234 ideas
| 23889 | Among the Greeks Aristotle is the only philosopher in the modern style [Weil] |
| 23881 | All thought about values is philosophical, and thought about anything else is not philosophy [Weil] |
| 23885 | Philosophy aims to change the soul, not to accumulate knowledge [Weil] |
| 23886 | Systems are not unique to each philosopher. The platonist tradition is old and continuous [Weil] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 23884 | Truth is a value of thought [Weil] |
| 23755 | Genius and love of truth are always accompanied by great humility [Weil] |
| 23825 | We seek truth only because it is good [Weil] |
| 23853 | Truth is not a object we love - it is the radiant manifestation of reality [Weil] |
| 23877 | Most people won't question an idea's truth if they depend on it [Weil] |
| 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
| 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
| 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [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] |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,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] |
| 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [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] |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| 13346 | Truth is the basic notion in classical logic [Bostock] |
| 13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
| 13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
| 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] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [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] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [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] |
| 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] |
| 13818 | If we allow empty domains, we must allow empty names [Bostock] |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [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] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| 13756 | A tree proof becomes too broad if its only rule is Modus Ponens [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] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
| 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] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 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] |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| 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] |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 23855 | Creation produced a network or web of determinations [Weil] |
| 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] |
| 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] |
| 23900 | Chance is compatible with necessity, and the two occur together [Weil] |
| 23888 | Knowledge is beyond question, as an unavoidable component of thinking [Weil] |
| 23747 | What is sacred is not a person, but the whole physical human being [Weil] |
| 23756 | The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 23878 | Weakness of will is the inadequacy of the original impetus to carry through the action [Weil] |
| 23832 | We both desire what is beautiful, and want it to remain as it is [Weil] |
| 23848 | The aesthete's treatment of beauty as amusement is sacreligious; beauty should nourish [Weil] |
| 23899 | The secret of art is that beauty is a just blend of unity and its opposite [Weil] |
| 23758 | Beauty is an attractive mystery, leaving nothing to be desired [Weil] |
| 23887 | Art (like philosophy) establishes a relation between world and self, and between oneself and others [Weil] |
| 23903 | When we admire a work, we see ourselves as its creator [Weil] |
| 23898 | Those who say immorality is not an aesthetic criterion must show that all criteria are aesthetic [Weil] |
| 23854 | Beauty is the proof of what is good [Weil] |
| 23814 | Every human yearns for an unattainable transcendent good [Weil] |
| 23826 | Beauty, goodness and truth are only achieved by applying full attention [Weil] |
| 23824 | Where human needs are satisfied we find happiness, friendship and beauty [Weil] |
| 23879 | In a violent moral disagreement, it can't be that both sides are just following social morality [Weil] |
| 23882 | Ends, unlike means, cannot be defined, which is why people tend to pursue means [Weil] |
| 23760 | All we need are the unity of justice, truth and beauty [Weil] |
| 23883 | Minds essentially and always strive towards value [Weil] |
| 23748 | The sacred in every human is their expectation of good rather than evil [Weil] |
| 23759 | Everything which originates in love is beautiful [Weil] |
| 23762 | Evil is transmitted by comforts and pleasures, but mostly by doing harm to people [Weil] |
| 23833 | The good is a nothingness, and yet real [Weil] |
| 23808 | There are two goods - the absolute good we want, and the reachable opposite of evil [Weil] |
| 23865 | Morality would improve if people could pursue private interests [Weil] |
| 23894 | The concept of character is at the centre of morality [Weil] |
| 23896 | We see our character as a restricting limit, but also as an unshakable support [Weil] |
| 23893 | We don't see character in a single moment, but only over a period of time [Weil] |
| 23895 | We modify our character by placing ourselves in situations, or by attending to what seems trivial [Weil] |
| 23837 | Respect is our only obligation, which can only be expressed through deeds, not words [Weil] |
| 23815 | We cannot equally respect what is unequal, so equal respect needs a shared ground [Weil] |
| 23834 | Friendship is partly universal - the love of a person is like the ideal of loving everyone [Weil] |
| 23823 | Life needs risks to avoid sickly boredom [Weil] |
| 23844 | The most important human need is to have multiple roots [Weil] |
| 23838 | The need for order stands above all others, and is understood via the other needs [Weil] |
| 23836 | Obligations only bind individuals, not collectives [Weil] |
| 7346 | Jeremiah implied a link between weakness and goodness, and the evil of the state [Jeremiah, by Johnson,P] |
| 23840 | A citizen should be able to understand the whole of society [Weil] |
| 23822 | We all need to partipate in public tasks, and take some initiative [Weil] |
| 23843 | Even the poorest should feel collective ownership, and participation in grand display [Weil] |
| 23846 | Culture is an instrument for creating an ongoing succession of teachers [Weil] |
| 23831 | The essence of power is illusory prestige [Weil] |
| 23857 | People in power always try to increase their power [Weil] |
| 23866 | In oppressive societies the scope of actual control is extended by a religion of power [Weil] |
| 23812 | Force is what turns man into a thing, and ultimately into a corpse [Weil] |
| 23839 | A lifelong head of society should only be a symbol, not a ruler [Weil] |
| 23871 | No central authority can initiate decentralisation [Weil] |
| 23856 | Spontaneous movements are powerless against organised repression [Weil] |
| 23867 | After a bloody revolution the group which already had the power comes to the fore [Weil] |
| 23830 | A group is only dangerous if it endorses an abstract entity [Weil] |
| 23870 | Decentralisation is only possible by co-operation between strong and weak - which is absurd [Weil] |
| 23809 | Our only social duty is to try to limit evil [Weil] |
| 23829 | National leaders want to preserve necessary order - but always the existing order [Weil] |
| 23817 | We need both equality (to attend to human needs) and hierarchy (as a scale of responsibilities) [Weil] |
| 23842 | Party politics in a democracy can't avoid an anti-democratic party [Weil] |
| 23859 | True democracy is the subordination of society to the individual [Weil] |
| 23863 | Only individual people of good will can achieve social progress [Weil] |
| 23869 | In the least evil societies people can think, control community life, and be autonomous [Weil] |
| 23750 | It is not more money which the wretched members of society need [Weil] |
| 23847 | Socialism tends to make a proletariat of the whole population [Weil] |
| 23749 | The problem of the collective is not suppression of persons, but persons erasing themselves [Weil] |
| 23807 | The collective is the one and only object of false idolatry [Weil] |
| 23861 | Marx showed that capitalist oppression, because of competition, is unstoppable [Weil] |
| 23897 | Once money is the main aim, society needs everyone to think wealth is possible [Weil] |
| 23845 | The capitalists neglect the people and the nation, and even their own interests [Weil] |
| 23828 | National prestige consists of behaving as if you could beat the others in a war [Weil] |
| 23810 | Charity is the only love, and you can feel that for a country (a place with traditions), but not a nation [Weil] |
| 23868 | The pleasure of completing tasks motivates just as well as the whip of slavery [Weil] |
| 23811 | If effort is from necessity rather than for a good, it is slavery [Weil] |
| 23819 | Deliberate public lying should be punished [Weil] |
| 23818 | We have liberty in the space between nature and accepted authority [Weil] |
| 23901 | Relationships depend on equality, so unequal treatment kills them [Weil] |
| 23753 | People absurdly claim an equal share of things which are essentially privileged [Weil] |
| 23841 | By making money the sole human measure, inequality has become universal [Weil] |
| 23864 | Inequality could easily be mitigated, if it were not for the struggle for power [Weil] |
| 23751 | Rights are asserted contentiously, and need the backing of force [Weil] |
| 23752 | Giving centrality to rights stifles all impulses of charity [Weil] |
| 23835 | People have duties, and only have rights because of the obligations of others to them [Weil] |
| 23820 | People need personal and collective property, and a social class lacking property is shameful [Weil] |
| 23813 | Only people who understand force, and don't respect it, are capable of justice [Weil] |
| 23757 | The spirit of justice needs the full attention of truth, and that attention is love [Weil] |
| 23761 | Justice (concerning harm) is distinct from rights (concerning inequality) [Weil] |
| 23852 | To punish people we must ourselves be innocent - but that undermines the desire to punish [Weil] |
| 23821 | Crime should be punished, to bring the perpetrator freely back to morality [Weil] |
| 23763 | Punishment aims at the good for men who don't desire it [Weil] |
| 23764 | The only thing in society worse than crime is repressive justice [Weil] |
| 23827 | Modern wars are fought in the name of empty words which are given capital letters [Weil] |
| 23880 | When war was a profession, customary morality justified any act of war [Weil] |
| 23850 | The soldier-civilian distinction should be abolished; every citizen is committed to a war [Weil] |
| 23858 | War is perpetuated by its continual preparations [Weil] |
| 23851 | Education is essentially motivation [Weil] |
| 23873 | Dividing history books into separate chapters is disastrous [Weil] |
| 23860 | Even if a drowning man is doomed, he should keep swimming to the last [Weil] |
| 22920 | Do I not fill heaven and earth? saith the Lord [Jeremiah] |
| 23816 | Attention to a transcendent reality motivates a duty to foster the good of humanity [Weil] |
| 23754 | The only choice is between supernatural good, or evil [Weil] |
| 23892 | The only legitimate proof of God by order derives from beauty [Weil] |
| 22089 | Am I a God afar off, and not a God close at hand? [Jeremiah] |
| 23904 | The cruelty of the Old Testament put me off Christianity [Weil] |
| 23849 | Religion should quietly suffuse all human life with its light [Weil] |
| 23902 | I attach little importance to immortality, which is an undecidable fact, and irrelevant to us [Weil] |
| 23765 | The soul is the intrinsic value of a human [Weil] |