40 ideas
| 6319 | Wise people choose inaction and silence [Laozi (Lao Tzu)] |
| 6325 | One who knows does not speak; one who speaks does not know [Laozi (Lao Tzu)] |
| 6321 | Vulgar people are alert; I alone am muddled [Laozi (Lao Tzu)] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 6328 | To know yet to think that one does not know is best [Laozi (Lao Tzu)] |
| 6323 | Pursuit of learning increases activity; the Way decreases it [Laozi (Lao Tzu)] |
| 6331 | Truth is not beautiful; beautiful speech is not truthful [Laozi (Lao Tzu)] |
| 2850 | How can emotivists explain someone who recognises morality but is indifferent to it? [Brink] |
| 2848 | Two people might agree in their emotional moral attitude while disagreeing in their judgement [Brink] |
| 2851 | Emotivists find it hard to analyse assertions of moral principles, rather than actual judgements [Brink] |
| 2853 | Emotivists claim to explain moral motivation by basing morality on non-cognitive attitudes [Brink] |
| 2852 | Emotivists tend to favour a redundancy theory of truth, making moral judgement meaningless [Brink] |
| 2849 | Emotivism implies relativism about moral meanings, but critics say disagreements are about moral reference [Brink] |
| 6330 | One with no use for life is wiser than one who values it [Laozi (Lao Tzu)] |
| 6327 | Do good to him who has done you an injury [Laozi (Lao Tzu)] |
| 23402 | The highest virtue is achieved without effort [Laozi (Lao Tzu)] |
| 6324 | To gain in goodness, treat as good those who are good, and those who are not [Laozi (Lao Tzu)] |
| 6322 | There is no crime greater than having too many desires [Laozi (Lao Tzu)] |
| 6320 | The best rulers are invisible, the next admired, the next feared, and the worst are exploited [Laozi (Lao Tzu)] |
| 6329 | People are hard to govern because authorities love to do things [Laozi (Lao Tzu)] |
| 6326 | The better known the law, the more criminals there are [Laozi (Lao Tzu)] |
| 23401 | A military victory is not a thing of beauty [Laozi (Lao Tzu)] |