78 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] |
3035 | Dialectic involves conversations with short questions and brief answers [Diog. Laertius] |
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] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [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] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [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] |
6504 | For physicalists, the only relations are spatial, temporal and causal [Robinson,H] |
6520 | If reality just has relational properties, what are its substantial ontological features? [Robinson,H] |
6485 | When a red object is viewed, the air in between does not become red [Robinson,H] |
6521 | Representative realists believe that laws of phenomena will apply to the physical world [Robinson,H] |
6509 | Representative realists believe some properties of sense-data are shared by the objects themselves [Robinson,H] |
6522 | Phenomenalism can be theistic (Berkeley), or sceptical (Hume), or analytic (20th century) [Robinson,H] |
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] |
6502 | Can we reduce perception to acquisition of information, which is reduced to causation or disposition? [Robinson,H] |
6513 | Would someone who recovered their sight recognise felt shapes just by looking? [Robinson,H] |
6512 | Secondary qualities have one sensory mode, but primary qualities can have more [Robinson,H] |
6497 | We say objects possess no intrinsic secondary qualities because physicists don't need them [Robinson,H] |
6494 | If objects are not coloured, and neither are sense-contents, we are left saying that nothing is coloured [Robinson,H] |
6499 | Shape can be experienced in different ways, but colour and sound only one way [Robinson,H] |
6500 | If secondary qualities match senses, would new senses create new qualities? [Robinson,H] |
6484 | Most moderate empiricists adopt Locke's representative theory of perception [Robinson,H] |
6508 | Sense-data leads to either representative realism or phenomenalism or idealism [Robinson,H] |
6480 | Sense-data do not have any intrinsic intentionality [Robinson,H] |
6482 | For idealists and phenomenalists sense-data are in objects; representative realists say they resemble objects [Robinson,H] |
6505 | Sense-data are rejected because they are a veil between us and reality, leading to scepticism [Robinson,H] |
6506 | 'Sense redly' sounds peculiar, but 'senses redly-squarely tablely' sounds far worse [Robinson,H] |
6507 | Adverbialism sees the contents of sense-experience as modes, not objects [Robinson,H] |
6511 | If there are only 'modes' of sensing, then an object can no more be red or square than it can be proud or lazy. [Robinson,H] |
1816 | Sceptics say demonstration depends on self-demonstrating things, or indemonstrable things [Diog. Laertius] |
1819 | Scepticism has two dogmas: that nothing is definable, and every argument has an opposite argument [Diog. Laertius] |
3064 | When sceptics say that nothing is definable, or all arguments have an opposite, they are being dogmatic [Diog. Laertius] |
3033 | Induction moves from some truths to similar ones, by contraries or consequents [Diog. Laertius] |
6515 | An explanation presupposes something that is improbable unless it is explained [Robinson,H] |
6517 | If all possibilities are equal, order seems (a priori) to need an explanation - or does it? [Robinson,H] |
6481 | If intentional states are intrinsically about other things, what are their own properties? [Robinson,H] |
6503 | Physicalism cannot allow internal intentional objects, as brain states can't be 'about' anything [Robinson,H] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
1838 | Cyrenaic pleasure is a motion, but Epicurean pleasure is a condition [Diog. Laertius] |
1769 | Cynics believe that when a man wishes for nothing he is like the gods [Diog. Laertius] |
6519 | Locke's solidity is not matter, because that is impenetrability and hardness combined [Robinson,H] |