116 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] |
| 3722 | Metaphysics goes beyond the empirical, so doesn't need examples [Kant] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 3738 | The hallmark of rationality is setting itself an end [Kant] |
| 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] |
| 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] |
| 3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 3726 | The categorical imperative is a practical synthetic a priori proposition [Kant] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 3739 | Free will is a kind of causality which works independently of other causes [Kant] |
| 3741 | We shall never be able to comprehend how freedom is possible [Kant] |
| 3740 | We cannot conceive of reason as being externally controlled [Kant] |
| 5296 | Kant made the political will into a pure self-determined "free" will [Kant, by Marx/Engels] |
| 24011 | Kant thought emotions are too random and passive to be part of morality [Kant, by Williams,B] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 5074 | Kant united religion and philosophy, by basing obedience to law on reason instead of faith [Taylor,R on Kant] |
| 8024 | The categorical imperative says nothing about what our activities and ends should be [MacIntyre on Kant] |
| 22390 | Kant thought human nature was pure hedonism, so virtue is only possible via the categorical imperative [Foot on Kant] |
| 9750 | We must only value what others find acceptable [Kant, by Korsgaard] |
| 20160 | Kant focuses exclusively on human values, and neglects cultural and personal values [Kekes on Kant] |
| 9749 | Our rational choices confer value, arising from the sense that we ourselves are important [Kant, by Korsgaard] |
| 7671 | Values are created by human choices, and are not some intrinsic quality, out there [Kant, by Berlin] |
| 3720 | We may claim noble motives, but we cannot penetrate our secret impulses [Kant] |
| 3717 | Reverence is awareness of a value which demolishes my self-love [Kant] |
| 3712 | A good will is not good because of what it achieves [Kant] |
| 3725 | The good of an action is in the mind of the doer, not the consequences [Kant] |
| 3733 | The 'golden rule' cannot be a universal law as it implies no duties [Kant] |
| 3736 | Virtue lets a rational being make universal law, and share in the kingdom of ends [Kant] |
| 3544 | Kant thinks virtue becomes passive, and hence morally unaccountable [Kant, by Annas] |
| 7674 | Generosity and pity are vices, because they falsely imply one person's superiority to another [Kant, by Berlin] |
| 21029 | Kantian respect is for humanity and reason (not from love or sympathy or solidarity) [Kant, by Sandel] |
| 3715 | Other causes can produce nice results, so morality must consist in the law, found only in rational beings [Kant] |
| 4024 | Kant follows Rousseau in defining freedom and morality in terms of each other [Taylor,C on Kant] |
| 7105 | If 'maxims' are deeper underlying intentions, Kant can be read as a virtue theorist [Kant, by Statman] |
| 3710 | The only purely good thing is a good will [Kant] |
| 7625 | We can ask how rational goodness is, but also why is rationality good [Putnam on Kant] |
| 3737 | The will is good if its universalised maxim is never in conflict with itself [Kant] |
| 20715 | It is basic that moral actions must be done from duty [Kant] |
| 3723 | There are no imperatives for a holy will, as the will is in harmony with moral law [Kant] |
| 5295 | Kant was happy with 'good will', even if it had no result [Kant, by Marx/Engels] |
| 3714 | Dutiful actions are judged not by purpose, but by the maxim followed [Kant] |
| 3718 | Telling the truth from duty is quite different from doing so to avoid inconvenience [Kant] |
| 6695 | Kant has to attribute high moral worth to some deeply unattractive human lives [Kant, by Graham] |
| 8028 | Kantian duty seems to imply conformism with authority [MacIntyre on Kant] |
| 3724 | A categorical imperative sees an action as necessary purely for its own sake [Kant] |
| 3735 | Men are subject to laws which are both self-made and universal [Kant] |
| 8026 | Almost any precept can be consistently universalized [MacIntyre on Kant] |
| 15673 | The intuition behind the categorical imperative is that one ought not to make an exception of oneself [Kant, by Finlayson] |
| 8068 | Universalising a maxim needs to first stipulate the right description for the action [Anscombe on Kant] |
| 22009 | Morality is the creation of the laws that enable a Kingdom of Ends [Kant] |
| 4413 | The categorical imperative smells of cruelty [Nietzsche on Kant] |
| 3719 | If lying were the universal law it would make promises impossible [Kant] |
| 3716 | Act according to a maxim you can will as a universal law [Kant] |
| 8025 | The categorical imperative will not suggest maxims suitable for testing [MacIntyre on Kant] |
| 8027 | I can universalize a selfish maxim, if it is expressed in a way that only applies to me [MacIntyre on Kant] |
| 3728 | Suicide, false promises, neglected talent, and lack of charity all involve contradictions of principle [Kant, by PG] |
| 22008 | Always treat yourself and others as an end, and never simply as a means [Kant] |
| 3727 | Act as if your maxim were to become a universal law of nature [Kant] |
| 3762 | Why couldn't all rational beings accept outrageously immoral rules of conduct? [Mill on Kant] |
| 22050 | The maxim of an action is chosen, and not externally imposed [Kant, by Bowie] |
| 3731 | Rational beings necessarily conceive their own existence as an end in itself [Kant] |
| 6694 | Always treat humanity as an end and never as a means only [Kant] |
| 4345 | For Kant, even a person who lacks all sympathy for others still has a motive for benevolence [Kant, by Hursthouse] |
| 3711 | Only a good will makes us worthy of happiness [Kant] |
| 4251 | If we are required to give moral thought the highest priority, this gives morality no content [Williams,B on Kant] |
| 16004 | If Kant lives by self-administered laws, this is as feeble as self-administered punishments [Kierkegaard on Kant] |
| 4344 | Actions where people spread happiness because they enjoy it have no genuine moral worth [Kant] |
| 3729 | Our inclinations are not innately desirable; in fact most rational beings would like to be rid of them [Kant] |
| 3713 | The function of reason is to produce a good will [Kant] |
| 3732 | Rational beings have a right to share in the end of an action, not just be part of the means [Kant] |
| 7670 | Kant is the father of the notion of exploitation as an evil [Kant, by Berlin] |
| 7591 | Kant completed Grotius's project of a non-religious basis for natural law [Scruton on Kant] |
| 7673 | Retributive punishment is better than being sent to hospital for your crimes [Kant, by Berlin] |
| 3730 | Non-rational beings only have a relative value, as means rather than as ends [Kant] |
| 3721 | We judge God to be good by a priori standards of moral perfection [Kant] |
| 8046 | We can only know we should obey God if we already have moral standards for judging God [Kant, by MacIntyre] |
| 20714 | God is not proved by reason, but is a postulate of moral thinking [Kant, by Davies,B] |