87 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] |
23000 | Vicious regresses force you to another level; non-vicious imply another level [Baron/Miller] |
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] |
23024 | A traveller takes a copy of a picture into the past, gives it the artist, who then creates the original! [Baron/Miller] |
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] |
23008 | Grounding is intended as a relation that fits dependences between things [Baron/Miller] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
23018 | How does a changing object retain identity or have incompatible properties over time? [Baron/Miller] |
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] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
3785 | You can't separate acts from the people performing them [Glover] |
3786 | Aggression in defence may be beneficial but morally corrupting [Glover] |
3784 | Duty prohibits some acts, whatever their consequences [Glover] |
3782 | Satisfaction of desires is not at all the same as achieving happiness [Glover, by PG] |
3787 | Rule-utilitarianism is either act-utilitarianism, or not really utilitarian [Glover] |
3783 | How can utilitarianism decide the ideal population size? [Glover] |
23011 | Modern accounts of causation involve either processes or counterfactuals [Baron/Miller] |
23013 | The main process theory of causation says it is transference of mass, energy, momentum or charge [Baron/Miller] |
23014 | If causes are processes, what is causation by omission? (Distinguish legal from scientific causes?) [Baron/Miller] |
23015 | The counterfactual theory of causation handles the problem no matter what causes actually are [Baron/Miller] |
23016 | Counterfactual theories struggle with pre-emption by a causal back-up system [Baron/Miller] |
23009 | There is no second 'law' of thermodynamics; it just reflects probabilities of certain microstates [Baron/Miller] |
23002 | In relativity space and time depend on one's motion, but spacetime gives an invariant metric [Baron/Miller] |
22988 | The block universe theory says entities of all times exist, and time is the B-series [Baron/Miller] |
22991 | How can we know this is the present moment, if other times are real? [Baron/Miller] |
22992 | If we are actually in the past then we shouldn't experience time passing [Baron/Miller] |
22994 | Erzatz Presentism allows the existence of other times, with only the present 'actualised' [Baron/Miller] |
22998 | How do presentists explain relations between things existing at different times? [Baron/Miller] |
23017 | Presentism needs endurantism, because other theories imply most of the object doesn't exist [Baron/Miller] |
23023 | How can presentists move to the next future moment, if that doesn't exist? [Baron/Miller] |
22995 | Most of the sciences depend on the concept of time [Baron/Miller] |
22993 | For abstractionists past times might still exist, althought their objects don't [Baron/Miller] |
23001 | The error theory of time's passage says it is either a misdescription or a false inference [Baron/Miller] |
22999 | It is meaningless to measure the rate of time using time itself, and without a rate there is no flow [Baron/Miller] |
22986 | The C-series rejects A and B, and just sees times as order by betweenness, without direction [Baron/Miller] |
22996 | The A-series has to treat being past, present or future as properties [Baron/Miller] |
23007 | The B-series can have a direction, as long as it does not arise from temporal flow [Baron/Miller] |
23003 | Static theories cannot account for time's obvious asymmetry, so time must be dynamic [Baron/Miller] |
23004 | The direction of time is either primitive, or reducible to something else [Baron/Miller] |
23005 | The kaon does not seem to be time-reversal invariant, unlike the rest of nature [Baron/Miller] |
23006 | Maybe the past is just the direction of decreasing entropy [Baron/Miller] |
23010 | We could explain time's direction by causation: past is the direction of causes, future of effects [Baron/Miller] |
22989 | Static time theory presents change as one property at t1, and a different property at t2 [Baron/Miller] |
23020 | If a time traveller kills his youthful grandfather, he both exists and fails to exist [Baron/Miller] |
23022 | Presentism means there no existing past for a time traveller to visit [Baron/Miller] |
22987 | The past (unlike the future) is fixed, along with truths about it, by the existence of past objects [Baron/Miller] |
22990 | The moving spotlight says entities can have properties of being present, past or future [Baron/Miller] |
22997 | The present moment is a matter of existence, not of acquiring a property [Baron/Miller] |