181 ideas
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
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] |
13354 | 'Conjunction' 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] |
13355 | 'Disjunction' 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] |
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] |
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] |
536 | We should follow the law in public, and nature in private [Antiphon] |
1557 | To gain the greatest advantage only treat law as important when other people are present [Antiphon] |
19956 | True goodness is political, and consists of love of and submission to the laws [Montesquieu] |
540 | The way you spend your time will form your character [Antiphon] |
19962 | Men do not desire to subjugate one another; domination is a complex and advanced idea [Montesquieu] |
19961 | Primitive people would be too vulnerable and timid to attack anyone, so peace would reign [Montesquieu] |
19963 | People are drawn into society by needs, shared fears, pleasure, and knowledge [Montesquieu] |
20008 | People are guided by a multitude of influences, from which the spirit of a nation emerges [Montesquieu] |
19993 | In small republics citizens identify with the public good, and abuses are fewer [Montesquieu] |
19992 | In a large republic there is too much wealth for individuals to manage it [Montesquieu] |
20005 | The rich would never submit to a lottery deciding which part of their society should be slaves [Montesquieu] |
19995 | All states aim at preservation, and then have distinctive individual purposes [Montesquieu] |
19964 | The natural power of a father suggests rule by one person, but that authority can be spread [Montesquieu] |
19972 | The nobility are an indispensable part of a monarchy [Montesquieu] |
19974 | Monarchs must not just have links to the people; they need a body which maintains the laws [Montesquieu] |
19976 | Ambition is good in a monarchy, because the monarch can always restrain it [Montesquieu] |
19978 | In monarchies, men's actions are judged by their grand appearance, not their virtues [Montesquieu] |
19985 | In a monarchy, the nobility must be hereditary, to bind them together [Montesquieu] |
19986 | Monarchies can act more quickly, because one person is in charge [Montesquieu] |
19988 | A despot's agents must be given power, so they inevitably become corrupt [Montesquieu] |
19975 | Despots are always lazy and ignorant, so they always delegate their power to a vizier [Montesquieu] |
19977 | Despotism and honour are incompatible, because honour scorns his power, and lives by rules [Montesquieu] |
20007 | Tyranny is either real violence, or the imposition of unpopular legislation [Montesquieu] |
19989 | The will of a despot is an enigma, so magistrates can only follow their own will [Montesquieu] |
19971 | Aristocracy is democratic if they resemble the people, but not if they resemble the monarch [Montesquieu] |
19984 | Great inequality between aristocrats and the rest is bad - and also among aristocrats themselves [Montesquieu] |
19970 | If the nobility is numerous, the senate is the artistocracy, and the nobles are a democracy [Montesquieu] |
19980 | If a government is to be preserved, it must first be loved [Montesquieu] |
19996 | A government has a legislature, an international executive, and a domestic executive [Montesquieu] |
19997 | The judiciary must be separate from the legislature, to avoid arbitrary power [Montesquieu] |
539 | Nothing is worse for mankind than anarchy [Antiphon] |
19965 | The fundamental laws of a democracy decide who can vote [Montesquieu] |
19968 | It is basic to a democracy that the people themselves must name their ministers [Montesquieu] |
19969 | Voting should be public, so the lower classes can be influenced by the example of notable people [Montesquieu] |
19999 | All citizens (apart from the very humble poor) should choose their representatives [Montesquieu] |
19967 | In a democracy the people should manage themselves, and only delegate what they can't do [Montesquieu] |
19966 | A democratic assembly must have a fixed number, to see whether everyone has spoken [Montesquieu] |
19998 | If deputies represent people, they are accountable, but less so if they represent places [Montesquieu] |
20000 | Slavery is entirely bad; the master abandons the virtues, and they are pointless in the slave [Montesquieu] |
20003 | Slaves are not members of the society, so no law can forbid them to run away [Montesquieu] |
20006 | The demand for slavery is just the masters' demand for luxury [Montesquieu] |
20009 | Freedom of speech and writing, within the law, is essential to preserve liberty [Montesquieu] |
19994 | Freedom in society is ability to do what is right, and not having to do what is wrong [Montesquieu] |
19981 | No one even thinks of equality in monarchies and despotism; they all want superiority [Montesquieu] |
19991 | Equality is not command by everyone or no one, but command and obedience among equals [Montesquieu] |
19990 | Democracy is corrupted by lack of equality, or by extreme equality (between rulers and ruled) [Montesquieu] |
19982 | Some equality can be achieved by social categories, combined with taxes and poor relief [Montesquieu] |
19983 | Democracies may sometimes need to restrict equality [Montesquieu] |
19959 | Prior to positive laws there is natural equity, of obedience, gratitude, dependence and merit [Montesquieu] |
19960 | Sensation gives animals natural laws, but knowledge can make them break them [Montesquieu] |
20002 | The death penalty is permissible, because its victims enjoyed the protection of that law [Montesquieu] |
20010 | If religion teaches determinism, penalties must be severe; if free will, then that is different [Montesquieu] |
20001 | The only right victors have over captives is the protection of the former [Montesquieu] |
19973 | The clergy are essential to a monarchy, but dangerous in a republic [Montesquieu] |
20011 | Religion can support the state when the law fails to do so [Montesquieu] |
19987 | Religion has the most influence in despotic states, and reinforces veneration for the ruler [Montesquieu] |
20004 | French slavery was accepted because it was the best method of religious conversion [Montesquieu] |
19979 | In monarchies education ennobles people, and in despotisms it debases them [Montesquieu] |
19957 | Teaching is the best practice of the general virtue that leads us to love everyone [Montesquieu] |
19958 | Laws are the necessary relations that derive from the nature of things [Montesquieu] |