199 ideas
| 20910 | Everything happens necessarily, and for a reason [Democritus] |
| 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] |
| 13803 | If we are to express that there at least two things, we need identity [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [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] |
| 16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
| 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] |
| 20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
| 20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
| 20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
| 20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
| 20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
| 16661 | There are two sorts of category - referring to things, and to circumstances of things [Boethius] |
| 20896 | It is not possible to know what sort each thing is [Democritus] |
| 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] |
| 15035 | If universals are not separate, we can isolate them by abstraction [Boethius, by Panaccio] |
| 14665 | We can call the quality of Plato 'Platonity', and say it is a quality which only he possesses [Boethius] |
| 3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
| 598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by 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] |
| 23308 | Reasoning relates to understanding as time does to eternity [Boethius, by Sorabji] |
| 1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
| 20894 | Man is separated from reality [Democritus] |
| 517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
| 20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
| 577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
| 20895 | We actually know nothing, and opinions are mere flux [Democritus] |
| 1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
| 492 | It is obviously impossible to understand the reality of each thing [Democritus] |
| 515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
| 20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
| 5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
| 6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
| 20912 | The soul is the same as the mind [Democritus, by Aristotle] |
| 20916 | Animals have a share of reason [Democritus, by Porphyry] |
| 20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
| 5771 | Knowledge of present events doesn't make them necessary, so future events are no different [Boethius] |
| 5767 | Rational natures require free will, in order to have power of judgement [Boethius] |
| 6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
| 5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
| 21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
| 5768 | God's universal foreknowledge seems opposed to free will [Boethius] |
| 5769 | Does foreknowledge cause necessity, or necessity cause foreknowledge? [Boethius] |
| 20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
| 24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
| 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] |
| 1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
| 5762 | The wicked want goodness, so they would not be wicked if they obtained it [Boethius] |
| 5770 | Rewards and punishments are not deserved if they don't arise from free movement of the mind [Boethius] |
| 5764 | When people fall into wickedness they lose their human nature [Boethius] |
| 495 | Wisdom creates a healthy passion-free soul [Democritus] |
| 1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
| 5756 | Happiness is a good which once obtained leaves nothing more to be desired [Boethius] |
| 5763 | The bad seek the good through desire, but the good through virtue, which is more natural [Boethius] |
| 20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
| 5759 | Varied aims cannot be good because they differ, but only become good when they unify [Boethius] |
| 13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
| 5754 | You can't control someone's free mind, only their body and possessions [Boethius] |
| 20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
| 20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
| 17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
| 1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
| 493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
| 5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
| 13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
| 1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
| 20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
| 1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
| 20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
| 1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
| 23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
| 20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
| 24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
| 5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
| 20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
| 16692 | Divine eternity is the all-at-once and complete possession of unending life [Boethius] |
| 5752 | Where does evil come from if there is a god; where does good come from if there isn't? [Boethius] |
| 5757 | God is the supreme good, so no source of goodness could take precedence over God [Boethius] |
| 5758 | God is the good [Boethius] |
| 5760 | The power through which creation remains in existence and motion I call 'God' [Boethius] |
| 5753 | The regular events of this life could never be due to chance [Boethius] |
| 1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
| 20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |
| 5765 | The reward of the good is to become gods [Boethius] |
| 5761 | God can do anything, but he cannot do evil, so evil must be nothing [Boethius] |
| 5766 | If you could see the plan of Providence, you would not think there was evil anywhere [Boethius] |