94 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] |
| 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] |
| 13479 | Given that thinking aims at truth, logic gives universal rules for how to do it [Burge] |
| 8132 | We now have a much more sophisticated understanding of logical form in language [Burge] |
| 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] |
| 17622 | We come to believe mathematical propositions via their grounding in the structure [Burge] |
| 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] |
| 16901 | The equivalent algebra model of geometry loses some essential spatial meaning [Burge] |
| 9159 | You can't simply convert geometry into algebra, as some spatial content is lost [Burge] |
| 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] |
| 16902 | Peano arithmetic requires grasping 0 as a primitive number [Burge] |
| 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] |
| 16892 | Is apriority predicated mainly of truths and proofs, or of human cognition? [Burge] |
| 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] |
| 9382 | Subjects may be unaware of their epistemic 'entitlements', unlike their 'justifications' [Burge] |
| 20043 | Evolutionary explanations look to the past or the group, not to the individual [Stout,R] |
| 20058 | Not all explanation is causal. We don't explain a painting's beauty, or the irrationality of root-2, that way [Stout,R] |
| 8126 | Anti-individualism says the environment is involved in the individuation of some mental states [Burge] |
| 8127 | Broad concepts suggest an extension of the mind into the environment (less computer-like) [Burge] |
| 8129 | Anti-individualism may be incompatible with some sorts of self-knowledge [Burge] |
| 8131 | Some qualities of experience, like blurred vision, have no function at all [Burge] |
| 3115 | Are meaning and expressed concept the same thing? [Burge, by Segal] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 20035 | Philosophy of action studies the nature of agency, and of deliberate actions [Stout,R] |
| 20084 | Agency is causal processes that are sensitive to justification [Stout,R] |
| 20061 | Mental states and actions need to be separate, if one is to cause the other [Stout,R] |
| 20079 | Are actions bodily movements, or a sequence of intention-movement-result? [Stout,R] |
| 20080 | If one action leads to another, does it cause it, or is it part of it? [Stout,R] |
| 20059 | I do actions, but not events, so actions are not events [Stout,R] |
| 20081 | Bicycle riding is not just bodily movement - you also have to be on the bicycle [Stout,R] |
| 20039 | The causal theory says that actions are intentional when intention (or belief-desire) causes the act [Stout,R] |
| 20047 | Deciding what to do usually involves consulting the world, not our own minds [Stout,R] |
| 20065 | Should we study intentions in their own right, or only as part of intentional action? [Stout,R] |
| 20067 | You can have incompatible desires, but your intentions really ought to be consistent [Stout,R] |
| 20078 | The normativity of intentions would be obvious if they were internal promises [Stout,R] |
| 20044 | The rationalistic approach says actions are intentional when subject to justification [Stout,R] |
| 20036 | Intentional agency is seen in internal precursors of action, and in external reasons for the act [Stout,R] |
| 20066 | Speech needs sustained intentions, but not prior intentions [Stout,R] |
| 20073 | Bratman has to treat shared intentions as interrelated individual intentions [Stout,R] |
| 20069 | A request to pass the salt shares an intention that the request be passed on [Stout,R] |
| 20070 | An individual cannot express the intention that a group do something like moving a piano [Stout,R] |
| 20071 | An intention is a goal to which behaviour is adapted, for an individual or for a group [Stout,R] |
| 20038 | If the action of walking is just an act of will, then movement of the legs seems irrelevant [Stout,R] |
| 20050 | Most philosophers see causation as by an event or state in the agent, rather than the whole agent [Stout,R] |
| 20052 | If you don't mention an agent, you aren't talking about action [Stout,R] |
| 20077 | If you can judge one act as best, then do another, this supports an inward-looking view of agency [Stout,R] |
| 20049 | Maybe your emotions arise from you motivations, rather than being their cause [Stout,R] |
| 20046 | For an ascetic a powerful desire for something is a reason not to implement it [Stout,R] |
| 20060 | Beliefs, desires and intentions are not events, so can't figure in causal relations [Stout,R] |
| 20055 | A standard view says that the explanation of an action is showing its rational justification [Stout,R] |
| 20056 | In order to be causal, an agent's reasons must be internalised as psychological states [Stout,R] |
| 20053 | An action is only yours if you produce it, rather than some state or event within you [Stout,R] |
| 20048 | There may be a justification relative to a person's view, and yet no absolute justification [Stout,R] |
| 20068 | Describing a death as a side-effect rather than a goal may just be good public relations [Stout,R] |
| 20083 | Aristotelian causation involves potentiality inputs into processes (rather than a pair of events) [Stout,R] |
| 14349 | If there are no finks or antidotes at the fundamental level, the laws can't be ceteris paribus [Burge, by Corry] |