88 ideas
7834 | Great philosophies are confessions by the author, growing out of moral intentions [Nietzsche] |
7080 | Metaphysics divided the old unified Greek world into two [Nietzsche, by Critchley] |
11090 | Why do we want truth, rather than falsehood or ignorance? The value of truth is a problem [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] |
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] |
7079 | Nietzsche resists nihilism through new values, for a world of becoming, without worship [Nietzsche, by Critchley] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
2878 | We see an approximation of a tree, not the full detail [Nietzsche] |
20140 | We shouldn't object to a false judgement, if it enhances and preserves life [Nietzsche] |
2877 | Morality becomes a problem when we compare many moralities [Nietzsche] |
20355 | The ranking of a person's innermost drives reveals their true nature [Nietzsche] |
2291 | A thought comes when 'it' wants, not when 'I' want [Nietzsche] |
2871 | Wanting 'freedom of will' is wanting to pull oneself into existence out of the swamp of nothingness by one's own hair [Nietzsche] |
20381 | It is psychology which reveals the basic problems [Nietzsche] |
2860 | The most boring and dangerous of all errors is Plato's invention of pure spirit and goodness [Nietzsche] |
1568 | Nietzsche felt that Plato's views downgraded the human body and its brevity of life [Nietzsche, by Roochnik] |
2883 | Noble people see themselves as the determiners of values [Nietzsche] |
23440 | Nietzsche's judgement of actions by psychology instead of outcome was poisonous [Foot on Nietzsche] |
2875 | That which is done out of love always takes place beyond good and evil [Nietzsche] |
2868 | Nature is totally indifferent, so you should try to be different from it, not live by it [Nietzsche] |
2882 | Morality originally judged people, and actions only later on [Nietzsche] |
2872 | In the earliest phase of human history only consequences mattered [Nietzsche] |
2885 | The noble soul has reverence for itself [Nietzsche] |
20134 | Moralities extravagantly address themselves to 'all', by falsely generalising [Nietzsche] |
2881 | Virtue has been greatly harmed by the boringness of its advocates [Nietzsche] |
20382 | The four virtues are courage, insight, sympathy, solitude [Nietzsche] |
2879 | In ancient Rome pity was considered neither good nor bad [Nietzsche] |
2859 | The idea of the categorical imperative is just that we should all be very obedient [Nietzsche] |
2884 | The morality of slaves is the morality of utility [Nietzsche] |
2880 | The greatest possibilities in man are still unexhausted [Nietzsche] |
2876 | The thought of suicide is a great reassurance on bad nights [Nietzsche] |
7078 | The freedom of the subject means the collapse of moral certainty [Nietzsche, by Critchley] |
6869 | Nietzsche thinks the human condition is to overcome and remake itself [Nietzsche, by Ansell Pearson] |
2874 | Man is the animal whose nature has not yet been fixed [Nietzsche] |
20137 | The great person engages wholly with life, and is happy to endlessly relive the life they created [Nietzsche] |
20139 | Only aristocratic societies can elevate the human species [Nietzsche] |
20373 | A healthy aristocracy has no qualms about using multitudes of men as instruments [Nietzsche] |
22394 | Democracy diminishes mankind, making them mediocre and lowering their value [Nietzsche] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
2867 | Christianity is Platonism for the people [Nietzsche] |