128 ideas
| 20262 | Don't use wisdom in order to become clever! [Nietzsche] |
| 20255 | Early 19th century German philosophers enjoyed concepts, rather than scientific explanations [Nietzsche] |
| 20260 | Carlyle spent his life vainly trying to make reason appear romantic [Nietzsche] |
| 20256 | What we think is totally dictated by the language available to express it [Nietzsche] |
| 20265 | The desire for a complete system requires making the weak parts look equal to the rest [Nietzsche] |
| 20380 | Why should truth be omnipotent? It is enough that it is very powerful [Nietzsche] |
| 20235 | Like animals, we seek truth because we want safety [Nietzsche] |
| 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] |
| 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] |
| 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] |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| 22246 | A train of reasoning must be treated as all happening simultaneously [Recanati] |
| 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] |
| 13803 | If we are to express that there at least two things, we need identity [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] |
| 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] |
| 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] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
| 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] |
| 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] |
| 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] |
| 20258 | Most people treat knowledge as a private possession [Nietzsche] |
| 20250 | We may be unable to remember, but we may never actually forget [Nietzsche] |
| 20270 | There is no one scientific method; we must try many approaches, and many emotions [Nietzsche] |
| 20131 | We can cultivate our drives, of anger, pity, curiosity, vanity, like a gardener, with good or bad taste [Nietzsche] |
| 20242 | Things are the boundaries of humanity, so all things must be known, for self-knowledge [Nietzsche] |
| 20249 | Our knowledge of the many drives that constitute us is hopelessly incomplete [Nietzsche] |
| 20231 | People used to think that outcomes were from God, rather than consequences of acts [Nietzsche] |
| 22247 | Indexicality is not just a feature of language; examples show it also occurs in thought [Recanati] |
| 22248 | How can we communicate indexical thoughts to people not in the right context? [Recanati] |
| 22242 | Mental files are concepts, which are either collections or (better) containers [Recanati] |
| 22243 | The Frege case of believing a thing is both F and not-F is explained by separate mental files [Recanati] |
| 22245 | A linguistic expression refers to what its associated mental file refers to [Recanati] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 22250 | There are speakers' thoughts and hearers' thoughts, but no further thought attached to the utterance [Recanati] |
| 20266 | It is essential that wise people learn to express their wisdom, possibly even as foolishness [Nietzsche] |
| 22249 | The Naive view of communication is that hearers acquire exactly the thoughts of the speaker [Recanati] |
| 20251 | Actions done for a purpose are least understood, because we complacently think it's obvious [Nietzsche] |
| 20271 | Beauty in art is the imitation of happiness [Nietzsche] |
| 20230 | The very idea of a critique of morality is regarded as immoral! [Nietzsche] |
| 20234 | Morality prevents us from developing better customs [Nietzsche] |
| 20237 | Moral feelings are entirely different from the moral concepts used to judge actions [Nietzsche] |
| 20238 | Treating morality as feelings is just obeying your ancestors [Nietzsche] |
| 20243 | Human beings are not majestic, either through divine origins, or through grand aims [Nietzsche] |
| 20268 | Most dying people have probably lost more important things than what they are about to lose [Nietzsche] |
| 20236 | Marriage upholds the idea that love, though a passion, can endure [Nietzsche] |
| 20263 | Fear reveals the natures of other people much more clearly than love does [Nietzsche] |
| 20252 | Marriage is too serious to be permitted for people in love! [Nietzsche] |
| 20233 | Punishment has distorted the pure innocence of the contingency of outcomes [Nietzsche] |
| 20248 | People do nothing for their real ego, but only for a phantom ego created by other people [Nietzsche] |
| 20246 | If you feel to others as they feel to themselves, you must hate a self-hater [Nietzsche] |
| 20272 | Honesty is a new young virtue, and we can promote it, or not [Nietzsche] |
| 20240 | The Jews treated great anger as holy, and were in awe of those who expressed it [Nietzsche] |
| 20244 | Christianity replaces rational philosophical virtues with great passions focused on God [Nietzsche] |
| 20274 | The cardinal virtues want us to be honest, brave, magnanimous and polite [Nietzsche] |
| 20257 | Cool courage and feverish bravery have one name, but are two very different virtues [Nietzsche] |
| 20259 | Teach youth to respect people who differ with them, not people who agree with them [Nietzsche] |
| 20267 | Seeing duty as a burden makes it a bit cruel, and it can thus never become a habit [Nietzsche] |
| 20275 | Most people think they are already complete, but we can cultivate ourselves [Nietzsche] |
| 20229 | No authority ever willingly accepts criticism [Nietzsche] |
| 20254 | People govern for the pleasure of it, or just to avoid being governed [Nietzsche] |
| 20273 | The French Revolution gave trusting Europe the false delusion of instant recovery [Nietzsche] |
| 20232 | Get rid of the idea of punishment! It is a noxious weed! [Nietzsche] |
| 20253 | Modern wars arise from the study of history [Nietzsche] |
| 20261 | History does not concern what really happened, but supposed events, which have all the influence [Nietzsche] |
| 20241 | Enquirers think finding our origin is salvation, but it turns out to be dull [Nietzsche] |
| 20245 | Christianity hoped for a short cut to perfection, that skipped the hard labour of morality [Nietzsche] |
| 20247 | Christianity was successful because of its heathen rituals [Nietzsche] |
| 20269 | 'I believe because it is absurd' - but how about 'I believe because I am absurd' [Nietzsche] |
| 20264 | The easy and graceful aspects of a person are called 'soul', and inner awkwardness is called 'soulless' [Nietzsche] |