139 ideas
15357 | Philosophy is the most general intellectual discipline [Horsten] |
14122 | Analysis gives us nothing but the truth - but never the whole truth [Russell] |
14109 | The study of grammar is underestimated in philosophy [Russell] |
14165 | Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell] |
15352 | A definition should allow the defined term to be eliminated [Horsten] |
14115 | Definition by analysis into constituents is useless, because it neglects the whole [Russell] |
14159 | In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell] |
14148 | Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell] |
18002 | As well as a truth value, propositions have a range of significance for their variables [Russell] |
15324 | Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles [Horsten] |
15323 | Truth is a property, because the truth predicate has an extension [Horsten] |
15374 | Truth has no 'nature', but we should try to describe its behaviour in inferences [Horsten] |
15348 | Propositions have sentence-like structures, so it matters little which bears the truth [Horsten] |
14102 | What is true or false is not mental, and is best called 'propositions' [Russell] |
15333 | Modern correspondence is said to be with the facts, not with true propositions [Horsten] |
15337 | The correspondence 'theory' is too vague - about both 'correspondence' and 'facts' [Horsten] |
15334 | The coherence theory allows multiple coherent wholes, which could contradict one another [Horsten] |
15336 | The pragmatic theory of truth is relative; useful for group A can be useless for group B [Horsten] |
15354 | Tarski's hierarchy lacks uniform truth, and depends on contingent factors [Horsten] |
15340 | Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa [Horsten] |
15345 | Semantic theories have a regress problem in describing truth in the languages for the models [Horsten] |
15373 | Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models [Horsten] |
15332 | 'Reflexive' truth theories allow iterations (it is T that it is T that p) [Horsten] |
15346 | Axiomatic approaches to truth avoid the regress problem of semantic theories [Horsten] |
15361 | A good theory of truth must be compositional (as well as deriving biconditionals) [Horsten] |
15371 | An axiomatic theory needs to be of maximal strength, while being natural and sound [Horsten] |
15350 | The Naïve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar [Horsten] |
15351 | Axiomatic theories take truth as primitive, and propose some laws of truth as axioms [Horsten] |
15367 | By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content! [Horsten] |
15330 | Friedman-Sheard theory keeps classical logic and aims for maximum strength [Horsten] |
15331 | Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength [Horsten] |
14176 | "The death of Caesar is true" is not the same proposition as "Caesar died" [Russell] |
15325 | Inferential deflationism says truth has no essence because no unrestricted logic governs the concept [Horsten] |
15344 | Deflationism skips definitions and models, and offers just accounts of basic laws of truth [Horsten] |
15356 | Deflationism concerns the nature and role of truth, but not its laws [Horsten] |
15368 | This deflationary account says truth has a role in generality, and in inference [Horsten] |
15358 | Deflationism says truth isn't a topic on its own - it just concerns what is true [Horsten] |
15359 | Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property [Horsten] |
15329 | Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well [Horsten] |
14113 | The null class is a fiction [Russell] |
15894 | Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine] |
14126 | Order rests on 'between' and 'separation' [Russell] |
14127 | Order depends on transitive asymmetrical relations [Russell] |
14121 | The part-whole relation is ultimate and indefinable [Russell] |
15326 | Doubt is thrown on classical logic by the way it so easily produces the liar paradox [Horsten] |
15341 | Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms [Horsten] |
14106 | Implication cannot be defined [Russell] |
14108 | It would be circular to use 'if' and 'then' to define material implication [Russell] |
14167 | The only classes are things, predicates and relations [Russell] |
14105 | There seem to be eight or nine logical constants [Russell] |
18722 | Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein on Russell] |
14104 | Constants are absolutely definite and unambiguous [Russell] |
14114 | Variables don't stand alone, but exist as parts of propositional functions [Russell] |
15328 | A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten] |
18946 | Unreflectively, we all assume there are nonexistents, and we can refer to them [Reimer] |
14137 | 'Any' is better than 'all' where infinite classes are concerned [Russell] |
15349 | It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) [Horsten] |
15366 | Satisfaction is a primitive notion, and very liable to semantical paradoxes [Horsten] |
15353 | The first incompleteness theorem means that consistency does not entail soundness [Horsten] |
14149 | The Achilles Paradox concerns the one-one correlation of infinite classes [Russell] |
15895 | Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine] |
15355 | Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated [Horsten] |
14151 | Pure geometry is deductive, and neutral over what exists [Russell] |
14152 | In geometry, Kant and idealists aimed at the certainty of the premisses [Russell] |
14154 | Geometry throws no light on the nature of actual space [Russell] |
14153 | In geometry, empiricists aimed at premisses consistent with experience [Russell] |
14155 | Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG] |
18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett] |
14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell] |
14128 | Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell] |
14129 | Ordinals presuppose two relations, where cardinals only presuppose one [Russell] |
14132 | Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell] |
14139 | Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell] |
14142 | Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell] |
14141 | Ordinals are defined through mathematical induction [Russell] |
14145 | For Cantor ordinals are types of order, not numbers [Russell] |
14146 | We aren't sure if one cardinal number is always bigger than another [Russell] |
14135 | Real numbers are a class of rational numbers (and so not really numbers at all) [Russell] |
15364 | English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten] |
14158 | Quantity is not part of mathematics, where it is replaced by order [Russell] |
14123 | Some quantities can't be measured, and some non-quantities are measurable [Russell] |
14120 | Counting explains none of the real problems about the foundations of arithmetic [Russell] |
14118 | We can define one-to-one without mentioning unity [Russell] |
14133 | There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell] |
14119 | We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell] |
14134 | Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell] |
14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell] |
14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell] |
14124 | Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell] |
7530 | Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk] |
18246 | Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell] |
14125 | Finite numbers, unlike infinite numbers, obey mathematical induction [Russell] |
14147 | Denying mathematical induction gave us the transfinite [Russell] |
14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
14117 | Numbers are properties of classes [Russell] |
15360 | ZFC showed that the concept of set is mathematical, not logical, because of its existence claims [Horsten] |
15369 | Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten] |
9977 | Ordinals can't be defined just by progression; they have intrinsic qualities [Russell] |
14162 | Mathematics doesn't care whether its entities exist [Russell] |
14103 | Pure mathematics is the class of propositions of the form 'p implies q' [Russell] |
21555 | For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell] |
18003 | In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor] |
15370 | Predicativism says mathematical definitions must not include the thing being defined [Horsten] |
11010 | Being is what belongs to every possible object of thought [Russell] |
14161 | Many things have being (as topics of propositions), but may not have actual existence [Russell] |
14173 | What exists has causal relations, but non-existent things may also have them [Russell] |
15338 | We may believe in atomic facts, but surely not complex disjunctive ones? [Horsten] |
15362 | If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't [Horsten] |
15363 | In the supervaluationist account, disjunctions are not determined by their disjuncts [Horsten] |
14163 | Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell] |
21341 | Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by Heil] |
10586 | 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell] |
10585 | Symmetrical and transitive relations are formally like equality [Russell] |
7781 | I call an object of thought a 'term'. This is a wide concept implying unity and existence. [Russell] |
14166 | Unities are only in propositions or concepts, and nothing that exists has unity [Russell] |
14164 | The only unities are simples, or wholes composed of parts [Russell] |
14112 | A set has some sort of unity, but not enough to be a 'whole' [Russell] |
14170 | Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell] |
14107 | Terms are identical if they belong to all the same classes [Russell] |
11849 | It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell] |
22303 | It makes no sense to say that a true proposition could have been false [Russell] |
15372 | Some claim that indicative conditionals are believed by people, even though they are not actually held true [Horsten] |
10583 | Abstraction principles identify a common property, which is some third term with the right relation [Russell] |
10582 | The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell] |
10584 | A certain type of property occurs if and only if there is an equivalence relation [Russell] |
15347 | A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding [Horsten] |
14110 | Proposition contain entities indicated by words, rather than the words themselves [Russell] |
19164 | If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell] |
19157 | Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson] |
14111 | A proposition is a unity, and analysis destroys it [Russell] |
14175 | We can drop 'cause', and just make inferences between facts [Russell] |
14172 | Moments and points seem to imply other moments and points, but don't cause them [Russell] |
14174 | The laws of motion and gravitation are just parts of the definition of a kind of matter [Russell] |
14168 | Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell] |
14171 | Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell] |
14160 | Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell] |
14156 | Mathematicians don't distinguish between instants of time and points on a line [Russell] |
14169 | The 'universe' can mean what exists now, what always has or will exist [Russell] |