43 ideas
| 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] |
| 15105 | F(x) walked into a bar. The barman said.. [Sommers,W] |
| 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] |
| 12408 | Sartre to Waitress: Coffee with no cream, please... [Sommers,W] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 12397 | Said Plato: 'The things that we feel... [Sommers,W] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 12407 | Barman to Descartes: Would you like another drink?... [Sommers,W] |
| 12399 | There was a young student called Fred... [Sommers,W] |
| 20963 | A philosopher and his wife are out for a drive... [Sommers,W] |
| 12404 | Dear Sir, Your astonishment's odd.... [Sommers,W] |
| 12403 | There once was a man who said: 'God... [Sommers,W] |
| 12402 | ..But if he's a student of Berkeley... [Sommers,W] |
| 12409 | The philosopher Berkeley once said.. [Sommers,W] |
| 14694 | "My dog's got synaesthesia." How does he smell? ..... [Sommers,W] |
| 12401 | A toper who spies in the distance... [Sommers,W] |
| 12410 | There once was a man who said 'Damn!... [Sommers,W] |
| 9392 | How do behaviourists greet each other? [Sommers,W] |
| 12405 | 'If you're aristocratic,' said Nietzsche... [Sommers,W] |
| 9391 | Why do anarchists drink herbal tea? [Sommers,W] |
| 12400 | Cries the maid: 'You must marry me Hume!'... [Sommers,W] |
| 16527 | Causation - we all thought we knew it/ Till Hume came along and saw through it/…. [Sommers,W] |
| 17592 | The barman called 'Time!', and Augustine said..... [Sommers,W] |
| 15208 | The past, present and future walked into a bar.... [Sommers,W] |