72 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] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 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] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 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] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 6890 | Quantifiers turn an open sentence into one to which a truth-value can be assigned [Mautner] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 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] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 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] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |