1975 | Which Logic is the Right Logic? |
§0 | p.35 | 10762 | In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' |
§1 | p.36 | 10763 | Completeness and compactness together give axiomatizability |
§2 | p.37 | 10765 | Soundness would seem to be an essential requirement of a proof procedure |
§2 | p.37 | 10766 | Logic is either for demonstration, or for characterizing structures |
§2 | p.37 | 10767 | Elementary logic is complete, but cannot capture mathematics |
§2 | p.37 | 10764 | A complete logic has an effective enumeration of the valid formulas |
§2 | p.38 | 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge |
§2 | p.38 | 10769 | Second-order logic isn't provable, but will express set-theory and classic problems |
§2 | p.38 | 10770 | If completeness fails there is no algorithm to list the valid formulas |
§2 | p.38 | 10772 | Compactness blocks infinite expansion, and admits non-standard models |
§2 | p.38 | 10771 | Compactness is important for major theories which have infinitely many axioms |
§2 | p.39 | 10773 | The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) |
§3 | p.39 | 10774 | There are at least five unorthodox quantifiers that could be used |
§3 | p.40 | 10775 | The axiom of choice now seems acceptable and obvious (if it is meaningful) |
§5 | p.41 | 10776 | The main quantifiers extend 'and' and 'or' to infinite domains |
§7 | p.43 | 10777 | Skolem mistakenly inferred that Cantor's conceptions were illusory |