84 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] |
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] |
14381 | A statue is essentially the statue, but its lump is not essentially a statue, so statue isn't lump [Yablo, by Rocca] |
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] |
23101 | Intuitions don't prove things; they just receptivity to interpretations [Kekes] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
23086 | Liberals say we are only responsible for fully autonomous actions [Kekes] |
23100 | Collective responsibility conflicts with responsibility's requirement of authonomy [Kekes] |
23087 | Much human evil is not autonomous, so moral responsibility need not be autonomous [Kekes] |
23089 | Evil people may not be autonomously aware, if they misjudge the situation [Kekes] |
23093 | Moral and causal responsibility are not clearly distinct [Kekes] |
23098 | Effects show the existence of moral responsibility, and mental states show the degree [Kekes] |
23096 | Morality should aim to prevent all evil actions, not just autonomous ones [Kekes] |
23095 | Why should moral responsibility depend on autonomy, rather than social role or experience? [Kekes] |
23094 | Ought implies can means moral responsibility needs autonomy [Kekes] |
23090 | Liberals assume people are naturally free, equal, rational, and morally good [Kekes] |
23117 | Love should be partial, and discriminate in favour of its object [Kekes] |
23119 | Sentimental love distorts its object [Kekes] |
23088 | Evil is not deviation from the good, any more than good is a deviation from evil [Kekes] |
23097 | What matters for morality is the effects of action, not the psychological causes [Kekes] |
23099 | It is said that if an agent is not autonomous then their evil actions don't reflect on their character [Kekes] |
23118 | Awareness of others' suffering doesn't create an obligation to help [Kekes] |
23109 | The veil of ignorance is only needed because people have bad motivations [Kekes] |
23114 | The chief function of the state is to arbitrate between contending visions of the good life [Kekes] |
23116 | Citizenship is easier than parenthood [Kekes] |
23103 | Power is meant to be confined to representatives, and subsequent delegation [Kekes] |
23107 | Prosperity is a higher social virtue than justice [Kekes] |
23081 | Liberal basics are pluralism, freedom, rights, equality, and distributive justice - for autonomy [Kekes] |
23085 | The key liberal values are explained by the one core value, which is autonomy [Kekes] |
23092 | Agents have little control over the capacities needed for liberal autonomy [Kekes] |
23102 | Liberals are egalitarians, but in varying degrees [Kekes] |
23084 | Are egalitarians too coercive, or not egalitarian enough, or lax over morality? [Kekes] |
23079 | Liberal justice ignores desert, which is the essence of justice [Kekes] |
23091 | Why do liberals not see a much wider range of values as basic? [Kekes] |
23112 | Liberals ignore contingency, and think people are good and equal, and institutions cause evil [Kekes] |
23082 | Liberal distribution cares more about recipients than donors [Kekes] |
23106 | To rectify the undeserved equality, we should give men longer and women shorter lives [Kekes] |
23121 | It is just a fact that some people are morally better than others [Kekes] |
23105 | It is not deplorable that billionaires have more than millionaires [Kekes] |
23120 | The problem is basic insufficiency of resources, not their inequality [Kekes] |
23108 | Justice combines consistency and desert; treat likes alike, judging likeness by desert [Kekes] |
23083 | Liberal welfare focuses on need rather than desert [Kekes] |
23113 | Sexual morality doesn't require monogamy, but it needs a group of sensible regulations [Kekes] |