85 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] |
| 13395 | If an analysis shows the features of a concept, it doesn't seem to 'reduce' the concept [Jubien] |
| 14092 | Philosophers are often too fussy about words, dismissing perfectly useful ordinary terms [Rosen] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 14100 | Figuring in the definition of a thing doesn't make it a part of that thing [Rosen] |
| 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] |
| 13378 | It is a mistake to think that the logic developed for mathematics can clarify language and philosophy [Jubien] |
| 13402 | We only grasp a name if we know whether to apply it when the bearer changes [Jubien] |
| 13405 | The baptiser picks the bearer of a name, but social use decides the category [Jubien] |
| 13399 | Examples show that ordinary proper names are not rigid designators [Jubien] |
| 13398 | We could make a contingent description into a rigid and necessary one by adding 'actual' to it [Jubien] |
| 13392 | Philosophers reduce complex English kind-quantifiers to the simplistic first-order quantifier [Jubien] |
| 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] |
| 14096 | Explanations fail to be monotonic [Rosen] |
| 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] |
| 13404 | To exist necessarily is to have an essence whose own essence must be instantiated [Jubien] |
| 14097 | Things could be true 'in virtue of' others as relations between truths, or between truths and items [Rosen] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 13386 | If objects are just conventional, there is no ontological distinction between stuff and things [Jubien] |
| 14095 | Facts are structures of worldly items, rather like sentences, individuated by their ingredients [Rosen] |
| 13403 | The category of Venus is not 'object', or even 'planet', but a particular class of good-sized object [Jubien] |
| 14093 | An 'intrinsic' property is one that depends on a thing and its parts, and not on its relations [Rosen] |
| 13375 | The idea that every entity must have identity conditions is an unfortunate misunderstanding [Jubien] |
| 13393 | Any entity has the unique property of being that specific entity [Jubien] |
| 13388 | It is incoherent to think that a given entity depends on its kind for its existence [Jubien] |
| 13384 | Objects need conventions for their matter, their temporal possibility, and their spatial possibility [Jubien] |
| 13385 | Basically, the world doesn't have ready-made 'objects'; we carve objects any way we like [Jubien] |
| 13383 | If the statue is loved and the clay hated, that is about the object first qua statue, then qua clay [Jubien] |
| 13400 | If one entity is an object, a statue, and some clay, these come apart in at least three ways [Jubien] |
| 13401 | The idea of coincident objects is a last resort, as it is opposed to commonsense naturalism [Jubien] |
| 13380 | Parts seem to matter when it is just an object, but not matter when it is a kind of object [Jubien] |
| 13376 | We should not regard essentialism as just nontrivial de re necessity [Jubien] |
| 13381 | Thinking of them as 'ships' the repaired ship is the original, but as 'objects' the reassembly is the original [Jubien] |
| 13382 | Rearranging the planks as a ship is confusing; we'd say it was the same 'object' with a different arrangement [Jubien] |
| 13379 | If two objects are indiscernible across spacetime, how could we decide whether or not they are the same? [Jubien] |
| 14094 | The excellent notion of metaphysical 'necessity' cannot be defined [Rosen] |
| 13394 | Entailment does not result from mutual necessity; mutual necessity ensures entailment [Jubien] |
| 13391 | Modality concerns relations among platonic properties [Jubien] |
| 13374 | To analyse modality, we must give accounts of objects, properties and relations [Jubien] |
| 14101 | Are necessary truths rooted in essences, or also in basic grounding laws? [Rosen] |
| 13389 | The love of possible worlds is part of the dream that technical logic solves philosophical problems [Jubien] |
| 13390 | Possible worlds don't explain necessity, because they are a bunch of parallel contingencies [Jubien] |
| 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] |
| 13396 | Analysing mental concepts points to 'inclusionism' - that mental phenomena are part of the physical [Jubien] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 13377 | First-order logic tilts in favour of the direct reference theory, in its use of constants for objects [Jubien] |
| 14099 | 'Bachelor' consists in or reduces to 'unmarried' male, but not the other way around [Rosen] |
| 14098 | An acid is just a proton donor [Rosen] |