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 2002,0631218688]].
Click on the Idea Number for the full details 
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10775

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
10766

Logic is either for demonstration, or for characterizing structures

5. Theory of Logic / A. Overview of Logic / 5. FirstOrder Logic
10767

Elementary logic is complete, but cannot capture mathematics

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
10769

Secondorder logic isn't provable, but will express settheory and classic problems

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
10762

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
10776

The main quantifiers extend 'and' and 'or' to infinite domains

5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
10774

There are at least five unorthodox quantifiers that could be used

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
10773

The LöwenheimSkolem property is a limitation (e.g. can't say there are uncountably many reals)

10777

Skolem mistakenly inferred that Cantor's conceptions were illusory

5. Theory of Logic / K. Features of Logics / 3. Soundness
10765

Soundness would seem to be an essential requirement of a proof procedure

5. Theory of Logic / K. Features of Logics / 4. Completeness
10763

Completeness and compactness together give axiomatizability

5. Theory of Logic / K. Features of Logics / 5. Incompleteness
10770

If completeness fails there is no algorithm to list the valid formulas

5. Theory of Logic / K. Features of Logics / 6. Compactness
10772

Compactness blocks infinite expansion, and admits nonstandard models

10771

Compactness is important for major theories which have infinitely many axioms

5. Theory of Logic / K. Features of Logics / 8. Enumerability
10764

A complete logic has an effective enumeration of the valid formulas

10768

Effective enumeration might be proved but not specified, so it won't guarantee knowledge
