41 ideas
| 7278 | Words of wisdom are precise and clear [Zhuangzi (Chuang Tzu)] |
| 7281 | Don't even start, let's just stay put [Zhuangzi (Chuang Tzu)] |
| 7282 | Disagreement means you do not understand at all [Zhuangzi (Chuang Tzu)] |
| 7284 | If you beat me in argument, does that mean you are right? [Zhuangzi (Chuang 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] |
| 7289 | Do not try to do things, or to master knowledge; just be empty [Zhuangzi (Chuang Tzu)] |
| 19544 | Closure says if you know P, and also know P implies Q, then you must know Q [Dretske] |
| 19545 | We needn't regret the implications of our regrets; regretting drinking too much implies the past is real [Dretske] |
| 19547 | Reasons for believing P may not transmit to its implication, Q [Dretske] |
| 19546 | Knowing by visual perception is not the same as knowing by implication [Dretske] |
| 19548 | The only way to preserve our homely truths is to abandon closure [Dretske] |
| 19549 | P may imply Q, but evidence for P doesn't imply evidence for Q, so closure fails [Dretske] |
| 19550 | We know past events by memory, but we don't know the past is real (an implication) by memory [Dretske] |
| 23403 | You know you were dreaming when you wake, but there might then be a greater awakening from that [Zhuangzi (Chuang Tzu)] |
| 7285 | Did Chuang Tzu dream he was a butterfly, or a butterfly dream he was Chuang Tzu? [Zhuangzi (Chuang Tzu)] |
| 7277 | The perfect man has no self [Zhuangzi (Chuang Tzu)] |
| 7286 | To see with true clarity, your self must be irrelevant [Zhuangzi (Chuang Tzu)] |
| 7279 | If words can't be defined, they may just be the chirruping of chicks [Zhuangzi (Chuang Tzu)] |
| 23404 | Words are for meaning, and once you have that you can forget the words [Zhuangzi (Chuang Tzu)] |
| 7283 | Great courage is not violent [Zhuangzi (Chuang Tzu)] |
| 7280 | As all life is one, what need is there for words? [Zhuangzi (Chuang Tzu)] |
| 7288 | Go with the flow, and be one with the void of Heaven [Zhuangzi (Chuang Tzu)] |
| 7287 | Fish forget about each other in the pond and forget each other in the Tao [Zhuangzi (Chuang Tzu)] |