76 ideas
3811 | Entailment and validity are relations, but inference is a human activity [Searle] |
3822 | Theory involves accepting conclusions, and so is a special case of practical reason [Searle] |
3806 | Rationality is built into the intentionality of the mind, and its means of expression [Searle] |
3812 | Rationality is the way we coordinate our intentionality [Searle] |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
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] |
3809 | If complex logic requires rules, then so does basic logic [Searle] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
3810 | In real reasoning semantics gives validity, not syntax [Searle] |
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] |
17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
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] |
3841 | Users of 'supervenience' blur its causal and constitutive meanings [Searle] |
3816 | Our beliefs are about things, not propositions (which are the content of the belief) [Searle] |
3833 | A belief is a commitment to truth [Searle] |
3837 | We can't understand something as a lie if beliefs aren't commitment to truth [Searle] |
3828 | Thinking must involve a self, not just an "it" [Searle] |
3831 | Reasons can either be facts in the world, or intentional states [Searle] |
3830 | In the past people had a reason not to smoke, but didn't realise it [Searle] |
3832 | Causes (usually events) are not the same as reasons (which are never events) [Searle] |
3823 | Being held responsible for past actions makes no sense without personal identity [Searle] |
3821 | Giving reasons for action requires reference to a self [Searle] |
3824 | A 'self' must be capable of conscious reasonings about action [Searle] |
3834 | An intentional, acting, rational being must have a self [Searle] |
3825 | Action requires a self, even though perception doesn't [Searle] |
3829 | Selfs are conscious, enduring, reasonable, active, free, and responsible [Searle] |
3826 | A self must at least be capable of consciousness [Searle] |
3827 | The self is neither an experience nor a thing experienced [Searle] |
3820 | The bundle must also have agency in order to act, and a self to act rationally [Searle] |
3817 | Free will is most obvious when we choose between several reasons for an action [Searle] |
3808 | Rational decision making presupposes free will [Searle] |
3818 | We freely decide whether to make a reason for action effective [Searle] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
3814 | Preferences can result from deliberation, not just precede it [Searle] |
3840 | We don't accept practical reasoning if the conclusion is unpalatable [Searle] |
3815 | The essence of humanity is desire-independent reasons for action [Searle] |
3839 | Only an internal reason can actually motivate the agent to act [Searle] |
3835 | If it is true, you ought to believe it [Searle] |
3836 | If this is a man, you ought to accept similar things as men [Searle] |
3838 | Promises hold because I give myself a reason, not because it is an institution [Searle] |
3813 | 'Ought' implies that there is a reason to do something [Searle] |