35 ideas
| 15957 | Essential definitions show the differences that discriminate things, and make them what they are [Boyle] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [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] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [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] |
| 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] |
| 12354 | A 'categorial' property is had by virtue of being or having an item from a category [Wedin] |
| 15965 | Boyle attacked a contemporary belief that powers were occult things [Boyle, by Alexander,P] |
| 16735 | In the 17th century, 'disposition' usually just means the spatial arrangement of parts [Boyle, by Pasnau] |
| 12358 | Substance is a principle and a kind of cause [Wedin] |
| 16034 | Form is not a separate substance, but just the manner, modification or 'stamp' of matter [Boyle] |
| 15953 | To cite a substantial form tells us what produced the effect, but not how it did it [Boyle] |
| 12346 | Form explains why some matter is of a certain kind, and that is explanatory bedrock [Wedin] |
| 15962 | Boyle's term 'texture' is not something you feel, but is unobservable structures of particles [Boyle, by Alexander,P] |
| 15964 | Boyle's secondary qualities are not illusory, or 'in the mind' [Boyle, by Alexander,P] |
| 16736 | Explanation is generally to deduce it from something better known, which comes in degrees [Boyle] |
| 15960 | Explanation is deducing a phenomenon from some nature better known to us [Boyle] |
| 16737 | The best explanations get down to primary basics, but others go less deep [Boyle] |
| 15952 | The corpuscles just have shape, size and motion, which explains things without 'sympathies' or 'forces' [Boyle, by Alexander,P] |
| 15972 | The corpuscular theory allows motion, but does not include forces between the particles [Boyle, by Alexander,P] |
| 15961 | I don't see how mere moving matter can lead to the bodies of men and animals, and especially their seeds [Boyle] |