73 ideas
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
15395 | Give up objects necessitating truths, and say their natures cause the truths? [Cameron] |
15394 | Truthmaker requires a commitment to tropes or states of affairs, for contingent truths [Cameron] |
13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
15401 | Essentialists say intrinsic properties arise from what the thing is, irrespective of surroundings [Cameron] |
15393 | An object's intrinsic properties are had in virtue of how it is, independently [Cameron] |
15396 | Most criteria for identity over time seem to leave two later objects identical to the earlier one [Cameron] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
23616 | Legal excuses are duress, ignorance, and diminished responsibility [McMahan] |
23606 | Liberty Rights are permissions, and Claim Rights are freedom from intervention [McMahan] |
23594 | Wars can be unjust, despite a just cause, if they are unnecessary or excessive or of mixed cause [McMahan] |
23620 | A person or state may be attacked if they are responsible for an unjustified threat [McMahan] |
23595 | The worst unjustified wars have no aim at all [McMahan] |
23597 | Just war theory says all and only persons posing a threat are liable to attack [McMahan] |
23598 | You (e.g. a police officer) are not liable to attack just because you pose a threat [McMahan] |
23619 | A defensive war is unjust, if it is responding to a just war [McMahan] |
23600 | Proportionality in fighting can't be judged independently of the justice of each side [McMahan] |
23603 | Can an army start an unjust war, and then fight justly to defend their own civilians? [McMahan] |
23611 | Soldiers cannot freely fight in unjust wars, just because they behave well when fighting [McMahan] |
23612 | The law of war differs from criminal law; attacking just combatants is immoral, but legal [McMahan] |
23617 | If the unjust combatants are morally excused they are innocent, so how can they be killed? [McMahan] |
23596 | If all combatants are seen as morally equal, that facilitates starting unjust wars [McMahan] |
23599 | You don't become a legitimate target, just because you violently resist an unjust attack [McMahan] |
23604 | Volunteer soldiers accept the risk of attack, but they don't agree to it, or to their deaths [McMahan] |
23605 | Soldiers cannot know enough facts to evaluate the justice of their war [McMahan] |
23608 | If being part of a big collective relieves soldiers of moral responsibility, why not the leaders too? [McMahan] |
23610 | If soldiers can't refuse to fight in unjust wars, can they choose to fight in just wars? [McMahan] |
23613 | Equality is both sides have permission, or both sides are justified, or one justified the other permitted [McMahan] |
23615 | Fighting unjustly under duress does not justify it, or permit it, but it may excuse it [McMahan] |
23602 | Innocence implies not being morally responsible, rather than merely being guiltless [McMahan] |
23618 | Unconditional surrender can't be demanded, since evil losers still have legitimate conditions [McMahan] |