83 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] |
14480 | Maybe analytic truths do not require truth-makers, as they place no demands on the world [Thomasson] |
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] |
14471 | Analytical entailments arise from combinations of meanings and inference rules [Thomasson] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [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] |
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] |
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] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
14493 | Existence might require playing a role in explanation, or in a causal story, or being composed in some way [Thomasson] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
14491 | Rival ontological claims can both be true, if there are analytic relationships between them [Thomasson] |
14489 | Theories do not avoid commitment to entities by avoiding certain terms or concepts [Thomasson] |
17377 | All descriptive language is classificatory [Dupré] |
17376 | We should aim for a classification which tells us as much as possible about the object [Dupré] |
14487 | The simple existence conditions for objects are established by our practices, and are met [Thomasson] |
14485 | Ordinary objects may be not indispensable, but they are nearly unavoidable [Thomasson] |
21651 | It is analytic that if simples are arranged chair-wise, then there is a chair [Thomasson, by Hofweber] |
14486 | Eliminativists haven't found existence conditions for chairs, beyond those of the word 'chair' [Thomasson] |
14467 | Ordinary objects are rejected, to avoid contradictions, or for greater economy in thought [Thomasson] |
14479 | To individuate people we need conventions, but conventions are made up by people [Thomasson] |
14481 | Wherever an object exists, there are intrinsic properties instantiating every modal profile [Thomasson] |
14482 | If the statue and the lump are two objects, they require separate properties, so we could add their masses [Thomasson] |
14483 | Given the similarity of statue and lump, what could possibly ground their modal properties? [Thomasson] |
17390 | Natural kinds don't need essentialism to be explanatory [Dupré] |
17389 | A species might have its essential genetic mechanism replaced by a new one [Dupré] |
17388 | It seems that species lack essential properties, so they can't be natural kinds [Dupré] |
14476 | Identity claims between objects are only well-formed if the categories are specified [Thomasson] |
14477 | Identical entities must be of the same category, and meet the criteria for the category [Thomasson] |
14478 | Modal Conventionalism says modality is analytic, not intrinsic to the world, and linguistic [Thomasson] |
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] |
14466 | A chief task of philosophy is making reflective sense of our common sense worldview [Thomasson] |
17374 | The possibility of prediction rests on determinism [Dupré] |
17378 | Presumably molecular structure seems important because we never have the Twin Earth experience [Dupré] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
14475 | How can causal theories of reference handle nonexistence claims? [Thomasson] |
14474 | Pure causal theories of reference have the 'qua problem', of what sort of things is being referred to [Thomasson] |
14488 | Analyticity is revealed through redundancy, as in 'He bought a house and a building' [Thomasson] |
17381 | Phylogenetics involves history, and cladism rests species on splits in lineage [Dupré] |
17385 | Kinds don't do anything (including evolve) because they are abstract [Dupré] |
17375 | Natural kinds are decided entirely by the intentions of our classification [Dupré] |
17379 | Borders between species are much less clear in vegetables than among animals [Dupré] |
17384 | Even atoms of an element differ, in the energy levels of their electrons [Dupré] |
17387 | Ecologists favour classifying by niche, even though that can clash with genealogy [Dupré] |
17382 | Cooks, unlike scientists, distinguish garlic from onions [Dupré] |
17380 | Wales may count as fish [Dupré] |
17383 | Species are the lowest-level classification in biology [Dupré] |
17386 | The theory of evolution is mainly about species [Dupré] |