45 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] |
23590 | Criminal responsibility can be fully assigned to each member of a group [Walzer] |
23578 | Double Effect needs a double intention - to achieve the good, and minimise the evil [Walzer] |
23564 | Deep ethical theory is very controversial, but we have to live with higher ethical practice [Walzer] |
467 | A virtue is a combination of intelligence, strength and luck [Ion] |
23568 | If whole states possess rights, there can be social relations between states [Walzer] |
23571 | States can rightly pre-empt real and serious threats [Walzer] |
23572 | Just wars are self-defence, or a rightful intercession in another's troubles [Walzer] |
23581 | The aim of reprisals is to enforce the rules of war [Walzer] |
23582 | Reprisal is defensible, as an alternative to war [Walzer] |
23588 | With nuclear weapons we have a permanent supreme emergency (which is unstable) [Walzer] |
23580 | States need not endure attacks passively, and successful reprisals are legitimate [Walzer] |
23567 | Even non-violent intrusive acts between states count as aggression, if they justify resistance [Walzer] |
23570 | The only good reason for fighting is in defence of rights [Walzer] |
23587 | Nuclear bombs are not for normal war; they undermine the 'just war', with a new morality [Walzer] |
23573 | For moral reasons, a just war must be a limited war [Walzer] |
23577 | Napoleon said 'I don't care about the deaths of a million men' [Walzer] |
23593 | Jus ad bellum and Jus in bello are independent; unjust wars can be fought in a just way [Walzer] |
23574 | The duties and moral status of loyal and obedient soldiers is the same in defence and aggression [Walzer] |
23575 | We can't blame soldiers for anything they do which clearly promotes victory [Walzer] |
23584 | Rejecting Combatant Equality allows just soldiers to be harsher, even to the extreme [Walzer] |
23614 | Even aggressor soldiers are not criminals, so they have equal rights with their opponents [Walzer] |
23589 | Kidnapped sailors and volunteers have different obligations to the passengers [Walzer] |
23579 | Soldiers will only protect civilians if they feel safe from them [Walzer] |
23586 | What matters in war is unacceptable targets, not unacceptable weapons [Walzer] |
23591 | If the oppressor is cruel, nonviolence is either surrender, or a mere gesture [Walzer] |
23592 | We can only lead war towards peace if we firmly enforce the rules of war [Walzer] |