107 ideas
| 14857 | The highest wisdom has the guise of simplicity [Nietzsche] |
| 14854 | Deep thinkers know that they are always wrong [Nietzsche] |
| 14833 | Comedy is a transition from fear to exuberance [Nietzsche] |
| 14853 | Truth finds fewest champions not when it is dangerous, but when it is boring [Nietzsche] |
| 24075 | Convictions, more than lies, are the great enemy of truth [Nietzsche] |
| 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] |
| 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] |
| 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] |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional 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] |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| 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] |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order 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] |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [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] |
| 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] |
| 14858 | Being certain presumes that there are absolute truths, and means of arriving at them [Nietzsche] |
| 14830 | Intuition only recognises what is possible, not what exists or is certain [Nietzsche] |
| 20378 | Just as skin hides the horrors of the body, vanity conceals the passions of the soul [Nietzsche] |
| 14820 | People always do what they think is right, according to the degree of their intellect [Nietzsche] |
| 14856 | Our judgment seems to cause our nature, but actually judgment arises from our nature [Nietzsche] |
| 14842 | Why are the strong tastes of other people so contagious? [Nietzsche] |
| 14835 | Artists are not especially passionate, but they pretend to be [Nietzsche] |
| 14807 | The history of morality rests on an error called 'responsibility', which rests on an error called 'free will' [Nietzsche] |
| 14823 | Ceasing to believe in human responsibility is bitter, if you had based the nobility of humanity on it [Nietzsche] |
| 14824 | It is absurd to blame nature and necessity; we should no more praise actions than we praise plants or artworks [Nietzsche] |
| 22473 | Nietzsche said the will doesn't exist, so it can't ground moral responsibility [Nietzsche, by Foot] |
| 14812 | Intellect is tied to morality, because it requires good memory and powerful imagination [Nietzsche] |
| 14810 | Originally it was the rulers who requited good for good and evil for evil who were called 'good' [Nietzsche] |
| 14831 | No one has ever done anything that was entirely for other people [Nietzsche] |
| 14855 | Simultaneous love and respect are impossible; love has no separation or rank, but respect admits power [Nietzsche] |
| 14815 | We get enormous pleasure from tales of noble actions [Nietzsche] |
| 14849 | We can only achieve happy moments, not happy eras [Nietzsche] |
| 14818 | First morality is force, then custom, then acceptance, then instinct, then a pleasure - and finally 'virtue' [Nietzsche] |
| 20103 | You are mastered by your own virtues, but you must master them, and turn them into tools [Nietzsche] |
| 14817 | The 'good' man does the moral thing as if by nature, easily and gladly, after a long inheritance [Nietzsche] |
| 14809 | All societies of good men give a priority to gratitude [Nietzsche] |
| 14816 | Justice (fairness) originates among roughly equal powers (as the Melian dialogues show) [Nietzsche] |
| 20112 | Pity consoles those who suffer, because they see that they still have the power to hurt [Nietzsche] |
| 14821 | Apart from philosophers, most people rightly have a low estimate of pity [Nietzsche] |
| 14841 | Many people are better at having good friends than being a good friend [Nietzsche] |
| 14843 | Women can be friends with men, but only some physical antipathy will maintain it [Nietzsche] |
| 14811 | In Homer it is the contemptible person, not the harmful person, who is bad [Nietzsche] |
| 20111 | We could live more naturally, relishing the spectacle, and not thinking we are special [Nietzsche] |
| 14844 | People do not experience boredom if they have never learned to work properly [Nietzsche] |
| 14808 | Over huge periods of time human character would change endlessly [Nietzsche] |
| 14822 | If self-defence is moral, then so are most expressions of 'immoral' egoism [Nietzsche] |
| 14838 | The state aims to protect individuals from one another [Nietzsche] |
| 14852 | Culture cannot do without passions and vices [Nietzsche] |
| 14846 | If we want the good life for the greatest number, we must let them decide on the good life [Nietzsche] |
| 14819 | Slavery cannot be judged by our standards, because the sense of justice was then less developed [Nietzsche] |
| 14847 | Laws that are well thought out, or laws that are easy to understand? [Nietzsche] |
| 14814 | Execution is worse than murder, because we are using the victim, and really we are the guilty [Nietzsche] |
| 14836 | People will enthusiastically pursue an unwanted war, once sacrifices have been made [Nietzsche] |
| 14845 | Don't crush girls with dull Gymnasium education, the way we have crushed boys! [Nietzsche] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 14839 | Interest in education gains strength when we lose interest in God [Nietzsche] |
| 14848 | Education in large states is mediocre, like cooking in large kitchens [Nietzsche] |
| 14834 | Teachers only gather knowledge for their pupils, and can't be serious about themselves [Nietzsche] |
| 14825 | In religious thought nature is a complex of arbitrary acts by conscious beings [Nietzsche] |
| 14826 | Modern man wants laws of nature in order to submit to them [Nietzsche] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 14827 | The Greeks saw the gods not as their masters, but as idealised versions of themselves [Nietzsche] |
| 14813 | Science rejecting the teaching of Christianity in favour of Epicurus shows the superiority of the latter [Nietzsche] |
| 14832 | The Sermon on the Mount is vanity - praying to one part of oneself, and demonising the rest [Nietzsche] |
| 14850 | Christ was the noblest human being [Nietzsche] |
| 14837 | Christ seems warm hearted, and suppressed intellect in favour of the intellectually weak [Nietzsche] |
| 14828 | Religion is tempting if your life is boring, but you can't therefore impose it on the busy people [Nietzsche] |