74 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] |
20768 | Like spiderswebs, dialectical arguments are clever but useless [Ariston, by Diog. Laertius] |
9959 | 'Real' definitions give the essential properties of things under a concept [Mautner] |
9961 | 'Contextual definitions' replace whole statements, not just expressions [Mautner] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
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] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
9957 | Ostensive definitions point to an object which an expression denotes [Mautner] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
6219 | The fallacy of composition is the assumption that what is true of the parts is true of the whole [Mautner] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
6888 | Fuzzy logic is based on the notion that there can be membership of a set to some degree [Mautner] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
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] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
6890 | Quantifiers turn an open sentence into one to which a truth-value can be assigned [Mautner] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
6884 | Counterfactuals say 'If it had been, or were, p, then it would be q' [Mautner] |
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] |
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] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
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] |
5452 | 'Essentialism' is opposed to existentialism, and claims there is a human nature [Mautner] |
3049 | The chief good is indifference to what lies midway between virtue and vice [Ariston, by Diog. Laertius] |
3549 | Ariston says rules are useless for the virtuous and the non-virtuous [Ariston, by Annas] |