112 ideas
| 21979 | Wisdom emerges at the end of a process [Hegel] |
| 22766 | Philosophy is exploration of the rational [Hegel] |
| 22768 | Subjective and objective are not firmly opposed, but merge into one another [Hegel] |
| 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] |
| 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] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [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] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |
| 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] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
| 17518 | Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers] |
| 17516 | If counting needs a sortal, what of things which fall under two sortals? [Ayers] |
| 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] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [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] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 22772 | Personality overcomes subjective limitations and posits Dasein as its own [Hegel] |
| 17520 | Events do not have natural boundaries, and we have to set them [Ayers] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 17519 | To express borderline cases of objects, you need the concept of an 'object' [Ayers] |
| 17510 | Speakers need the very general category of a thing, if they are to think about it [Ayers] |
| 17522 | We use sortals to classify physical objects by the nature and origin of their unity [Ayers] |
| 17515 | Seeing caterpillar and moth as the same needs continuity, not identity of sortal concepts [Ayers] |
| 17511 | Recognising continuity is separate from sortals, and must precede their use [Ayers] |
| 17517 | Could the same matter have more than one form or principle of unity? [Ayers] |
| 17513 | If there are two objects, then 'that marble, man-shaped object' is ambiguous [Ayers] |
| 17523 | Sortals basically apply to individuals [Ayers] |
| 17521 | You can't have the concept of a 'stage' if you lack the concept of an object [Ayers] |
| 17514 | Temporal 'parts' cannot be separated or rearranged [Ayers] |
| 17509 | Some say a 'covering concept' completes identity; others place the concept in the reference [Ayers] |
| 17512 | If diachronic identities need covering concepts, why not synchronic identities too? [Ayers] |
| 22780 | It is a rejection of intellectual dignity to say that we cannot know the truth [Hegel] |
| 22770 | A person is a being which is aware of its own self-directed and free subjectivity [Hegel] |
| 22788 | A human only become a somebody as a member of a social estate [Hegel] |
| 22792 | Individuals attain their right by discovering their self-consciousness in institutions [Hegel] |
| 21780 | A free will primarily wills its own freedoom [Hegel, by Houlgate] |
| 22769 | The concept of the will is the free will which wills its freedom [Hegel] |
| 21787 | Evil enters a good will when we believe we are doing right, but allow no criticism of our choice [Hegel, by Houlgate] |
| 21786 | Conscience is the right of the self to know what is right and obligatory, and thus make them true [Hegel] |
| 22784 | Love is ethical life in its natural form [Hegel] |
| 8029 | You can't have a morality which is supplied by the individual, but is also genuinely universal [Hegel, by MacIntyre] |
| 22051 | The categorical imperative lacks roots in a historical culture [Hegel, by Bowie] |
| 22771 | Be a person, and respect other persons [Hegel] |
| 22781 | The categorical imperative is fine if you already have a set of moral principles [Hegel] |
| 22779 | The good is realised freedom [Hegel] |
| 22787 | The family is the first basis of the state, but estates are a necessary second [Hegel] |
| 22790 | We cannot assert rights which are unnatural [Hegel] |
| 22767 | I aim to portray the state as a rational entity [Hegel] |
| 22789 | Society draws people, and requires their work, making them wholly dependent on it [Hegel] |
| 22791 | The state is the march of God in the world [Hegel] |
| 22778 | Individuals can't leave the state, because they are natural citizens, and humans require a state [Hegel] |
| 22794 | A fully developed state is conscious and knows what it wills [Hegel] |
| 22799 | The people do not have the ability to know the general will [Hegel] |
| 22801 | The great man of the ages is the one who reveals and accomplishes the will of his time [Hegel] |
| 22796 | A constitution embodies a nation's rights and condition [Hegel] |
| 22777 | Individuals must dedicate themselves to the ethical whole, and give their lives when asked [Hegel] |
| 21791 | Social groups must focus on the state, which must in turn respect their inclusion and their will [Hegel] |
| 22795 | People can achieve respect for their state by insight into its essence [Hegel] |
| 21988 | In the 1840s Hegel seemed to defend society being right as it is, as a manifestation of Mind [Hegel, by Singer] |
| 22800 | Majority rule means obligations can be imposed on me [Hegel] |
| 21792 | The state should reflect all interests, and not just popular will, or a popular party [Hegel, by Houlgate] |
| 22797 | In modern states an individual's actions should be their choice [Hegel] |
| 21790 | Moral individuals become ethical when they see the social aspect of a matter [Hegel, by Houlgate] |
| 8030 | For Hegel, the moral life can only be led within a certain type of community [Hegel, by MacIntyre] |
| 22785 | Even educated women are unsuited to science, philosophy, art and government [Hegel] |
| 21789 | Slaves have no duties because they have no rights [Hegel] |
| 22776 | Slaves are partly responsible for their own condition [Hegel] |
| 21778 | True liberal freedom is to pursue something, while being free to cease the pursuit [Hegel, by Houlgate] |
| 21779 | People assume they are free, but the options available are not under their control [Hegel] |
| 22085 | Freedom requires us to submit to a family, or a corporation, or a state [Hegel, by Houlgate] |
| 22798 | Money is the best way to achieve just equality [Hegel] |
| 22783 | Rights imply duties, and duties imply rights [Hegel] |
| 21782 | Man has an absolute right to appropriate things [Hegel] |
| 22773 | Because only human beings can own property, everything else can become our property [Hegel] |
| 22774 | A community does not have the property-owning rights that a person has [Hegel] |
| 22775 | The owner of a thing is obviously the first person to freely take possession of it [Hegel] |
| 22802 | Wars add strength to a nation, and cure internal dissension [Hegel] |
| 22786 | Children need discipline, to break their self-will and eradicate sensuousness [Hegel] |
| 22782 | To have pagan beliefs and be a pagan are quite different [Hegel] |
| 22793 | Some religions lead to harsh servitude and the debasement of human beings [Hegel] |