least five unorthodox quantifiers that could be used
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
The L÷wenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals)
Skolem mistakenly inferred that Cantor's conceptions were illusory
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness would seem to be an essential requirement of a proof procedure
5. Theory of Logic / K. Features of Logics / 4. Completeness
Completeness and compactness together give axiomatizability
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
If completeness fails there is no algorithm to list the valid formulas
5. Theory of Logic / K. Features of Logics / 6. Compactness
Compactness is important for major theories which have infinitely many axioms
Compactness blocks infinite expansion, and admits non-standard models
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A complete logic has an effective enumeration of the valid formulas
Effective enumeration might be proved but not specified, so it won't guarantee knowledge