160 ideas
5361 | Philosophers must get used to absurdities [Russell] |
5368 | Philosophy verifies that our hierarchy of instinctive beliefs is harmonious and consistent [Russell] |
5432 | Metaphysics cannot give knowledge of the universe as a whole [Russell] |
5434 | Philosophy is similar to science, and has no special source of wisdom [Russell] |
5405 | The law of contradiction is not a 'law of thought', but a belief about things [Russell] |
5396 | Three Laws of Thought: identity, contradiction, and excluded middle [Russell] |
5420 | Truth is a property of a belief, but dependent on its external relations, not its internal qualities [Russell] |
5419 | Truth and falsehood are properties of beliefs and statements [Russell] |
5417 | A good theory of truth must make falsehood possible [Russell] |
7395 | Truth as congruence may work for complex beliefs, but not for simple beliefs about existence [Joslin on Russell] |
5428 | Beliefs are true if they have corresponding facts, and false if they don't [Russell] |
5421 | The coherence theory says falsehood is failure to cohere, and truth is fitting into a complete system of Truth [Russell] |
5422 | More than one coherent body of beliefs seems possible [Russell] |
5423 | If we suspend the law of contradiction, nothing will appear to be incoherent [Russell] |
5424 | Coherence is not the meaning of truth, but an important test for truth [Russell] |
5401 | The mortality of Socrates is more certain from induction than it is from deduction [Russell] |
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] |
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] |
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] |
5395 | Demonstration always relies on the rule that anything implied by a truth is true [Russell] |
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] |
5386 | Proper names are really descriptions, and can be replaced by a description in a person's mind [Russell] |
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] |
5385 | The phrase 'a so-and-so' is an 'ambiguous' description'; 'the so-and-so' (singular) is a 'definite' description [Russell] |
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] |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [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] |
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] |
5399 | Maths is not known by induction, because further instances are not needed to support it [Russell] |
5370 | Space is neutral between touch and sight, so it cannot really be either of them [Russell] |
5418 | In a world of mere matter there might be 'facts', but no truths [Russell] |
5371 | Because we depend on correspondence, we know relations better than we know the items that relate [Russell] |
5407 | That Edinburgh is north of London is a non-mental fact, so relations are independent universals [Russell] |
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] |
5383 | Every complete sentence must contain at least one word (a verb) which stands for a universal [Russell] |
4428 | Propositions express relations (prepositions and verbs) as well as properties (nouns and adjectives) [Russell] |
5406 | Confused views of reality result from thinking that only nouns and adjectives represent universals [Russell] |
4479 | All universals are like the relation "is north of", in having no physical location at all [Russell, by Loux] |
4030 | Russell claims that universals are needed to explain a priori knowledge (as their relations) [Russell, by Mellor/Oliver] |
4427 | Every sentence contains at least one word denoting a universal, so we need universals to know truth [Russell] |
5409 | Normal existence is in time, so we must say that universals 'subsist' [Russell] |
5408 | If we identify whiteness with a thought, we can never think of it twice; whiteness is the object of a thought [Russell] |
4441 | 'Resemblance Nominalism' won't work, because the theory treats resemblance itself as a universal [Russell] |
4429 | If we consider whiteness to be merely a mental 'idea', we rob it of its universality [Russell] |
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] |
5400 | In any possible world we feel that two and two would be four [Russell] |
5431 | Knowledge cannot be precisely defined, as it merges into 'probable opinion' [Russell] |
5426 | Belief relates a mind to several things other than itself [Russell] |
5366 | We have an 'instinctive' belief in the external world, prior to all reflection [Russell] |
5359 | Descartes showed that subjective things are the most certain [Russell] |
5377 | 'Acquaintance' is direct awareness, without inferences or judgements [Russell] |
6510 | Russell (1912) said phenomena only resemble reality in abstract structure [Russell, by Robinson,H] |
5372 | There is no reason to think that objects have colours [Russell] |
5373 | 'Idealism' says that everything which exists is in some sense mental [Russell] |
5362 | It is not illogical to think that only myself and my mental events exist [Russell] |
5412 | Some propositions are self-evident, but their implications may also be self-evident [Russell] |
5413 | Particular instances are more clearly self-evident than any general principles [Russell] |
5415 | As shown by memory, self-evidence comes in degrees [Russell] |
5416 | If self-evidence has degrees, we should accept the more self-evident as correct [Russell] |
5397 | The rationalists were right, because we know logical principles without experience [Russell] |
4430 | All a priori knowledge deals with the relations of universals [Russell] |
5411 | We can know some general propositions by universals, when no instance can be given [Russell] |
6514 | Russell's representationalism says primary qualities only show the structure of reality [Russell, by Robinson,H] |
6415 | After 1912, Russell said sense-data are last in analysis, not first in experience [Russell, by Grayling] |
5358 | 'Sense-data' are what are immediately known in sensation, such as colours or roughnesses [Russell] |
7290 | If Russell rejects innate ideas and direct a priori knowledge, he is left with a tabula rasa [Russell, by Thompson] |
5357 | It is natural to begin from experience, and presumably that is the basis of knowledge [Russell] |
5382 | We are acquainted with outer and inner sensation, memory, Self, and universals [Russell, by PG] |
5389 | Knowledge by descriptions enables us to transcend private experience [Russell] |
5376 | I can know the existence of something with which nobody is acquainted [Russell] |
5414 | Images are not memory, because they are present, and memories are of the past [Russell] |
5430 | A true belief is not knowledge if it is reached by bad reasoning [Russell] |
5429 | True belief is not knowledge when it is deduced from false belief [Russell] |
5378 | All knowledge (of things and of truths) rests on the foundations of acquaintance [Russell] |
5365 | Dreams can be explained fairly scientifically if we assume a physical world [Russell] |
5391 | Science aims to find uniformities to which (within the limits of experience) there are no exceptions [Russell] |
5390 | Chickens are not very good at induction, and are surprised when their feeder wrings their neck [Russell] |
5394 | We can't prove induction from experience without begging the question [Russell] |
5392 | It doesn't follow that because the future has always resembled the past, that it always will [Russell] |
5363 | If the cat reappears in a new position, presumably it has passed through the intermediate positions [Russell] |
5367 | Belief in real objects makes our account of experience simpler and more systematic [Russell] |
5364 | It is hard not to believe that speaking humans are expressing thoughts, just as we do ourselves [Russell] |
5379 | If we didn't know our own minds by introspection, we couldn't know that other people have minds [Russell] |
5410 | I learn the universal 'resemblance' by seeing two shades of green, and their contrast with red [Russell] |
5381 | In seeing the sun, we are acquainted with our self, but not as a permanent person [Russell] |
5380 | In perceiving the sun, I am aware of sun sense-data, and of the perceiver of the data [Russell] |
5369 | It is rational to believe in reality, despite the lack of demonstrative reasons for it [Russell] |
5375 | Knowledge of truths applies to judgements; knowledge by acquaintance applies to sensations and things [Russell] |
21711 | Russell's 'multiple relations' theory says beliefs attach to ingredients, not to propositions [Russell, by Linsky,B] |
5427 | Truth is when a mental state corresponds to a complex unity of external constituents [Russell] |
5425 | In order to explain falsehood, a belief must involve several terms, not two [Russell] |
5384 | A universal of which we are aware is called a 'concept' [Russell] |
13468 | Russell started philosophy of language, by declaring some plausible sentences to be meaningless [Russell, by Hart,WD] |
5388 | Every understood proposition is composed of constituents with which we are acquainted [Russell] |
5387 | It is pure chance which descriptions in a person's mind make a name apply to an individual [Russell] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
21724 | The main aim of the multiple relations theory of judgement was to dispense with propositions [Russell, by Linsky,B] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
5398 | Judgements of usefulness depend on judgements of value [Russell] |
5393 | We can't know that our laws are exceptionless, or even that there are any laws [Russell] |