114 ideas
| 21267 | Supposing many principles is superfluous if a few will do it [Aquinas] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 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] |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| 23173 | If a syllogism admits one absurdity, others must follow [Aquinas] |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 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] |
| 10464 | A trope is a bit of a property or relation (not an exemplification or a quality) [Bacon,John] |
| 10465 | Trope theory is ontologically parsimonious, with possibly only one-category [Bacon,John] |
| 10467 | Individuals consist of 'compresent' tropes [Bacon,John] |
| 16765 | Humans only have a single substantial form, which contains the others and acts for them [Aquinas] |
| 10466 | Maybe possible worlds are just sets of possible tropes [Bacon,John] |
| 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] |
| 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] |
| 20224 | Sensation prepares the way for intellectual knowledge, which needs the virtues of reason [Aquinas] |
| 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] |
| 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] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 23180 | The will is the rational appetite [Aquinas] |
| 22112 | For humans good is accordance with reason, and bad is contrary to reason [Aquinas] |
| 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] |
| 22114 | Tyrannical laws are irrational, and so not really laws [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] |
| 7291 | For Aquinas a war must be in a just cause, have proper authority, and aim at good [Aquinas, by Grayling] |
| 5508 | Aquinas says a fertilized egg is not human, and has no immortal soul [Aquinas, by Martin/Barresi] |
| 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] |
| 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] |