80 ideas
13252 | Some truths have true negations [Beall/Restall] |
13247 | A truthmaker is an object which entails a sentence [Beall/Restall] |
13249 | (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically [Beall/Restall] |
13243 | Excluded middle must be true for some situation, not for all situations [Beall/Restall] |
13242 | It's 'relevantly' valid if all those situations make it true [Beall/Restall] |
13246 | Relevant logic does not abandon classical logic [Beall/Restall] |
13245 | Relevant consequence says invalidity is the conclusion not being 'in' the premises [Beall/Restall] |
13254 | A doesn't imply A - that would be circular [Beall/Restall] |
13255 | Relevant logic may reject transitivity [Beall/Restall] |
13250 | Free logic terms aren't existential; classical is non-empty, with referring names [Beall/Restall] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
13235 | Logic studies consequence; logical truths are consequences of everything, or nothing [Beall/Restall] |
13238 | Syllogisms are only logic when they use variables, and not concrete terms [Beall/Restall] |
13234 | The view of logic as knowing a body of truths looks out-of-date [Beall/Restall] |
13232 | Logic studies arguments, not formal languages; this involves interpretations [Beall/Restall] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
13241 | The model theory of classical predicate logic is mathematics [Beall/Restall] |
13253 | There are several different consequence relations [Beall/Restall] |
13240 | A sentence follows from others if they always model it [Beall/Restall] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
13236 | Logical truth is much more important if mathematics rests on it, as logicism claims [Beall/Restall] |
13237 | Preface Paradox affirms and denies the conjunction of propositions in the book [Beall/Restall] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
13244 | Relevant necessity is always true for some situation (not all situations) [Beall/Restall] |
13239 | Judgement is always predicating a property of a subject [Beall/Restall] |
13248 | We can rest truth-conditions on situations, rather than on possible worlds [Beall/Restall] |
13233 | Propositions commit to content, and not to any way of spelling it out [Beall/Restall] |
20547 | We should respect the right of people to live in their own way, even if it is irrational [Swift] |
20564 | Anti-colonial movements usually invoke the right of their 'people' to self-determination [Swift] |
20535 | Isn't it more rational to maximise the average position, but with a safety net? [Swift] |
20537 | Hypothetical contracts have no binding force [Swift] |
20542 | Cosmopolitans reject the right of different states to distribute resources in different ways [Swift] |
20559 | Democracy is bad, but the other systems are worse [Swift] |
20561 | Since all opinions are treated as equal in democracy, it implies there are no right answers [Swift] |
20562 | Design your democracy to treat citizens equally, or to produce better citizens? [Swift] |
20563 | Design your democracy to yield political stability, or good decisions? [Swift] |
20560 | Teledemocracy omits debate and deliberation, which are important parts of good decisions [Swift] |
20554 | Multiculturalism is a barrier to the whole state being a community [Swift] |
20553 | Liberals mistakenly think individuals choose their values, without reference to the community [Swift] |
20556 | The best way to build a cohesive community is to be involved in a war [Swift] |
20557 | Membership and inclusion in a community implies non-membership and exclusion [Swift] |
20555 | Liberals are concerned to protect individuals from too much community [Swift] |
20540 | Redistributing wealth treats some people as means, rather than as ends [Swift] |
20551 | Men have had the power to structure all of our social institutions [Swift] |
20545 | Maybe a freedom is from a restraint, and also in order to do something [Swift] |
20550 | Opportunity should ignore extraneous factors, or foster competence, or ignore all disadvantages [Swift] |
20536 | Inequalities are needed, as incentives to do the most important jobs [Swift] |
20548 | A person can desire redistibution of wealth, without it being for reasons of equality [Swift] |
20541 | You can't necessarily sell your legitimate right to something, even if you produced it [Swift] |
20546 | Libertarians about property ignore the fact that private property is a denial of freedoms [Swift] |
20533 | Justice can be seen as fairness or entitlement or desert [Swift] |