86 ideas
20388 | 'Necessary' conditions are requirements, and 'sufficient' conditions are guarantees [Davies,S] |
20389 | A definition of a thing gives all the requirements which add up to a guarantee of it [Davies,S] |
20391 | Feminists warn that ideologies use timeless objective definitions as a tool of repression [Davies,S] |
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] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [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] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [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] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [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] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [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] |
5311 | If observation goes up a level, we expect the laws of the lower level to remain in force [Wilson,EO] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
5312 | A child first sees objects as distinct, and later as members of groups [Wilson,EO] |
5309 | Beliefs are really enabling mechanisms for survival [Wilson,EO] |
20387 | Aesthetic experience involves perception, but also imagination and understanding [Davies,S] |
20385 | The faculty of 'taste' was posited to explain why only some people had aesthetic appreciation [Davies,S] |
20386 | The sublime is negative in awareness of insignificance, and positive in showing understanding [Davies,S] |
20384 | The idea that art forms are linked into a single concept began in the 1740s [Davies,S] |
20390 | Defining art as representation or expression or form were all undermined by the avant-garde [Davies,S] |
20392 | 'Aesthetic functionalism' says art is what is intended to create aesthetic experiences [Davies,S] |
20405 | Music may be expressive by being 'associated' with other emotional words or events [Davies,S] |
20403 | It seems unlikely that sad music expresses a composer's sadness; it takes ages to write [Davies,S] |
20393 | The 'institutional' theory says art is just something appropriately placed in the 'artworld' [Davies,S] |
20402 | Music is too definite to be put into words (not too indefinite!) [Davies,S] |
20395 | The title of a painting can be vital, and the artist decrees who the portrait represents [Davies,S] |
20396 | We must know what the work is meant to be, to evaluate the artist's achievement [Davies,S] |
20399 | Intentionalism says either meaning just is intention, or ('moderate') meaning is successful intention [Davies,S] |
20401 | The meaning is given by the audience's best guess at the author's intentions [Davies,S] |
20397 | If we could perfectly clone the Mona Lisa, the original would still be special [Davies,S] |
20398 | Art that is multiply instanced may require at least one instance [Davies,S] |
20404 | Music isn't just sad because it makes the listener feel sad [Davies,S] |
22705 | If the depiction of evil is glorified, that is an artistic flaw [Davies,S] |
22707 | It is an artistic defect if excessive moral outrage distorts the story, and narrows our sympathies [Davies,S] |
22706 | A work which seeks approval for immorality, but alienates the audience, is a failure [Davies,S] |
22704 | Immorality may or may not be an artistic defect [Davies,S] |
5310 | Philosophers study the consequences of ethics instead of its origins [Wilson,EO] |
5313 | The rules of human decision-making converge and overlap in a 'human nature' [Wilson,EO] |
5316 | We undermine altruism by rewarding it, but we reward it to encourage it [Wilson,EO] |
5318 | Pure hard-core altruism based on kin selection is the enemy of civilisation [Wilson,EO] |
5317 | The actor is most convincing who believes that his performance is real [Wilson,EO] |
5308 | The only human purpose is that created by our genetic history [Wilson,EO] |
5314 | Cultural evolution is Lamarckian and fast, biological evolution is Darwinian and slow [Wilson,EO] |
5315 | Over 99 percent of human evolution has been in the hunter-gatherer phase [Wilson,EO] |
5320 | It is estimated that mankind has produced 100,000 religions [Wilson,EO] |