141 ideas
| 2136 | Philosophers become as divine and orderly as possible, by studying divinity and order [Plato] |
| 23767 | The winds of the discussion should decide its destination [Plato] |
| 23682 | It would be absurd to be precise about the small things, but only vague about the big things [Plato] |
| 2151 | Dialectic is the only method of inquiry which uproots the things which it takes for granted [Plato] |
| 2154 | The ability to take an overview is the distinguishing mark of a dialectician [Plato] |
| 4011 | For Plato, rationality is a vision of and love of a cosmic rational order [Plato, by Taylor,C] |
| 2093 | You must never go against what you actually believe [Plato] |
| 2130 | People often merely practice eristic instead of dialectic, because they don't analyse the subject-matter [Plato] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 2145 | In mathematics certain things have to be accepted without further explanation [Plato] |
| 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] |
| 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] |
| 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] |
| 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] |
| 8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
| 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] |
| 9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
| 9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
| 8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
| 9861 | The same thing is both one and an unlimited number at the same time [Plato] |
| 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] |
| 9862 | To become rational, philosophers must rise from becoming into being [Plato] |
| 21818 | Being depends on the Good, which is not itself being, but superior to being [Plato] |
| 2061 | The best things (gods, healthy bodies, good souls) are least liable to change [Plato] |
| 6562 | Plato's reality has unchanging Parmenidean forms, and Heraclitean flux [Plato, by Fogelin] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 2142 | The plurality of beautiful things must belong to a single class, because they have a single particular character [Plato] |
| 5094 | Plato's Forms are said to have no location in space [Plato, by Aristotle] |
| 12043 | Forms are not universals, as they don't cover every general term [Plato, by Annas] |
| 2159 | Craftsmen making furniture refer to the form, but no one manufactures the form of furniture [Plato] |
| 17 | A Form applies to a set of particular things with the same name [Plato] |
| 12122 | Plato mistakenly thought forms were totally abstracted away from matter [Bacon on Plato] |
| 5574 | Plato's Forms not only do not come from the senses, but they are beyond possibility of sensing [Plato, by Kant] |
| 2133 | Knowledge must be of the permanent unchanging nature of things [Plato] |
| 2162 | If theory and practice conflict, the best part of the mind accepts theory, so the other part is of lower grade [Plato] |
| 2140 | True belief without knowledge is like blind people on the right road [Plato] |
| 2096 | Is the function of the mind management, authority and planning - or is it one's whole way of life? [Plato] |
| 6009 | Psychic conflict is clear if appetite is close to the body and reason fairly separate [Plato, by Modrak] |
| 6041 | There is a third element to the mind - spirit - lying between reason and appetite [Plato] |
| 2127 | The mind has parts, because we have inner conflicts [Plato] |
| 1737 | The soul seems to have an infinity of parts [Aristotle on Plato] |
| 5945 | The 'Republic' is a great work of rhetorical theory [Lawson-Tancred on Plato] |
| 23316 | For Plato and Aristotle there is no will; there is only rational desire for what is seen as good [Plato, by Frede,M] |
| 16 | We avoid evil either through a natural aversion, or because we have acquired knowledge [Plato] |
| 16565 | Without the surface decoration, poetry shows only appearances and nothing of what is real [Plato] |
| 2160 | Representation is two steps removed from the truth [Plato] |
| 2163 | Artists should be excluded from a law-abiding community, because they destroy the rational mind [Plato] |
| 2135 | Truth is closely related to proportion [Plato] |
| 2141 | I suggest that we forget about trying to define goodness itself for the time being [Plato] |
| 1869 | The good cannot be expressed in words, but imprints itself upon the soul [Plato, by Celsus] |
| 4115 | Plato found that he could only enforce rational moral justification by creating an authoritarian society [Williams,B on Plato] |
| 4547 | Plato measured the degree of reality by the degree of value [Nietzsche on Plato] |
| 2094 | A thing's function is what it alone can do, or what it does better than other things [Plato] |
| 2095 | If something has a function then it has a state of being good [Plato] |
| 2129 | Goodness is mental health, badness is mental sickness [Plato] |
| 2168 | Clever criminals do well at first, but not in the long run [Plato] |
| 12 | If we were invisible, would the just man become like the unjust? [Plato] |
| 4007 | For Plato we abandon honour and pleasure once we see the Good [Plato, by Taylor,C] |
| 2137 | The main aim is to understand goodness, which gives everything its value and advantage [Plato] |
| 2139 | Every person, and every activity, aims at the good [Plato] |
| 2144 | Goodness makes truth and knowledge possible [Plato] |
| 2164 | Bad is always destructive, where good preserves and benefits [Plato] |
| 2143 | Good has the same role in the world of knowledge as the sun has in the physical world [Plato] |
| 2147 | The sight of goodness leads to all that is fine and true and right [Plato] |
| 2138 | Pleasure is commonly thought to be the good, though the more ingenious prefer knowledge [Plato] |
| 2070 | Even people who think pleasure is the good admit that there are bad pleasures [Plato] |
| 2157 | Nice smells are intensive, have no preceding pain, and no bad after-effect [Plato] |
| 2134 | Philosophers are concerned with totally non-physical pleasures [Plato] |
| 2156 | There are three types of pleasure, for reason, for spirit and for appetite [Plato] |
| 2158 | Pleasure-seekers desperately seek illusory satisfaction, like filling a leaky vessel [Plato] |
| 2123 | Excessive pleasure deranges people, making the other virtues impossible [Plato] |
| 2166 | We should behave well even if invisible, for the health of the mind [Plato] |
| 2097 | Isn't it better to have a reputation for goodness than to actually be good? [Plato] |
| 19946 | Morality is a compromise, showing restraint, to avoid suffering wrong without compensation [Plato] |
| 5 | Justice is merely the interests of the stronger party [Plato] |
| 7 | Surely you don't return a borrowed weapon to a mad friend? [Plato] |
| 8 | Is right just the interests of the powerful? [Plato] |
| 15 | Sin first, then sacrifice to the gods from the proceeds [Plato] |
| 5944 | For Plato, virtue is its own reward [Lawson-Tancred on Plato] |
| 2155 | True goodness requires mental unity and harmony [Plato] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 2126 | A good community necessarily has wisdom, courage, self-discipline and morality [Plato] |
| 2092 | Simonides said morality is helping one's friends and harming one's enemies [Plato] |
| 23562 | If the parts of our soul do their correct work, we will be just people, and will act justly [Plato] |
| 19889 | People need society because the individual has too many needs [Plato] |
| 19890 | All exchanges in a community are for mutual benefit [Plato] |
| 10 | After a taste of mutual harm, men make a legal contract to avoid it [Plato] |
| 23561 | People doing their jobs properly is the fourth cardinal virtue for a city [Plato] |
| 2149 | Reluctant rulers make a better and more unified administration [Plato] |
| 2132 | Only rule by philosophers of integrity can keep a community healthy [Plato] |
| 2131 | Is there anything better for a community than to produce excellent people? [Plato] |
| 2152 | Dialectic is the highest and most important part of the curriculum [Plato] |
| 2148 | To gain knowledge, turn away from the world of change, and focus on true goodness [Plato] |
| 2153 | Compulsory intellectual work never remains in the mind [Plato] |
| 2630 | If Plato's God is immaterial, he will lack consciousness, wisdom, pleasure and movement, which are essential to him [Cicero on Plato] |
| 14 | If the gods are non-existent or indifferent, why bother to deceive them? [Plato] |
| 2165 | Something is unlikely to be immortal if it is imperfectly made from diverse parts [Plato] |
| 13 | Is the supreme reward for virtue to be drunk for eternity? [Plato] |
| 2120 | God is responsible for the good things, but we must look elsewhere for the cause of the bad things [Plato] |