166 ideas
2845 | Free and great-souled men do not keep asking "what is the use of it?" [Aristotle] |
22521 | Our method of inquiry is to examine the smallest parts that make up the whole [Aristotle] |
1570 | Human beings, alone of the animals, have logos [Aristotle] |
1574 | Reasoning distinguishes what is beneficial, and hence what is right [Aristotle] |
2801 | Intelligence which looks ahead is a natural master, while bodily strength is a natural slave [Aristotle] |
11178 | The essence or definition of an essence involves either a class of properties or a class of propositions [Fine,K] |
22529 | Men are natural leaders (apart from the unnatural ones) [Aristotle] |
22571 | 'If each is small, so too are all' is in one way false, for the whole composed of all is not small [Aristotle] |
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] |
11175 | Logical concepts rest on certain inferences, not on facts about implications [Fine,K] |
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] |
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] |
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] |
11176 | The property of Property Abstraction says any suitable condition must imply a property [Fine,K] |
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] |
11174 | A logical truth is true in virtue of the nature of the logical concepts [Fine,K] |
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] |
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] |
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] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
22525 | The whole is prior to its parts, because parts are defined by their role [Aristotle] |
11177 | Can the essence of an object circularly involve itself, or involve another object? [Fine,K] |
11173 | Being a man is a consequence of his essence, not constitutive of it [Fine,K] |
11179 | If there are alternative definitions, then we have three possibilities for essence [Fine,K] |
22587 | Understanding is the aim of our nature [Aristotle] |
22522 | To grasp something, trace it back to its natural origins [Aristotle] |
22524 | The nature of each thing is its mature state [Aristotle] |
22528 | The nature of all animate things is to have one part which rules it [Aristotle] |
22570 | Rhetoric now enables good speakers to become popular leaders [Aristotle] |
22574 | A community can lack self-control [Aristotle] |
2837 | Nothing contrary to nature is beautiful [Aristotle] |
2824 | The collective judgement of many people on art is better than that of an individual [Aristotle] |
2846 | Music can mould the character to be virtuous (just as gymnastics trains the body) [Aristotle] |
2807 | Some say slavery is unnatural and created by convention, and is therefore forced, and unjust [Aristotle] |
22582 | Spirit [thumos] is the capacity by which we love [Aristotle] |
2810 | Selfishness is wrong not because it is self-love, but because it is excessive [Aristotle] |
22555 | The function of good men is to confer benefits [Aristotle] |
22557 | Virtuous people are like the citizens of the best city [Aristotle] |
2841 | People become good because of nature, habit and reason [Aristotle] |
2829 | The law is the mean [Aristotle] |
22590 | Virtue is concerned with correct feelings [Aristotle] |
2813 | It is quite possible to live a moderate life and yet be miserable [Aristotle] |
22553 | Justice is a virtue of communities [Aristotle] |
22561 | The rich are seen as noble, because they don't need to commit crimes [Aristotle] |
2808 | Master and slave can have friendship through common interests [Aristotle] |
2803 | Man is by nature a political animal [Aristotle] |
2820 | People want to live together, even when they don't want mutual help [Aristotle] |
22586 | Only humans have reason [Aristotle] |
22523 | The community (of villages) becomes a city when it is totally self-sufficient [Aristotle] |
2805 | A community must share a common view of good and justice [Aristotle] |
22526 | People who are anti-social or wholly self-sufficient are no part of a city [Aristotle] |
22535 | Friendship is the best good for cities, because it reduces factions [Aristotle] |
22584 | A community should all share to some extent in something like land or food [Aristotle] |
22532 | A city can't become entirely one, because its very nature is to be a multitude [Aristotle] |
22581 | The size of a city is decided by the maximum self-sufficient community that can be surveyed [Aristotle] |
22577 | What is the best life for everyone, and is that a communal or an individual problem? [Aristotle] |
22578 | The happiest city is the one that acts most nobly [Aristotle] |
22579 | The same four cardinal virtues which apply to individuals also apply to a city [Aristotle] |
2431 | Every state is an association formed for some good purpose [Aristotle] |
22548 | A city aims at living well [Aristotle] |
2832 | The state aims to consist as far as possible of those who are like and equal [Aristotle] |
2973 | We must decide the most desirable human life before designing a constitution [Aristotle] |
22580 | The best constitution enables everyone to live the best life [Aristotle] |
22546 | A city is a community of free people, and the constitution should aim at the common advantage [Aristotle] |
2821 | The six constitutions are monarchy/tyranny, aristocracy/oligarchy, and polity/democracy [Aristotle] |
22558 | Constitutions specify distribution of offices, the authorities, and the community's aim [Aristotle] |
22566 | The greed of the rich is more destructive than the greed of the people [Aristotle] |
2835 | Any constitution can be made to last for a day or two [Aristotle] |
22544 | A citizen is someone who is allowed to hold official posts in a city [Aristotle] |
2818 | The virtues of a good citizen are relative to a particular constitution [Aristotle] |
22545 | A person can be an excellent citizen without being an excellent man [Aristotle] |
22563 | The middle classes are neither ambitious nor anarchic, which is good [Aristotle] |
22541 | Kings should be selected according to character [Aristotle] |
22542 | People who buy public office will probably expect to profit from it [Aristotle] |
22583 | The guardians should not be harsh to strangers, as no one should behave like that [Aristotle] |
2819 | The only virtue special to a ruler is practical wisdom [Aristotle] |
22552 | The rich can claim to rule, because of land ownership, and being more trustworthy [Aristotle] |
22543 | In large communities it is better if more people participate in the offices [Aristotle] |
22539 | Election of officials by the elected is dangerous, because factions can control it [Aristotle] |
22572 | Officers should like the constitution, be capable, and have appropriate virtues and justice [Aristotle] |
22567 | Democracy arises when people who are given equal freedom assume unconditional equality [Aristotle] |
22560 | Popular leaders only arise in democracies that are not in accord with the law [Aristotle] |
22562 | Choosing officials by lot is democratic [Aristotle] |
2826 | Like water, large numbers of people are harder to corrupt than a few [Aristotle] |
2823 | The many may add up to something good, even if they are inferior as individuals [Aristotle] |
22533 | If the people are equal in nature, then they should all share in ruling [Aristotle] |
2817 | It is wrong that a worthy officer of state should seek the office [Aristotle] |
22576 | No office is permanent in a democracy [Aristotle] |
22549 | In many cases, the claim that the majority is superior would apply equally to wild beasts [Aristotle] |
22575 | Ultimate democracy is tyranny [Aristotle] |
22531 | We aim to understand the best possible community for free people [Aristotle] |
22589 | Citizens do not just own themselves, but are also parts of the city [Aristotle] |
22585 | Look at all of the citizens before judging a city to be happy [Aristotle] |
22564 | Community is based on friends, who are equal and similar, and share things [Aristotle] |
22565 | The best communities rely on a large and strong middle class [Aristotle] |
22534 | People care less about what is communal, and more about what is their own [Aristotle] |
22538 | Owning and sharing property communally increases disagreements [Aristotle] |
22536 | There could be private land and public crops, or public land and private crops, or both public [Aristotle] |
22530 | Both women and children should be educated, as this contributes to a city's excellence [Aristotle] |
21358 | Natural slaves are those naturally belonging to another, or who can manage no more than labouring [Aristotle] |
20092 | One principle of liberty is to take turns ruling and being ruled [Aristotle] |
2833 | Equality is obviously there to help people who do not get priority in the constitution [Aristotle] |
2834 | It is always the weak who want justice and equality, not the strong [Aristotle] |
2830 | We can claim an equal right to aristocratic virtue, as well as to wealth or freedom [Aristotle] |
22569 | The Heraeans replaced election with lot, to thwart campaigning [Aristotle] |
22550 | It is dreadful to neither give a share nor receive a share [Aristotle] |
22568 | Faction is for inferiors to be equal, and equals to become superior [Aristotle] |
2814 | Phaleas proposed equality of property, provided there is equality of education [Aristotle] |
22540 | Wealth could be quickly leveled by only the rich giving marriage dowries [Aristotle] |
2828 | Law is intelligence without appetite [Aristotle] |
22537 | Property should be owned privately, but used communally [Aristotle] |
2825 | The good is obviously justice, which benefits the whole community, and involves equality in some sense [Aristotle] |
22527 | Justice is the order in a political community [Aristotle] |
22547 | Justice is equality for equals, and inequality for unequals [Aristotle] |
22573 | The virtue of justice may be relative to a particular constitution [Aristotle] |
22556 | Laws that match people's habits are more effective than mere written rules [Aristotle] |
2806 | Man is the worst of all animals when divorced from law and justice [Aristotle] |
2816 | If it is easy to change the laws, that makes them weaker [Aristotle] |
22551 | Correct law should be in control, with rulers only deciding uncertain issues [Aristotle] |
22554 | It is said that we should not stick strictly to written law, as it is too vague [Aristotle] |
2827 | It is preferable that law should rule rather than any single citizen [Aristotle] |
2840 | The whole state should pay for the worship of the gods [Aristotle] |
2811 | A state is plural, and needs education to make it a community [Aristotle] |
22588 | A city has a single end, so education must focus on that, and be communal, not private [Aristotle] |
2847 | The aim of serious childhood play is the amusement of the complete adult [Aristotle] |
2842 | Men learn partly by habit, and partly by listening [Aristotle] |
2844 | Abortions should be procured before the embryo has acquired life and sensation [Aristotle] |
2809 | If nature makes everything for a purpose, then plants and animals must have been made for man [Aristotle] |
2800 | The best instruments have one purpose, not many [Aristotle] |
2836 | God is not blessed and happy because of external goods, but because of his own nature [Aristotle] |
2802 | Men imagine gods to be of human shape, with a human lifestyle [Aristotle] |