137 ideas
| 22396 | We take courage, temperance, wisdom and justice as moral, but Aristotle takes wisdom as intellectual [Foot] |
| 22496 | Wisdom only implies the knowledge achievable in any normal lifetime [Foot] |
| 22397 | Wisdom is open to all, and not just to the clever or well trained [Foot] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 22462 | We should speak the truth, but also preserve and pursue it [Foot] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 22449 | When we say 'is red' we don't mean 'seems red to most people' [Foot] |
| 22371 | Determinism threatens free will if actions can be causally traced to external factors [Foot] |
| 23438 | Full rationality must include morality [Foot] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 23437 | Practical reason is goodness in choosing actions [Foot] |
| 22480 | Possessing the virtue of justice disposes a person to good practical rationality [Foot] |
| 23694 | All criterions of practical rationality derive from goodness of will [Foot] |
| 22372 | Not all actions need motives, but it is irrational to perform troublesome actions with no motive [Foot] |
| 22393 | I don't understand the idea of a reason for acting, but it is probably the agent's interests or desires [Foot] |
| 23436 | It is an odd Humean view to think a reason to act must always involve caring [Foot] |
| 22481 | There is no restitution after a dilemma, if it only involved the agent, or just needed an explanation [Foot, by PG] |
| 22482 | I can't understand how someone can be necessarily wrong whatever he does [Foot] |
| 22384 | A 'double effect' is a foreseen but not desired side-effect, which may be forgivable [Foot] |
| 22465 | We see a moral distinction between doing and allowing to happen [Foot] |
| 22385 | The doctrine of double effect can excuse an outcome because it wasn't directly intended [Foot] |
| 22386 | Double effect says foreseeing you will kill someone is not the same as intending it [Foot] |
| 22387 | Without double effect, bad men can make us do evil by threatening something worse [Foot] |
| 22388 | Double effect seems to rely on a distinction between what we do and what we allow [Foot] |
| 22466 | We see a moral distinction between our aims and their foreseen consequences [Foot] |
| 22467 | Acts and omissions only matter if they concern doing something versus allowing it [Foot] |
| 4692 | It is not true that killing and allowing to die (or acts and omissions) are morally indistinguishable [Foot] |
| 4694 | Making a runaway tram kill one person instead of five is diverting a fatal sequence, not initiating one [Foot] |
| 22445 | Morality shows murder is wrong, but not what counts as a murder [Foot] |
| 22444 | A moral system must deal with the dangers and benefits of life [Foot] |
| 23683 | Moral norms are objective, connected to facts about human goods [Foot, by Hacker-Wright] |
| 22392 | Morality is inescapable, in descriptive words such as 'dishonest', 'unjust' and 'uncharitable' [Foot] |
| 22451 | All people need affection, cooperation, community and help in trouble [Foot] |
| 22485 | Non-cognitivists give the conditions of use of moral sentences as facts about the speaker [Foot] |
| 22474 | Unlike aesthetic evaluation, moral evaluation needs a concept of responsibility [Foot] |
| 23684 | Morality gives everyone reasons to act, irrespective of their desires [Foot, by Hacker-Wright] |
| 23690 | We all have reason to cultivate the virtues, even when we lack the desire [Foot, by Hacker-Wright] |
| 23685 | Reason is not a motivator of morality [Foot, by Hacker-Wright] |
| 23691 | Rejecting moral rules may be villainous, but it isn't inconsistent [Foot] |
| 23686 | Moral reason is not just neutral, because morality is part of the standard of rationality [Foot, by Hacker-Wright] |
| 23693 | Practical rationality must weigh both what is morally and what is non-morally required [Foot] |
| 23431 | Human defects are just like plant or animal defects [Foot] |
| 23687 | Moral virtues arise from human nature, as part of what makes us good human beings [Foot, by Hacker-Wright] |
| 22477 | Calling a knife or farmer or speech or root good does not involve attitudes or feelings [Foot] |
| 22486 | The mistake is to think good grounds aren't enough for moral judgement, which also needs feelings [Foot] |
| 23432 | Concepts such as function, welfare, flourishing and interests only apply to living things [Foot] |
| 22375 | Moral judgements need more than the relevant facts, if the same facts lead to 'x is good' and 'x is bad' [Foot] |
| 22493 | Sterility is a human defect, but the choice to be childless is not [Foot] |
| 22492 | Virtues are as necessary to humans as stings are to bees [Foot] |
| 23433 | Humans need courage like a plant needs roots [Foot] |
| 22378 | We can't affirm a duty without saying why it matters if it is not performed [Foot] |
| 22487 | Moral arguments are grounded in human facts [Foot] |
| 22377 | Whether someone is rude is judged by agreed criteria, so the facts dictate the value [Foot] |
| 22376 | Facts and values are connected if we cannot choose what counts as evidence of rightness [Foot] |
| 22491 | Moral evaluations are not separate from facts, but concern particular facts about functioning [Foot] |
| 23434 | There is no fact-value gap in 'owls should see in the dark' [Foot] |
| 22447 | Saying something 'just is' right or wrong creates an illusion of fact and objectivity [Foot] |
| 23439 | Principles are not ultimate, but arise from the necessities of human life [Foot] |
| 22452 | Do we have a concept of value, other than wanting something, or making an effort to get it? [Foot] |
| 23435 | If you demonstrate the reason to act, there is no further question of 'why should I?' [Foot] |
| 22381 | Being a good father seems to depend on intentions, rather than actual abilities [Foot] |
| 22379 | The meaning of 'good' and other evaluations must include the object to which they attach [Foot] |
| 22458 | Consequentialists can hurt the innocent in order to prevent further wickedness [Foot] |
| 22460 | Why might we think that a state of affairs can be morally good or bad? [Foot] |
| 22461 | Good outcomes are not external guides to morality, but a part of virtuous actions [Foot] |
| 22464 | The idea of a good state of affairs has no role in the thought of Aristotle, Rawls or Scanlon [Foot] |
| 22497 | Deep happiness usually comes from the basic things in life [Foot] |
| 22498 | Happiness is enjoying the pursuit and attainment of right ends [Foot] |
| 23695 | Good actions can never be justified by the good they brings to their agent [Foot] |
| 22470 | A good moral system benefits its participants, and so demands reciprocity [Foot] |
| 22499 | We all know that just pretending to be someone's friend is not the good life [Foot] |
| 22402 | Most people think virtues can be displayed in bad actions [Foot] |
| 23145 | Virtues are intended to correct design flaws in human beings [Foot, by Driver] |
| 22401 | Actions can be in accordance with virtue, but without actually being virtuous [Foot] |
| 22398 | Virtues are corrective, to resist temptation or strengthen motivation [Foot] |
| 22478 | The essential thing is the 'needs' of plants and animals, and their operative parts [Foot] |
| 23692 | Good and bad are a matter of actions, not of internal dispositions [Foot] |
| 22468 | Virtues can have aims, but good states of affairs are not among them [Foot] |
| 22495 | Someone is a good person because of their rational will, not their body or memory [Foot] |
| 22373 | People can act out of vanity without being vain, or even vain about this kind of thing [Foot] |
| 22456 | Maybe virtues conflict with each other, if some virtue needs a vice for its achievement [Foot] |
| 22469 | Some virtues imply rules, and others concern attachment [Foot] |
| 22403 | Temperance is not a virtue if it results from timidity or excessive puritanism [Foot] |
| 22472 | The practice of justice may well need a recognition of human equality [Foot] |
| 22479 | Observing justice is necessary to humans, like hunting to wolves or dancing to bees [Foot] |
| 22400 | Courage overcomes the fears which should be overcome, and doesn't overvalue personal safety [Foot] |
| 7071 | Life and rationality are pointless if we can only contemplate the freedom of our own ego [Jacobi] |
| 22391 | Saying we 'ought to be moral' makes no sense, unless it relates to some other system [Foot] |
| 22389 | Morality no more consists of categorical imperatives than etiquette does [Foot] |
| 22395 | Moral judgements are hypothetical, because they depend on interests and desires [Foot] |
| 22448 | We sometimes just use the word 'should' to impose a rule of conduct on someone [Foot] |
| 22463 | Morality is seen as tacit legislation by the community [Foot] |
| 22459 | For consequentialism, it is irrational to follow a rule which in this instance ends badly [Foot] |
| 7072 | Jacobi was the first philosopher to talk of nihilism [Jacobi, by Critchley] |
| 22502 | Refraining from murder is not made good by authenticity or self-fulfilment [Foot] |
| 4693 | The right of non-interference (with a 'negative duty'), and the right to goods/services ('positive') [Foot] |
| 22383 | Abortion is puzzling because we do and don't want the unborn child to have rights [Foot] |
| 22446 | In the case of something lacking independence, calling it a human being is a matter of choice [Foot] |
| 22380 | Some words, such as 'knife', have a meaning which involves its function [Foot] |