79 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
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] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |