82 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] |
17663 | If you know what it is, investigation is pointless. If you don't, investigation is impossible [Armstrong] |
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] |
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] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
17688 | Negative facts are supervenient on positive facts, suggesting they are positive facts [Armstrong] |
17691 | Nothing is genuinely related to itself [Armstrong] |
17679 | All instances of some property are strictly identical [Armstrong] |
12677 | Armstrong holds that all basic properties are categorical [Armstrong, by Ellis] |
17666 | Actualism means that ontology cannot contain what is merely physically possible [Armstrong] |
17667 | Dispositions exist, but their truth-makers are actual or categorical properties [Armstrong] |
17687 | If everything is powers there is a vicious regress, as powers are defined by more powers [Armstrong] |
17678 | Universals are just the repeatable features of a world [Armstrong] |
17669 | Realist regularity theories of laws need universals, to pick out the same phenomena [Armstrong] |
17677 | Past, present and future must be equally real if universals are instantiated [Armstrong] |
15442 | Universals are abstractions from their particular instances [Armstrong, by Lewis] |
17686 | Universals are abstractions from states of affairs [Armstrong] |
17668 | It is likely that particulars can be individuated by unique conjunctions of properties [Armstrong] |
16625 | In hylomorphism all the explanation of actions is in the form, and the matter doesn't do anything [Bacon] |
17680 | The identity of a thing with itself can be ruled out as a pseudo-property [Armstrong] |
17693 | The necessary/contingent distinction may need to recognise possibilities as real [Armstrong] |
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] |
17685 | Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities [Armstrong] |
17683 | Science suggests that the predicate 'grue' is not a genuine single universal [Armstrong] |
17675 | Unlike 'green', the 'grue' predicate involves a time and a change [Armstrong] |
17674 | The raven paradox has three disjuncts, confirmed by confirming any one of them [Armstrong] |
17672 | A good reason for something (the smoke) is not an explanation of it (the fire) [Armstrong] |
17684 | To explain observations by a regular law is to explain the observations by the observations [Armstrong] |
17676 | Best explanations explain the most by means of the least [Armstrong] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
17664 | Each subject has an appropriate level of abstraction [Armstrong] |
16624 | Stripped and passive matter is just a human invention [Bacon] |
17692 | We can't deduce the phenomena from the One [Armstrong] |
17689 | Absences might be effects, but surely not causes? [Armstrong] |
17682 | A universe couldn't consist of mere laws [Armstrong] |
17662 | Science depends on laws of nature to study unobserved times and spaces [Armstrong] |
17690 | Oaken conditional laws, Iron universal laws, and Steel necessary laws [Armstrong, by PG] |
17670 | Newton's First Law refers to bodies not acted upon by a force, but there may be no such body [Armstrong] |
8582 | Regularities are lawful if a second-order universal unites two first-order universals [Armstrong, by Lewis] |
17671 | A naive regularity view says if it never occurs then it is impossible [Armstrong] |
17681 | The laws of nature link properties with properties [Armstrong] |
16246 | Rather than take necessitation between universals as primitive, just make laws primitive [Maudlin on Armstrong] |
9480 | Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything [Bird on Armstrong] |