97 ideas
15357 | Philosophy is the most general intellectual discipline [Horsten] |
15352 | A definition should allow the defined term to be eliminated [Horsten] |
9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
15323 | Truth is a property, because the truth predicate has an extension [Horsten] |
15324 | Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles [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] |
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] |
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
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] |
15373 | Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models [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] |
15325 | Inferential deflationism says truth has no essence because no unrestricted logic governs the concept [Horsten] |
15356 | Deflationism concerns the nature and role of truth, but not its laws [Horsten] |
15344 | Deflationism skips definitions and models, and offers just accounts of basic laws of truth [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] |
9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
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] |
9605 | If a proposition is false, then its negation is true [Brown,JR] |
15328 | A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten] |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
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] |
9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
15353 | The first incompleteness theorem means that consistency does not entail soundness [Horsten] |
9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
15355 | Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated [Horsten] |
9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
15364 | English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten] |
9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
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] |
9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
9630 | The most brilliant formalist was Hilbert [Brown,JR] |
9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
15370 | Predicativism says mathematical definitions must not include the thing being defined [Horsten] |
9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
15338 | We may believe in atomic facts, but surely not complex disjunctive ones? [Horsten] |
15363 | In the supervaluationist account, disjunctions are not determined by their disjuncts [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] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
15372 | Some claim that indicative conditionals are believed by people, even though they are not actually held true [Horsten] |
9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
15347 | A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding [Horsten] |
9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |