84 ideas
11832 | We learn a concept's relations by using it, without reducing it to anything [Wiggins] |
10073 | There cannot be a set theory which is complete [Smith,P] |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
11863 | (λx)[Man x] means 'the property x has iff x is a man'. [Wiggins] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
10613 | No nice theory can define truth for its own language [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
14746 | What exists can't depend on our conceptual scheme, and using all conceptual schemes is too liberal [Sider on Wiggins] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
11900 | We can accept criteria of distinctness and persistence, without making the counterfactual claims [Mackie,P on Wiggins] |
11870 | Activity individuates natural things, functions do artefacts, and intentions do artworks [Wiggins] |
11866 | The idea of 'thisness' is better expressed with designation/predication and particular/universal [Wiggins] |
11896 | A sortal essence is a thing's principle of individuation [Wiggins, by Mackie,P] |
15835 | Wiggins's sortal essentialism rests on a thing's principle of individuation [Wiggins, by Mackie,P] |
11841 | The evening star is the same planet but not the same star as the morning star, since it is not a star [Wiggins] |
10679 | 'Sortalism' says parts only compose a whole if it falls under a sort or kind [Wiggins, by Hossack] |
14363 | Identity a=b is only possible with some concept to give persistence and existence conditions [Wiggins, by Strawson,P] |
14364 | A thing is necessarily its highest sortal kind, which entails an essential constitution [Wiggins, by Strawson,P] |
11851 | Many predicates are purely generic, or pure determiners, rather than sortals [Wiggins] |
11865 | The possibility of a property needs an essential sortal concept to conceive it [Wiggins] |
14744 | Objects can only coincide if they are of different kinds; trees can't coincide with other trees [Wiggins, by Sider] |
11852 | Is the Pope's crown one crown, if it is made of many crowns? [Wiggins] |
11875 | Boundaries are not crucial to mountains, so they are determinate without a determinate extent [Wiggins] |
14749 | Identity is an atemporal relation, but composition is relative to times [Wiggins, by Sider] |
11844 | If I destroy an item, I do not destroy each part of it [Wiggins] |
11861 | We can forget about individual or particularized essences [Wiggins] |
11871 | Essences are not explanations, but individuations [Wiggins] |
11879 | Essentialism is best represented as a predicate-modifier: □(a exists → a is F) [Wiggins, by Mackie,P] |
11835 | The nominal essence is the idea behind a name used for sorting [Wiggins] |
11876 | It is easier to go from horses to horse-stages than from horse-stages to horses [Wiggins] |
11858 | The question is not what gets the title 'Theseus' Ship', but what is identical with the original [Wiggins] |
11843 | Identity over a time and at a time aren't different concepts [Wiggins] |
11864 | Hesperus=Hesperus, and Phosphorus=Hesperus, so necessarily Phosphorus=Hesperus [Wiggins] |
11831 | The formal properties of identity are reflexivity and Leibniz's Law [Wiggins] |
14362 | Relative Identity is incompatible with the Indiscernibility of Identicals [Wiggins, by Strawson,P] |
11838 | Relativity of Identity makes identity entirely depend on a category [Wiggins] |
11847 | To identify two items, we must have a common sort for them [Wiggins] |
11839 | Do both 'same f as' and '=' support Leibniz's Law? [Wiggins] |
11845 | Substitutivity, and hence most reasoning, needs Leibniz's Law [Wiggins] |
11869 | Possible worlds rest on the objects about which we have suppositions [Wiggins] |
11850 | Not every story corresponds to a possible world [Wiggins] |
11848 | Asking 'what is it?' nicely points us to the persistence of a continuing entity [Wiggins] |
11859 | The mind conceptualizes objects; yet objects impinge upon the mind [Wiggins] |
11836 | We can use 'concept' for the reference, and 'conception' for sense [Wiggins] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
11860 | Lawlike propensities are enough to individuate natural kinds [Wiggins] |