71 ideas
| 18005 | Philosophy aims to become more disciplined about categories [Ryle] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 18004 | We can't do philosophy without knowledge of types and categories [Ryle] |
| 18009 | Chomsky established the view that category mistakes are well-formed but meaningless [Chomsky, by Magidor] |
| 13985 | A true proposition seems true of one fact, but a false proposition seems true of nothing at all. [Ryle] |
| 13984 | Two maps might correspond to one another, but they are only 'true' of the country they show [Ryle] |
| 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] |
| 13979 | Logic studies consequence, compatibility, contradiction, corroboration, necessitation, grounding.... [Ryle] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 10800 | The values of variables can't determine existence, because they are just expressions [Ryle, by Quine] |
| 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] |
| 13988 | Many sentences do not state facts, but there are no facts which could not be stated [Ryle] |
| 14297 | A dispositional property is not a state, but a liability to be in some state, given a condition [Ryle] |
| 14300 | No physical scientist now believes in an occult force-exerting agency [Ryle] |
| 13983 | Representation assumes you know the ideas, and the reality, and the relation between the two [Ryle] |
| 2622 | Can one movement have a mental and physical cause? [Ryle] |
| 1353 | Reporting on myself has the same problems as reporting on you [Ryle] |
| 1354 | We cannot introspect states of anger or panic [Ryle] |
| 2624 | I cannot prepare myself for the next thought I am going to think [Ryle] |
| 2620 | Dualism is a category mistake [Ryle] |
| 2388 | Behaviour depends on desires as well as beliefs [Chalmers on Ryle] |
| 3354 | You can't explain mind as dispositions, if they aren't real [Benardete,JA on Ryle] |
| 2387 | How can behaviour be the cause of behaviour? [Chalmers on Ryle] |
| 13980 | If you like judgments and reject propositions, what are the relata of incoherence in a judgment? [Ryle] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 6649 | Chomsky now says concepts are basically innate, as well as syntax [Chomsky, by Lowe] |
| 13978 | Husserl and Meinong wanted objective Meanings and Propositions, as subject-matter for Logic [Ryle] |
| 13977 | When I utter a sentence, listeners grasp both my meaning and my state of mind [Ryle] |
| 18007 | Syntax is independent of semantics; sentences can be well formed but meaningless [Chomsky, by Magidor] |
| 18006 | Chomsky's 'interpretative semantics' says syntax comes first, and is then interpreted [Chomsky, by Magidor] |
| 13976 | 'Propositions' name what is thought, because 'thoughts' and 'judgments' are too ambiguous [Ryle] |
| 13981 | Several people can believe one thing, or make the same mistake, or share one delusion [Ryle] |
| 13987 | We may think in French, but we don't know or believe in French [Ryle] |
| 13989 | There are no propositions; they are just sentences, used for thinking, which link to facts in a certain way [Ryle] |
| 13982 | If we accept true propositions, it is hard to reject false ones, and even nonsensical ones [Ryle] |