20 ideas
13838 | A decent modern definition should always imply a semantics [Hacking] |
10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W] |
13834 | Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking] |
13835 | Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking] |
13833 | 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking] |
13845 | The various logics are abstractions made from terms like 'if...then' in English [Hacking] |
13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
13842 | Second-order completeness seems to need intensional entities and possible worlds [Hacking] |
10477 | |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W] |
13837 | With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking] |
13839 | Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking] |
10474 | |= should be read as 'is a model for' or 'satisfies' [Hodges,W] |
10475 | A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W] |
10473 | Model theory studies formal or natural language-interpretation using set-theory [Hodges,W] |
10481 | Models in model theory are structures, not sets of descriptions [Hodges,W] |
13843 | If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking] |
10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
22200 | If you eliminate the impossible, the truth will remain, even if it is weird [Conan Doyle] |