136 ideas
| 19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
| 1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
| 1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
| 21267 | Supposing many principles is superfluous if a few will do it [Aquinas] |
| 8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
| 4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
| 23176 | Truth is universal, but knowledge of it is not [Aquinas] |
| 20621 | Types of lying: Speak lies, intend lies, intend deception, aim at deceptive goal? [Aquinas, by Tuckness/Wolf] |
| 21248 | If the existence of truth is denied, the 'Truth does not exist' must be true! [Aquinas] |
| 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] |
| 13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
| 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] |
| 23173 | If a syllogism admits one absurdity, others must follow [Aquinas] |
| 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] |
| 4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
| 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] |
| 15812 | Being implies distinctness, which implies division, unity, and multitude [Aquinas] |
| 21268 | Non-human things are explicable naturally, and voluntary things by the will, so God is not needed [Aquinas] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
| 13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
| 16765 | Humans only have a single substantial form, which contains the others and acts for them [Aquinas] |
| 5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
| 23175 | The conclusions of speculative reason about necessities are certain [Aquinas] |
| 21337 | A knowing being possesses a further reality, the 'presence' of the thing known [Aquinas] |
| 11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
| 21249 | Some things are self-evident to us; others are only self-evident in themselves [Aquinas] |
| 21250 | A proposition is self-evident if the predicate is included in the essence of the subject [Aquinas] |
| 23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
| 20224 | Sensation prepares the way for intellectual knowledge, which needs the virtues of reason [Aquinas] |
| 16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
| 16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
| 11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
| 3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
| 12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
| 22107 | Sensations are transmitted to 'internal senses' in the brain, chiefly to 'phantasia' and 'imagination' [Aquinas, by Kretzmann/Stump] |
| 9098 | Mental activity combines what we sense with imagination of what is not present [Aquinas] |
| 9092 | Abstracting A from B generates truth, as long as the connection is not denied [Aquinas] |
| 9093 | We understand the general nature of things by ignoring individual peculiarities [Aquinas] |
| 9097 | The mind abstracts generalities from images, but also uses images for understanding [Aquinas] |
| 9095 | Very general ideas (being, oneness, potentiality) can be abstracted from thought matter in general [Aquinas] |
| 9099 | Particular instances come first, and (pace Plato) generalisations are abstracted from them [Aquinas] |
| 10508 | Species are abstracted from appearances by ignoring individual conditions [Aquinas] |
| 22111 | Aquinas attributes freedom to decisions and judgements, and not to the will alone [Aquinas, by Kretzmann/Stump] |
| 22105 | The human intellectual soul is an incorporeal, subsistent principle [Aquinas] |
| 23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
| 22108 | First grasp what it is, then its essential features; judgement is their compounding and division [Aquinas] |
| 10503 | We abstract forms from appearances, and acquire knowledge of immaterial things [Aquinas] |
| 10509 | Understanding consists entirely of grasping abstracted species [Aquinas] |
| 10506 | Mathematics can be abstracted from sensible matter, and from individual intelligible matter [Aquinas] |
| 9094 | Mathematical objects abstract both from perceived matter, and from particular substance [Aquinas] |
| 10505 | We can just think of an apple's colour, because the apple is not part of the colour's nature [Aquinas] |
| 10504 | Abstracting either treats something as separate, or thinks of it separately [Aquinas] |
| 10507 | Numbers and shapes are abstracted by ignoring their sensible qualities [Aquinas] |
| 9096 | The mind must produce by its own power an image of the individual species [Aquinas] |
| 11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
| 23180 | The will is the rational appetite [Aquinas] |
| 22112 | For humans good is accordance with reason, and bad is contrary to reason [Aquinas] |
| 6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
| 22494 | We must know the end, know that it is the end, and know how to attain it [Aquinas] |
| 23181 | All acts of virtue relate to justice, which is directed towards the common good [Aquinas] |
| 8009 | Aquinas wanted, not to escape desire, but to transform it for moral ends [Aquinas, by MacIntyre] |
| 23182 | Legal justice is supreme, because it directs the other virtues to the common good [Aquinas] |
| 22399 | Temperance prevents our passions from acting against reason [Aquinas] |
| 23177 | Justice directs our relations with others, because it denotes a kind of equality [Aquinas] |
| 23179 | People differ in their social degrees, and a particular type of right applies to each [Aquinas] |
| 23174 | Natural law is a rational creature's participation in eternal law [Aquinas] |
| 22113 | Right and wrong actions pertain to natural law, as perceived by practical reason [Aquinas] |
| 22114 | Tyrannical laws are irrational, and so not really laws [Aquinas] |
| 7291 | For Aquinas a war must be in a just cause, have proper authority, and aim at good [Aquinas, by Grayling] |
| 11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
| 3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
| 5508 | Aquinas says a fertilized egg is not human, and has no immortal soul [Aquinas, by Martin/Barresi] |
| 8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
| 12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
| 16687 | Bodies are three-dimensional substances [Aquinas] |
| 23178 | Divine law commands some things because they are good, while others are good because commanded [Aquinas] |
| 21251 | We can't know God's essence, so his existence can't be self-evident for us [Aquinas] |
| 5614 | If you assume that there must be a necessary being, you can't say which being has this quality [Kant on Aquinas] |
| 21269 | Way 1: the infinite chain of potential-to-actual movement has to have a first mover [Aquinas] |
| 21270 | Way 2: no effect without a cause, and this cannot go back to infinity, so there is First Cause [Aquinas] |
| 21271 | Way 3: contingent beings eventually vanish, so continuity needs a necessary being [Aquinas] |
| 21272 | Way 4: the source of all qualities is their maximum, so something (God) causes all perfections [Aquinas] |
| 21273 | Way 5: mindless things act towards an obvious end, so there is an intelligent director [Aquinas] |
| 22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |
| 20211 | Life aims at the Beatific Vision - of perfect happiness, and revealed truth [Aquinas, by Zagzebski] |
| 22106 | Aquinas saw angels as separated forms, rather than as made of 'spiritual matter' [Aquinas, by Kretzmann/Stump] |
| 23306 | Humans have a non-physical faculty of reason, so they can be immortal [Aquinas, by Sorabji] |
| 4412 | Those in bliss have their happiness increased by seeing the damned punished [Aquinas] |
| 21266 | God does not exist, because He is infinite and good, and so no evil should be discoverable [Aquinas] |
| 21274 | It is part of God's supreme goodness that He brings good even out of evil [Aquinas] |