82 ideas
| 21267 | Supposing many principles is superfluous if a few will do it [Aquinas] |
| 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] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 23173 | If a syllogism admits one absurdity, others must follow [Aquinas] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 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] |
| 16765 | Humans only have a single substantial form, which contains the others and acts for them [Aquinas] |
| 3570 | Maybe knowledge is belief which 'tracks' the truth [Nozick, by Williams,M] |
| 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] |
| 2748 | A true belief isn't knowledge if it would be believed even if false. It should 'track the truth' [Nozick, by Dancy,J] |
| 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] |
| 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] |
| 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] |
| 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] |