73 ideas
6887 | Linguistic philosophy approaches problems by attending to actual linguistic usage [Mautner] |
6881 | Analytic philosophy studies the unimportant, and sharpens tools instead of using them [Mautner] |
5439 | The 'hermeneutic circle' says parts and wholes are interdependent, and so cannot be interpreted [Mautner] |
9959 | 'Real' definitions give the essential properties of things under a concept [Mautner] |
9961 | 'Contextual definitions' replace whole statements, not just expressions [Mautner] |
9958 | Recursive definition defines each instance from a previous instance [Mautner] |
9960 | A stipulative definition lays down that an expression is to have a certain meaning [Mautner] |
9957 | Ostensive definitions point to an object which an expression denotes [Mautner] |
6219 | The fallacy of composition is the assumption that what is true of the parts is true of the whole [Mautner] |
6888 | Fuzzy logic is based on the notion that there can be membership of a set to some degree [Mautner] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
6877 | Entailment is logical requirement; it may be not(p and not-q), but that has problems [Mautner] |
6880 | Strict implication says false propositions imply everything, and everything implies true propositions [Mautner] |
6879 | 'Material implication' is defined as 'not(p and not-q)', but seems to imply a connection between p and q [Mautner] |
6878 | A person who 'infers' draws the conclusion, but a person who 'implies' leaves it to the audience [Mautner] |
6889 | Vagueness seems to be inconsistent with the view that every proposition is true or false [Mautner] |
6890 | Quantifiers turn an open sentence into one to which a truth-value can be assigned [Mautner] |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
18176 | Pure mathematics is pure set theory [Cantor] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
6882 | Counterfactuals presuppose a belief (or a fact) that the condition is false [Mautner] |
6886 | Counterfactuals are not true, they are merely valid [Mautner] |
6885 | Counterfactuals are true if in every world close to actual where p is the case, q is also the case [Mautner] |
6884 | Counterfactuals say 'If it had been, or were, p, then it would be q' [Mautner] |
6883 | Maybe counterfactuals are only true if they contain valid inference from premisses [Mautner] |
5449 | Essentialism is often identified with belief in 'de re' necessary truths [Mautner] |
6898 | Fallibilism is the view that all knowledge-claims are provisional [Mautner] |
6452 | 'Sense-data' arrived in 1910, but it denotes ideas in Locke, Berkeley and Hume [Mautner] |
4783 | Observing lots of green x can confirm 'all x are green' or 'all x are grue', where 'grue' is arbitrary [Mautner, by PG] |
4782 | 'All x are y' is equivalent to 'all non-y are non-x', so observing paper is white confirms 'ravens are black' [Mautner, by PG] |
8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
6899 | The references of indexicals ('there', 'now', 'I') depend on the circumstances of utterance [Mautner] |
6896 | Double effect is the distinction between what is foreseen and what is intended [Mautner] |
6897 | Double effect acts need goodness, unintended evil, good not caused by evil, and outweighing [Mautner] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
5452 | 'Essentialism' is opposed to existentialism, and claims there is a human nature [Mautner] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |