75 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] |
| 6891 | Quine's naturalistic and empirical view is based entirely on first-order logic and set theory [Quine, by Mautner] |
| 6310 | Enquiry needs a conceptual scheme, so we should retain the best available [Quine] |
| 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] |
| 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] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [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] |
| 12798 | Plurals can in principle be paraphrased away altogether [Quine] |
| 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] |
| 13513 | Models are ways the world might be from a first-order point of view [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] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 17905 | Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [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] |
| 9556 | Nearly all of mathematics has to quantify over abstract objects [Quine] |
| 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] |
| 16462 | The quest for ultimate categories is the quest for a simple clear pattern of notation [Quine] |
| 15723 | Either dispositions rest on structures, or we keep saying 'all things being equal' [Quine] |
| 15490 | Explain unmanifested dispositions as structural similarities to objects which have manifested them [Quine, by Martin,CB] |
| 15034 | Are genera and species real or conceptual? bodies or incorporeal? in sensibles or separate from them? [Porphyry] |
| 8504 | Quine aims to deal with properties by the use of eternal open sentences, or classes [Quine, by Devitt] |
| 8464 | Physical objects in space-time are just events or processes, no matter how disconnected [Quine] |
| 7924 | The notion of a physical object is by far the most useful one for science [Quine] |
| 8482 | Mathematicians must be rational but not two-legged, cyclists the opposite. So a mathematical cyclist? [Quine] |
| 12136 | Cyclist are not actually essentially two-legged [Brody on Quine] |
| 17594 | We can paraphrase 'x=y' as a sequence of the form 'if Fx then Fy' [Quine] |
| 15725 | Normal conditionals have a truth-value gap when the antecedent is false. [Quine] |
| 15722 | Conditionals are pointless if the truth value of the antecedent is known [Quine] |
| 15719 | We feign belief in counterfactual antecedents, and assess how convincing the consequent is [Quine] |
| 15721 | Counterfactuals are plausible when dispositions are involved, as they imply structures [Quine] |
| 15724 | Counterfactuals have no place in a strict account of science [Quine] |
| 15720 | What stays the same in assessing a counterfactual antecedent depends on context [Quine] |
| 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] |
| 4630 | Two theories can be internally consistent and match all the facts, yet be inconsistent with one another [Quine, by Baggini /Fosl] |
| 3131 | Quine expresses the instrumental version of eliminativism [Quine, by Rey] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 3988 | Indeterminacy of translation also implies indeterminacy in interpreting people's mental states [Dennett on Quine] |
| 6311 | The firmer the links between sentences and stimuli, the less translations can diverge [Quine] |
| 6312 | We can never precisely pin down how to translate the native word 'Gavagai' [Quine] |
| 6313 | Stimulus synonymy of 'Gavagai' and 'Rabbit' does not even guarantee they are coextensive [Quine] |
| 6317 | Dispositions to speech behaviour, and actual speech, are never enough to fix any one translation [Quine] |
| 6315 | We should be suspicious of a translation which implies that a people have very strange beliefs [Quine] |
| 6314 | Weird translations are always possible, but they improve if we impose our own logic on them [Quine] |