73 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] |
16440 | I don't think Lewis's cost-benefit reflective equilibrium approach offers enough guidance [Stalnaker] |
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] |
16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
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] |
16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
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] |
16464 | We regiment to get semantic structure, for evaluating arguments, and understanding complexities [Stalnaker] |
16465 | In 'S was F or some other than S was F', the disjuncts need S, but the whole disjunction doesn't [Stalnaker] |
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] |
9117 | The smallest heap has four objects: three on the bottom, one on the top [Hart,WD, by Sorensen] |
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] |
16434 | Some say what exists must do so, and nothing else could possible exist [Stalnaker] |
16439 | A nominalist view says existence is having spatio-temporal location [Stalnaker] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
16443 | Properties are modal, involving possible situations where they are exemplified [Stalnaker] |
16471 | I accept a hierarchy of properties of properties of properties [Stalnaker] |
16452 | Dispositions have modal properties, of which properties things would have counterfactually [Stalnaker] |
16467 | 'Socrates is essentially human' seems to say nothing could be Socrates if it was not human [Stalnaker] |
16453 | The bundle theory makes the identity of indiscernibles a necessity, since the thing is the properties [Stalnaker] |
16466 | Strong necessity is always true; weak necessity is cannot be false [Stalnaker] |
16438 | Necessity and possibility are fundamental, and there can be no reductive analysis of them [Stalnaker] |
16436 | Modal concepts are central to the actual world, and shouldn't need extravagant metaphysics [Stalnaker] |
16433 | Given actualism, how can there be possible individuals, other than the actual ones? [Stalnaker] |
16437 | Possible worlds are properties [Stalnaker] |
16444 | Possible worlds don't reduce modality, they regiment it to reveal its structure [Stalnaker] |
16445 | I think of worlds as cells (rather than points) in logical space [Stalnaker] |
16454 | Modal properties depend on the choice of a counterpart, which is unconstrained by metaphysics [Stalnaker] |
16450 | Anti-haecceitism says there is no more to an individual than meeting some qualitative conditions [Stalnaker] |
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] |
16474 | How can we know what we are thinking, if content depends on something we don't know? [Stalnaker] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
16461 | We still lack an agreed semantics for quantifiers in natural language [Stalnaker] |
16448 | Possible world semantics may not reduce modality, but it can explain it [Stalnaker] |
16442 | I take propositions to be truth conditions [Stalnaker] |
16447 | A theory of propositions at least needs primitive properties of consistency and of truth [Stalnaker] |
16446 | Propositions presumably don't exist if the things they refer to don't exist [Stalnaker] |