65 ideas
| 20388 | 'Necessary' conditions are requirements, and 'sufficient' conditions are guarantees [Davies,S] |
| 20389 | A definition of a thing gives all the requirements which add up to a guarantee of it [Davies,S] |
| 20391 | Feminists warn that ideologies use timeless objective definitions as a tool of repression [Davies,S] |
| 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] |
| 458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
| 5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
| 13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
| 457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
| 462 | One vision is produced by both eyes [Empedocles] |
| 22765 | Wisdom and thought are shared by all things [Empedocles] |
| 1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
| 20387 | Aesthetic experience involves perception, but also imagination and understanding [Davies,S] |
| 20385 | The faculty of 'taste' was posited to explain why only some people had aesthetic appreciation [Davies,S] |
| 20386 | The sublime is negative in awareness of insignificance, and positive in showing understanding [Davies,S] |
| 20384 | The idea that art forms are linked into a single concept began in the 1740s [Davies,S] |
| 20390 | Defining art as representation or expression or form were all undermined by the avant-garde [Davies,S] |
| 20392 | 'Aesthetic functionalism' says art is what is intended to create aesthetic experiences [Davies,S] |
| 20405 | Music may be expressive by being 'associated' with other emotional words or events [Davies,S] |
| 20403 | It seems unlikely that sad music expresses a composer's sadness; it takes ages to write [Davies,S] |
| 20393 | The 'institutional' theory says art is just something appropriately placed in the 'artworld' [Davies,S] |
| 20402 | Music is too definite to be put into words (not too indefinite!) [Davies,S] |
| 20395 | The title of a painting can be vital, and the artist decrees who the portrait represents [Davies,S] |
| 20396 | We must know what the work is meant to be, to evaluate the artist's achievement [Davies,S] |
| 20399 | Intentionalism says either meaning just is intention, or ('moderate') meaning is successful intention [Davies,S] |
| 20401 | The meaning is given by the audience's best guess at the author's intentions [Davies,S] |
| 20397 | If we could perfectly clone the Mona Lisa, the original would still be special [Davies,S] |
| 20398 | Art that is multiply instanced may require at least one instance [Davies,S] |
| 20404 | Music isn't just sad because it makes the listener feel sad [Davies,S] |
| 22705 | If the depiction of evil is glorified, that is an artistic flaw [Davies,S] |
| 22707 | It is an artistic defect if excessive moral outrage distorts the story, and narrows our sympathies [Davies,S] |
| 22706 | A work which seeks approval for immorality, but alienates the audience, is a failure [Davies,S] |
| 22704 | Immorality may or may not be an artistic defect [Davies,S] |
| 552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
| 589 | 'Nature' is just a word invented by people [Empedocles] |
| 21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
| 2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
| 6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
| 13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
| 459 | All change is unity through love or division through hate [Empedocles] |
| 13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
| 13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
| 460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
| 5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
| 461 | God is a pure, solitary, and eternal sphere [Empedocles] |
| 466 | God is pure mind permeating the universe [Empedocles] |
| 1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
| 1522 | It is wretched not to want to think clearly about the gods [Empedocles] |