78 ideas
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
12204 | The logic of metaphysical necessity is S5 [Rumfitt] |
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 in which a∈A and b∈B [George/Velleman] |
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] |
12195 | Soundness in argument varies with context, and may be achieved very informally indeed [Rumfitt] |
12199 | There is a modal element in consequence, in assessing reasoning from suppositions [Rumfitt] |
12201 | We reject deductions by bad consequence, so logical consequence can't be deduction [Rumfitt] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
12194 | Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B' [Rumfitt] |
12198 | Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt] |
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] |
6504 | For physicalists, the only relations are spatial, temporal and causal [Robinson,H] |
6520 | If reality just has relational properties, what are its substantial ontological features? [Robinson,H] |
14532 | A distinctive type of necessity is found in logical consequence [Rumfitt, by Hale/Hoffmann,A] |
12193 | Logical necessity is when 'necessarily A' implies 'not-A is contradictory' [Rumfitt] |
12200 | A logically necessary statement need not be a priori, as it could be unknowable [Rumfitt] |
12202 | Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not [Rumfitt] |
12203 | If a world is a fully determinate way things could have been, can anyone consider such a thing? [Rumfitt] |
6485 | When a red object is viewed, the air in between does not become red [Robinson,H] |
6521 | Representative realists believe that laws of phenomena will apply to the physical world [Robinson,H] |
6509 | Representative realists believe some properties of sense-data are shared by the objects themselves [Robinson,H] |
6522 | Phenomenalism can be theistic (Berkeley), or sceptical (Hume), or analytic (20th century) [Robinson,H] |
6502 | Can we reduce perception to acquisition of information, which is reduced to causation or disposition? [Robinson,H] |
6513 | Would someone who recovered their sight recognise felt shapes just by looking? [Robinson,H] |
6512 | Secondary qualities have one sensory mode, but primary qualities can have more [Robinson,H] |
6497 | We say objects possess no intrinsic secondary qualities because physicists don't need them [Robinson,H] |
6494 | If objects are not coloured, and neither are sense-contents, we are left saying that nothing is coloured [Robinson,H] |
6499 | Shape can be experienced in different ways, but colour and sound only one way [Robinson,H] |
6500 | If secondary qualities match senses, would new senses create new qualities? [Robinson,H] |
6484 | Most moderate empiricists adopt Locke's representative theory of perception [Robinson,H] |
6508 | Sense-data leads to either representative realism or phenomenalism or idealism [Robinson,H] |
6480 | Sense-data do not have any intrinsic intentionality [Robinson,H] |
6482 | For idealists and phenomenalists sense-data are in objects; representative realists say they resemble objects [Robinson,H] |
6505 | Sense-data are rejected because they are a veil between us and reality, leading to scepticism [Robinson,H] |
6506 | 'Sense redly' sounds peculiar, but 'senses redly-squarely tablely' sounds far worse [Robinson,H] |
6507 | Adverbialism sees the contents of sense-experience as modes, not objects [Robinson,H] |
6511 | If there are only 'modes' of sensing, then an object can no more be red or square than it can be proud or lazy. [Robinson,H] |
6515 | An explanation presupposes something that is improbable unless it is explained [Robinson,H] |
6517 | If all possibilities are equal, order seems (a priori) to need an explanation - or does it? [Robinson,H] |
6481 | If intentional states are intrinsically about other things, what are their own properties? [Robinson,H] |
6503 | Physicalism cannot allow internal intentional objects, as brain states can't be 'about' anything [Robinson,H] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
6519 | Locke's solidity is not matter, because that is impenetrability and hardness combined [Robinson,H] |