194 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] |
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] |
13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
13346 | Truth is the basic notion in classical logic [Bostock] |
13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
14249 | Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley] |
10830 | Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos] |
13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
10829 | A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos] |
13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
13803 | If we are to express that there at least two things, we need identity [Bostock] |
13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
10832 | '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos] |
13818 | If we allow empty domains, we must allow empty names [Bostock] |
13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
13760 | A sequent calculus is good for comparing proof systems [Bostock] |
13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
10834 | Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos] |
13841 | Why should compactness be definitive of logic? [Boolos, by Hacking] |
13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
10833 | Many concepts can only be expressed by second-order logic [Boolos] |
13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
22525 | The whole is prior to its parts, because parts are defined by their role [Aristotle] |
13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
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] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
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] |
22586 | Only humans have reason [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] |
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] |
2431 | Every state is an association formed for some good purpose [Aristotle] |
22548 | A city aims at living well [Aristotle] |
22578 | The happiest city is the one that acts most nobly [Aristotle] |
22577 | What is the best life for everyone, and is that a communal or an individual problem? [Aristotle] |
22579 | The same four cardinal virtues which apply to individuals also apply to a city [Aristotle] |
2832 | The state aims to consist as far as possible of those who are like and equal [Aristotle] |
22566 | The greed of the rich is more destructive than the greed of the people [Aristotle] |
22580 | The best constitution enables everyone to live the best life [Aristotle] |
2835 | Any constitution can be made to last for a day or two [Aristotle] |
2973 | We must decide the most desirable human life before designing a constitution [Aristotle] |
22558 | Constitutions specify distribution of offices, the authorities, and the community's aim [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] |
22563 | The middle classes are neither ambitious nor anarchic, which is good [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] |
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] |
22572 | Officers should like the constitution, be capable, and have appropriate virtues and justice [Aristotle] |
22539 | Election of officials by the elected is dangerous, because factions can control it [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] |
2823 | The many may add up to something good, even if they are inferior as individuals [Aristotle] |
2826 | Like water, large numbers of people are harder to corrupt than a few [Aristotle] |
22567 | Democracy arises when people who are given equal freedom assume unconditional equality [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] |
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] |
22569 | The Heraeans replaced election with lot, to thwart campaigning [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] |
22573 | The virtue of justice may be relative to a particular constitution [Aristotle] |
22547 | Justice is equality for equals, and inequality for unequals [Aristotle] |
22556 | Laws that match people's habits are more effective than mere written rules [Aristotle] |
2816 | If it is easy to change the laws, that makes them weaker [Aristotle] |
2806 | Man is the worst of all animals when divorced from law and justice [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] |
2847 | The aim of serious childhood play is the amusement of the complete adult [Aristotle] |
22588 | A city has a single end, so education must focus on that, and be communal, not private [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] |