35 ideas
| 21959 | Metaphysics is the most general attempt to make sense of things [Moore,AW] |
| 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] |
| 21958 | Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW] |
| 21233 | The beautiful is whatever it is intrinsically good to admire [Moore,GE] |
| 8039 | Moore tries to show that 'good' is indefinable, but doesn't understand what a definition is [MacIntyre on Moore,GE] |
| 22151 | The Open Question argument leads to anti-realism and the fact-value distinction [Boulter on Moore,GE] |
| 11056 | The naturalistic fallacy claims that natural qualties can define 'good' [Moore,GE] |
| 8033 | Moore cannot show why something being good gives us a reason for action [MacIntyre on Moore,GE] |
| 8032 | Can learning to recognise a good friend help us to recognise a good watch? [MacIntyre on Moore,GE] |
| 11050 | Moore's combination of antinaturalism with strong supervenience on the natural is incoherent [Hanna on Moore,GE] |
| 23726 | Despite Moore's caution, non-naturalists incline towards intuitionism [Moore,GE, by Smith,M] |
| 18676 | We should ask what we would judge to be good if it existed in absolute isolation [Moore,GE] |
| 11057 | It is always an open question whether anything that is natural is good [Moore,GE] |
| 5925 | The three main values are good, right and beauty [Moore,GE, by Ross] |
| 5902 | For Moore, 'right' is what produces good [Moore,GE, by Ross] |
| 5903 | 'Right' means 'cause of good result' (hence 'useful'), so the end does justify the means [Moore,GE] |
| 5907 | Relationships imply duties to people, not merely the obligation to benefit them [Ross on Moore,GE] |