156 ideas
20383 | The wisest man is full of contradictions, and attuned to other people, with occasional harmony [Nietzsche] |
4520 | I don't want to persuade anyone to be a philosopher; they should be rare plants [Nietzsche] |
4545 | Could not the objective character of things be merely a difference of degree within the subjective? [Nietzsche] |
4530 | Reason is a mere idiosyncrasy of a certain species of animal [Nietzsche] |
4523 | What can be 'demonstrated' is of little worth [Nietzsche] |
4531 | Our inability to both affirm and deny a single thing is merely an inability, not a 'necessity' [Nietzsche] |
4541 | Everything simple is merely imaginary [Nietzsche] |
20357 | Truth was given value by morality, but eventually turned against its own source [Nietzsche] |
4534 | 'Truth' is the will to be master over the multiplicity of sensations [Nietzsche] |
4548 | Only because there is thought is there untruth [Nietzsche] |
5652 | True beliefs are those which augment one's power [Nietzsche, by Scruton] |
4508 | The truth is what gives us the minimum of spiritual effort, and avoids the exhaustion of lying [Nietzsche] |
4538 | Judgements can't be true and known in isolation; the only surety is in connections and relations [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] |
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] |
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] |
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] |
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] |
13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [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] |
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] |
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] |
13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [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] |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
13762 | Tableau rules are all elimination rules, gradually shortening formulae [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] |
13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [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] |
20361 | We need 'unities' for reckoning, but that does not mean they exist [Nietzsche] |
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] |
4533 | Logic and maths refer to fictitious entities which we have created [Nietzsche] |
20359 | The nature of being, of things, is much easier to understand than is becoming [Nietzsche] |
4525 | There are no facts in themselves, only interpretations [Nietzsche] |
4543 | There are no 'facts-in-themselves', since a sense must be projected into them to make them 'facts' [Nietzsche] |
4484 | Nihilism results from valuing the world by the 'categories of reason', because that is fiction [Nietzsche] |
13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
4546 | We realise that properties are sensations of the feeling subject, not part of the thing [Nietzsche] |
4544 | A thing has no properties if it has no effect on other 'things' [Nietzsche] |
20362 | We saw unity in things because our ego seemed unified (but now we doubt the ego!) [Nietzsche] |
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] |
4528 | For me, a priori 'truths' are just provisional assumptions [Nietzsche] |
4537 | We can't know whether there is knowledge if we don't know what it is [Nietzsche] |
4485 | Every belief is a considering-something-true [Nietzsche] |
4487 | A note for asses: What convinces is not necessarily true - it is merely convincing [Nietzsche] |
4539 | The forms of 'knowledge' about logic which precede experience are actually regulations of belief [Nietzsche] |
4529 | All sense perceptions are permeated with value judgements (useful or harmful) [Nietzsche] |
4532 | We can have two opposite sensations, like hard and soft, at the same time [Nietzsche] |
4486 | The extreme view is there are only perspectives, no true beliefs, because there is no true world [Nietzsche] |
4536 | It is a major blunder to think of consciousness as a unity, and hence as an entity, a thing [Nietzsche] |
4551 | Great self-examination is to become conscious of oneself not as an individual, but as mankind [Nietzsche] |
4527 | Perhaps we are not single subjects, but a multiplicity of 'cells', interacting to create thought [Nietzsche] |
20374 | Consciousness is a terminal phenomenon, and causes nothing [Nietzsche] |
23938 | Passions are ranked, as if they are non-rational and animal pleasure seeking [Nietzsche] |
23939 | We fail to see that reason is a network of passions, and every passion contains some reason [Nietzsche] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
4554 | The concept of the 'will' is just a false simplification by our understanding [Nietzsche] |
4552 | There is no such things a pure 'willing' on its own; the aim must always be part of it [Nietzsche] |
4521 | None of the ancients had the courage to deny morality by denying free will [Nietzsche] |
4496 | 'Conscience' is invented to value actions by intention and conformity to 'law', rather than consequences [Nietzsche] |
20136 | There is an extended logic to a great man's life, achieved by a sustained will [Nietzsche] |
20358 | The highest man can endure and control the greatest combination of powerful drives [Nietzsche] |
20369 | The highest man directs the values of the highest natures over millenia [Nietzsche] |
4506 | There is a conspiracy (a will to power) to make morality dominate other values, like knowledge and art [Nietzsche] |
4514 | The basic tendency of the weak has always been to pull down the strong, using morality [Nietzsche] |
20370 | All evaluation is from some perspective, and aims at survival [Nietzsche] |
20354 | The ruling drives of our culture all want to be the highest court of our values [Nietzsche] |
4505 | How can it be that I should prefer my neighbour to myself, but he should prefer me to himself? [Nietzsche] |
4509 | Utilitarians prefer consequences because intentions are unknowable - but so are consequences! [Nietzsche] |
4500 | It is a sign of degeneration when eudaimonistic values begin to prevail [Nietzsche] |
4558 | We have no more right to 'happiness' than worms [Nietzsche] |
4550 | Pleasure and pain are mere epiphenomena, and achievement requires that one desire both [Nietzsche] |
4517 | Egoism is inescapable, and when it grows weak, the power of love also grows weak [Nietzsche] |
4518 | The question about egoism is: what kind of ego? since not all egos are equal [Nietzsche] |
4519 | The ego is only a fiction, and doesn't exist at all [Nietzsche] |
4560 | The Golden Rule prohibits harmful actions, with the premise that actions will be requited [Nietzsche] |
4555 | The great error is to think that happiness derives from virtue, which in turn derives from free will [Nietzsche] |
4494 | Not "return to nature", for there has never yet been a natural humanity [Nietzsche] |
4498 | 'Love your enemy' is unnatural, for the natural law says 'love your neighbour and hate your enemy' [Nietzsche] |
4493 | Be natural! But how, if one happens to be "unnatural"? [Nietzsche] |
4511 | We would avoid a person who always needed reasons for remaining decent [Nietzsche] |
4512 | Virtue is pursued from self-interest and prudence, and reduces people to non-entities [Nietzsche] |
20372 | The instinct of the herd, the majority, aims for the mean, in the middle [Nietzsche] |
4510 | A path to power: to introduce a new virtue under the name of an old one [Nietzsche] |
4515 | Modesty, industriousness, benevolence and temperance are the virtues of a good slave [Nietzsche] |
4516 | Many virtues are merely restraints on the most creative qualities of a human being [Nietzsche] |
4559 | When powerless one desires freedom; if power is too weak, one desires equal power ('justice') [Nietzsche] |
4557 | The supposed great lovers of honour (Alexander etc) were actually great despisers of honour [Nietzsche] |
4507 | The categorical imperative needs either God behind it, or a metaphysic of the unity of reason [Nietzsche] |
4501 | Utilitarianism criticises the origins of morality, but still believes in it as much as Christians [Nietzsche] |
4489 | If faith is lost, people seek other authorities, in order to avoid the risk of willing personal goals [Nietzsche] |
4513 | Virtuous people are inferior because they are not 'persons', but conform to a fixed pattern [Nietzsche] |
4504 | Morality used to be for preservation, but now we can only experiment, giving ourselves moral goals [Nietzsche] |
22817 | Citizenship involves a group of mutually supporting rights, which create community and equality [Miller,D] |
4495 | The high points of culture and civilization do not coincide [Nietzsche] |
22816 | Socialists reject nationality as a false source of identity [Miller,D] |
4491 | In modern society virtue is 'equal rights', but only because everyone is zero, so it is a sum of zeroes [Nietzsche] |
4542 | Science has taken the meaning out of causation; cause and effect are two equal sides of an equation [Nietzsche] |
4553 | We derive the popular belief in cause and effect from our belief that our free will causes things [Nietzsche] |
4535 | A 'species' is a stable phase of evolution, implying the false notion that evolution has a goal [Nietzsche] |
4497 | The concept of 'God' represents a turning away from life, and a critique of life [Nietzsche] |
4488 | Those who have abandoned God cling that much more firmly to the faith in morality [Nietzsche] |
4502 | Morality cannot survive when the God who sanctions it is missing [Nietzsche] |
4499 | Primitive Christianity is abolition of the state; it is opposed to defence, justice, patriotism and class [Nietzsche] |