32 ideas
| 18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
| 18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
| 18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
| 18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
| 18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
| 10775 | The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp] |
| 10766 | Logic is either for demonstration, or for characterizing structures [Tharp] |
| 18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
| 18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
| 10767 | Elementary logic is complete, but cannot capture mathematics [Tharp] |
| 10769 | Second-order logic isn't provable, but will express set-theory and classic problems [Tharp] |
| 18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
| 10762 | In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' [Tharp] |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
| 10776 | The main quantifiers extend 'and' and 'or' to infinite domains [Tharp] |
| 10774 | There are at least five unorthodox quantifiers that could be used [Tharp] |
| 18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
| 10777 | Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp] |
| 10773 | The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp] |
| 10765 | Soundness would seem to be an essential requirement of a proof procedure [Tharp] |
| 10763 | Completeness and compactness together give axiomatizability [Tharp] |
| 10770 | If completeness fails there is no algorithm to list the valid formulas [Tharp] |
| 10771 | Compactness is important for major theories which have infinitely many axioms [Tharp] |
| 10772 | Compactness blocks infinite expansion, and admits non-standard models [Tharp] |
| 10764 | A complete logic has an effective enumeration of the valid formulas [Tharp] |
| 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp] |
| 18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
| 18957 | Nominalism only makes sense if it is materialist [Putnam] |
| 18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
| 18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |