36 ideas
17651 | Without words or other symbols, we have no world [Goodman] |
17652 | Truth is irrelevant if no statements are involved [Goodman] |
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] |
17656 | Being primitive or prior always depends on a constructional system [Goodman] |
17661 | We don't recognise patterns - we invent them [Goodman] |
17659 | Reality is largely a matter of habit [Goodman] |
17657 | We build our world, and ignore anything that won't fit [Goodman] |
16657 | Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent] |
17654 | A world can be full of variety or not, depending on how we sort it [Goodman] |
16658 | The only reality in the category of Relation is things from another category [Henry of Ghent] |
16645 | Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent] |
17653 | Things can only be judged the 'same' by citing some respect of sameness [Goodman] |
22012 | Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard] |
17660 | Discovery is often just finding a fit, like a jigsaw puzzle [Goodman] |
17658 | Users of digital thermometers recognise no temperatures in the gaps [Goodman] |
17650 | We lack frames of reference to transform physics, biology and psychology into one another [Goodman] |
17655 | Grue and green won't be in the same world, as that would block induction entirely [Goodman] |
17649 | If the world is one it has many aspects, and if there are many worlds they will collect into one [Goodman] |