99 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] |
| 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] |
| 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] |
| 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] |
| 1635 | Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper [Quine] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 14562 | A process is unified as an expression of a collection of causal powers [Mumford/Anjum] |
| 14541 | Events are essentially changes; property exemplifications are just states of affairs [Mumford/Anjum] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 14553 | Weak emergence is just unexpected, and strong emergence is beyond all deduction [Mumford/Anjum] |
| 14538 | Powers explain properties, causes, modality, events, and perhaps even particulars [Mumford/Anjum] |
| 14555 | Powers offer no more explanation of nature than laws do [Mumford/Anjum] |
| 14557 | Powers are not just basic forces, since they combine to make new powers [Mumford/Anjum] |
| 14583 | Dispositionality is a natural selection function, picking outcomes from the range of possibilities [Mumford/Anjum] |
| 14536 | We say 'power' and 'disposition' are equivalent, but some say dispositions are manifestable [Mumford/Anjum] |
| 14584 | The simple conditional analysis of dispositions doesn't allow for possible prevention [Mumford/Anjum] |
| 14582 | Might dispositions be reduced to normativity, or to intentionality? [Mumford/Anjum] |
| 14542 | If statue and clay fall and crush someone, the event is not overdetermined [Mumford/Anjum] |
| 14535 | Pandispositionalists say structures are clusters of causal powers [Mumford/Anjum] |
| 14561 | Perdurantism imposes no order on temporal parts, so sequences of events are contingent [Mumford/Anjum] |
| 14579 | Dispositionality is the core modality, with possibility and necessity as its extreme cases [Mumford/Anjum] |
| 14580 | Dispositions may suggest modality to us - as what might not have been, and what could have been [Mumford/Anjum] |
| 14552 | Relations are naturally necessary when they are generated by the essential mechanisms of the world [Mumford/Anjum] |
| 14578 | Possibility might be non-contradiction, or recombinations of the actual, or truth in possible worlds [Mumford/Anjum] |
| 14549 | Maybe truths are necessitated by the facts which are their truthmakers [Mumford/Anjum] |
| 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] |
| 14585 | We have more than five senses; balance and proprioception, for example [Mumford/Anjum] |
| 7627 | You can't reduce epistemology to psychology, because that presupposes epistemology [Maund on Quine] |
| 8871 | We should abandon a search for justification or foundations, and focus on how knowledge is acquired [Quine, by Davidson] |
| 8826 | If we abandon justification and normativity in epistemology, we must also abandon knowledge [Kim on Quine] |
| 8827 | Without normativity, naturalized epistemology isn't even about beliefs [Kim on Quine] |
| 8899 | Epistemology is a part of psychology, studying how our theories relate to our evidence [Quine] |
| 14576 | Smoking disposes towards cancer; smokers without cancer do not falsify this claim [Mumford/Anjum] |
| 14551 | If causation were necessary, the past would fix the future, and induction would be simple [Mumford/Anjum] |
| 14571 | The only full uniformities in nature occur from the essences of fundamental things [Mumford/Anjum] |
| 14570 | Nature is not completely uniform, and some regular causes sometimes fail to produce their effects [Mumford/Anjum] |
| 14569 | It is tempting to think that only entailment provides a full explanation [Mumford/Anjum] |
| 14568 | A structure won't give a causal explanation unless we know the powers of the structure [Mumford/Anjum] |
| 14556 | Strong emergence seems to imply top-down causation, originating in consciousness [Mumford/Anjum] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 8898 | Inculcations of meanings of words rests ultimately on sensory evidence [Quine] |
| 8900 | In observation sentences, we could substitute community acceptance for analyticity [Quine] |
| 14566 | Causation by absence is not real causation, but part of our explanatory practices [Mumford/Anjum] |
| 14577 | Causation may not be transitive. Does a fire cause itself to be extinguished by the sprinklers? [Mumford/Anjum] |
| 14563 | Causation is the passing around of powers [Mumford/Anjum] |
| 14587 | We take causation to be primitive, as it is hard to see how it could be further reduced [Mumford/Anjum] |
| 14533 | Causation doesn't have two distinct relata; it is a single unfolding process [Mumford/Anjum] |
| 14558 | A collision is a process, which involves simultaneous happenings, but not instantaneous ones [Mumford/Anjum] |
| 14559 | Does causation need a third tying ingredient, or just two that meet, or might there be a single process? [Mumford/Anjum] |
| 14565 | Sugar dissolving is a process taking time, not one event and then another [Mumford/Anjum] |
| 14567 | Privileging one cause is just an epistemic or pragmatic matter, not an ontological one [Mumford/Anjum] |
| 14537 | Coincidence is conjunction without causation; smoking causing cancer is the reverse [Mumford/Anjum] |
| 14572 | Is a cause because of counterfactual dependence, or is the dependence because there is a cause? [Mumford/Anjum] |
| 14573 | Occasionally a cause makes no difference (pre-emption, perhaps) so the counterfactual is false [Mumford/Anjum] |
| 14574 | Cases of preventing a prevention may give counterfactual dependence without causation [Mumford/Anjum] |
| 14539 | Nature can be interfered with, so a cause never necessitates its effects [Mumford/Anjum] |
| 14550 | We assert causes without asserting that they necessitate their effects [Mumford/Anjum] |
| 14546 | Necessary causation should survive antecedent strengthening, but no cause can always survive that [Mumford/Anjum] |
| 14575 | A 'ceteris paribus' clause implies that a conditional only has dispositional force [Mumford/Anjum] |
| 14548 | There may be necessitation in the world, but causation does not supply it [Mumford/Anjum] |
| 14554 | Laws are nothing more than descriptions of the behaviour of powers [Mumford/Anjum] |
| 14564 | If laws are equations, cause and effect must be simultaneous (or the law would be falsified)! [Mumford/Anjum] |