70 ideas
6267 | A culture needs to admit that knowledge is more extensive than just 'science' [Putnam] |
6272 | 'True' and 'refers' cannot be made scientically precise, but are fundamental to science [Putnam] |
6276 | 'The rug is green' might be warrantedly assertible even though the rug is not green [Putnam] |
6266 | We need the correspondence theory of truth to understand language and science [Putnam] |
6277 | Correspondence between concepts and unconceptualised reality is impossible [Putnam] |
6264 | In Tarski's definition, you understand 'true' if you accept the notions of the object language [Putnam] |
6265 | Tarski has given a correct account of the formal logic of 'true', but there is more to the concept [Putnam] |
6269 | Only Tarski has found a way to define 'true' [Putnam] |
8956 | What is a singleton set, if a set is meant to be a collection of objects? [Szabó] |
10073 | There cannot be a set theory which is complete [Smith,P] |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
10613 | No nice theory can define truth for its own language [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
8953 | Abstract entities don't depend on their concrete entities ...but maybe on the totality of concrete things [Szabó] |
6280 | Realism is a theory, which explains the convergence of science and the success of language [Putnam] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
6284 | If a tautology is immune from revision, why would that make it true? [Putnam] |
6273 | Knowledge depends on believing others, which must be innate, as inferences are not strong enough [Putnam] |
6274 | Empathy may not give knowledge, but it can give plausibility or right opinion [Putnam] |
17084 | You can't decide which explanations are good if you don't attend to the interest-relative aspects [Putnam] |
8954 | Geometrical circles cannot identify a circular paint patch, presumably because they lack something [Szabó] |
8955 | Abstractions are imperceptible, non-causal, and non-spatiotemporal (the third explaining the others) [Szabó] |
6282 | Theory of meaning presupposes theory of understanding and reference [Putnam] |
6281 | Truth conditions can't explain understanding a sentence, because that in turn needs explanation [Putnam] |
6278 | We should reject the view that truth is prior to meaning [Putnam] |
6271 | How reference is specified is not what reference is [Putnam] |
6268 | The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions [Putnam] |
6279 | A private language could work with reference and beliefs, and wouldn't need meaning [Putnam] |
6270 | The correct translation is the one that explains the speaker's behaviour [Putnam] |
6283 | Language maps the world in many ways (because it maps onto other languages in many ways) [Putnam] |
6275 | You can't say 'most speaker's beliefs are true'; in some areas this is not so, and you can't count beliefs [Putnam] |