33 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] |
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] |
8628 | I hold that algebra and number are developments of logic [Jevons] |
23529 | Conduct is not isolated from its effect on the moral code [Hart,HLA] |
23530 | The great danger of democracy is that the oppression of the minority becomes unobjectionable [Hart,HLA] |
23522 | In an organised society all actions have some effect on other people [Hart,HLA] |
23528 | The value of liberty allows freedom of action, even if that distresses other people [Hart,HLA] |
21004 | Hart (against Bentham) says human rights are what motivate legal rights [Hart,HLA, by Sen] |
23523 | The principle of legality requires crimes to be precisely defined in advance of any action [Hart,HLA] |
23524 | Some private moral issues are no concern of the law [Hart,HLA] |
23521 | Do morals influence law? Is morality an aspect of law? Can law be morally criticised? [Hart,HLA] |
23525 | Is the enforcement of morality morally justifiable? [Hart,HLA] |
23526 | Modern law still suppresses practices seen as immoral, and yet harmless [Hart,HLA] |
20932 | Positive law needs secondary 'rules of recognition' for their correct application [Hart,HLA, by Zimmermann,J] |
20931 | Hart replaced positivism with the democratic requirement of the people's acceptance [Hart,HLA, by Zimmermann,J] |
23527 | Moral wickedness of an offence is always relevant to the degree of punishment [Hart,HLA] |