113 ideas
| 18009 | Chomsky established the view that category mistakes are well-formed but meaningless [Chomsky, by Magidor] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 22355 | In the realist view, the real external world explains how it (and perceptions of it) are possible [Williams,B] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 23283 | Necessity implies possibility, but in experience it matters which comes first [Williams,B] |
| 4244 | It is very confused to deduce a nonrelativist morality of universal toleration from relativism [Williams,B] |
| 4243 | Our ability to react to an alien culture shows that ethical thought extends beyond cultural boundaries [Williams,B] |
| 3238 | 'Dead person' isn't a contradiction, so 'person' is somewhat vague [Williams,B] |
| 3239 | You can only really love a person as a token, not as a type [Williams,B] |
| 7946 | The memory criterion has a problem when one thing branches into two things [Williams,B, by Macdonald,C] |
| 2181 | It is an absurd Kantian idea that at the limit rationality and freedom coincide [Williams,B] |
| 2176 | There is only a problem of free will if you think the notion of 'voluntary' can be metaphysically deepened [Williams,B] |
| 24008 | Reference to a person's emotions is often essential to understanding their actions [Williams,B] |
| 24009 | Moral education must involve learning about various types of feeling towards things [Williams,B] |
| 6649 | Chomsky now says concepts are basically innate, as well as syntax [Chomsky, by Lowe] |
| 18007 | Syntax is independent of semantics; sentences can be well formed but meaningless [Chomsky, by Magidor] |
| 18006 | Chomsky's 'interpretative semantics' says syntax comes first, and is then interpreted [Chomsky, by Magidor] |
| 4317 | We judge weakness of will by an assessment after the event is concluded [Williams,B, by Cottingham] |
| 9284 | Reasons are 'internal' if they give a person a motive to act, but 'external' otherwise [Williams,B] |
| 2174 | Responsibility involves cause, intention, state of mind, and response after the event [Williams,B] |
| 22455 | Many ethical theories neglect the power of regretting the ought not acted upon [Williams,B] |
| 4114 | Philosophers try to produce ethical theories because they falsely assume that ethics can be simple [Williams,B] |
| 22453 | Moral conflicts have a different feeling and structure from belief conflicts [Williams,B, by Foot] |
| 22454 | We tolerate inconsistency in ethics but not in other beliefs (which reflect an independent order) [Williams,B, by Foot] |
| 22450 | If moral systems can't judge other moral systems, then moral relativism is true [Williams,B, by Foot] |
| 2178 | In bad actions, guilt points towards victims, and shame to the agent [Williams,B] |
| 20168 | Blame usually has no effect if the recipient thinks it unjustified [Williams,B] |
| 20167 | Blame partly rests on the fiction that blamed agents always know their obligations [Williams,B] |
| 4128 | Intuitionism has been demolished by critics, and no longer looks interesting [Williams,B] |
| 4366 | We can't accept Aristotle's naturalism about persons, because it is normative and unscientific [Williams,B, by Hursthouse] |
| 4132 | The category of person is a weak basis for ethics, because it is not fixed but comes in degrees [Williams,B] |
| 24007 | Emotivism saw morality as expressing emotions, and influencing others' emotions [Williams,B] |
| 4134 | The weakness of prescriptivism is shown by "I simply don't like staying at good hotels" [Williams,B] |
| 4135 | Some ethical ideas, such as 'treachery' and 'promise', seem to express a union of facts and values [Williams,B] |
| 22410 | Maybe the unthinkable is a moral category, and considering some options is dishonourable or absurd [Williams,B] |
| 22408 | Consequentialism assumes that situations can be compared [Williams,B] |
| 22411 | For a consequentialist massacring 7 million must be better than massacring 7 million and one [Williams,B] |
| 4120 | It is an error of consequentialism to think we just aim at certain states of affairs; we also want to act [Williams,B] |
| 23282 | If all that matters in morality is motive and intention, that makes moral luck irrelevant [Williams,B] |
| 4252 | Promise keeping increases reliability, by making deliberation focus on something which would be overlooked [Williams,B] |
| 4116 | A weakness of contractual theories is the position of a person of superior ability and power [Williams,B] |
| 2169 | Greek moral progress came when 'virtue' was freed from social status [Williams,B] |
| 4112 | A crucial feature of moral thought is second-order desire - the desire to have certain desires [Williams,B] |
| 24010 | An admirable human being should have certain kinds of emotional responses [Williams,B] |
| 23279 | It is important that a person can change their character, and not just be successive 'selves' [Williams,B] |
| 23280 | Kantians have an poor account of individuals, and insist on impartiality, because they ignore character [Williams,B] |
| 3236 | Equality of opportunity without equality of respect would create a very inhuman society [Williams,B] |
| 4113 | 'Deon' in Greek means what one must do; there was no word meaning 'duty' [Williams,B] |
| 4250 | The concept of a 'duty to myself' is fraudulent [Williams,B] |
| 4110 | Obligation and duty look backwards (because of a promise or job), although the acts are in the future [Williams,B] |
| 4249 | "Ought implies can" is a famous formula in connection with moral obligation [Williams,B] |
| 2172 | The modern idea of duty is unknown in archaic Greece [Williams,B] |
| 4248 | Not all moral deliberations lead to obligations; some merely reveal what 'may' be done [Williams,B] |
| 4121 | Why should I think of myself as both the legislator and the citizen who follows the laws? [Williams,B] |
| 22409 | We don't have a duty to ensure that others do their duty [Williams,B] |
| 4122 | If the self becomes completely impartial, it no longer has enough identity to worry about its interests [Williams,B] |
| 2180 | If reason cannot lead people to good, we must hope they have an internal voice [Williams,B] |
| 24012 | Kant's love of consistency is too rigid, and it even overrides normal fairness [Williams,B] |
| 2179 | If the moral self is seen as characterless, then other people have a very limited role in our moral lives [Williams,B] |
| 22407 | Utilitarianism cannot make any serious sense of integrity [Williams,B] |
| 23278 | For utilitarians states of affairs are what have value, not matter who produced them [Williams,B] |
| 4124 | Utilitarian benevolence involves no particular attachments, and is immune to the inverse square law [Williams,B] |
| 4245 | Ethical conviction must be to some extent passive, and can't just depend on the will and decisions [Williams,B] |
| 4246 | Taking responsibility won't cure ethical uncertainty by; we are uncertain what to decide [Williams,B] |
| 3233 | Equality implies that people are alike in potential as well as in needs [Williams,B] |
| 3234 | Equality seems to require that each person be acknowledged as having a significant point of view [Williams,B] |
| 3235 | It is a mark of extreme exploitation that the sufferers do not realise their plight [Williams,B] |
| 4247 | It is a mark of our having ethical values that we aim to reproduce them in our children [Williams,B] |
| 4131 | Most women see an early miscarriage and a late stillbirth as being very different in character [Williams,B] |
| 4133 | Speciesism isn't like racism, because the former implies a viewpoint which belongs to no one [Williams,B] |
| 2175 | There is a problem of evil only if you expect the world to be good [Williams,B] |