31 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] |
10429 | It is best to say that a name designates iff there is something for it to designate [Sainsbury] |
10425 | Definite descriptions may not be referring expressions, since they can fail to refer [Sainsbury] |
10438 | Definite descriptions are usually rigid in subject, but not in predicate, position [Sainsbury] |
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] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [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] |
8983 | If 'red' is vague, then membership of the set of red things is vague, so there is no set of red things [Sainsbury] |
8986 | We should abandon classifying by pigeon-holes, and classify around paradigms [Sainsbury] |
8982 | Vague concepts are concepts without boundaries [Sainsbury] |
8984 | If concepts are vague, people avoid boundaries, can't spot them, and don't want them [Sainsbury] |
8985 | Boundaryless concepts tend to come in pairs, such as child/adult, hot/cold [Sainsbury] |
10432 | A new usage of a name could arise from a mistaken baptism of nothing [Sainsbury] |
10434 | Even a quantifier like 'someone' can be used referentially [Sainsbury] |
20930 | The existence of law is one thing, its merits and demerits another [Austin,J] |
10431 | Things are thought to have a function, even when they can't perform them [Sainsbury] |