17 ideas
| 13373 | Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong [Priest,G] |
| Full Idea: A natural principle is the same kind of paradox will have the same kind of solution. Standardly Ramsey's first group are solved by denying the existence of some totality, and the second group are less clear. But denial of the groups sink both. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §5) | |
| A reaction: [compressed] This sums up the argument of Priest's paper, which is that it is Ramsey's division into two kinds (see Idea 13334) which is preventing us from getting to grips with the paradoxes. Priest, notoriously, just lives with them. |
| 13368 | The 'least indefinable ordinal' is defined by that very phrase [Priest,G] |
| Full Idea: König: there are indefinable ordinals, and the least indefinable ordinal has just been defined in that very phrase. (Recall that something is definable iff there is a (non-indexical) noun-phrase that refers to it). | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: Priest makes great subsequent use of this one, but it feels like a card trick. 'Everything indefinable has now been defined' (by the subject of this sentence)? König, of course, does manage to pick out one particular object. |
| 13370 | 'x is a natural number definable in less than 19 words' leads to contradiction [Priest,G] |
| Full Idea: Berry: if we take 'x is a natural number definable in less than 19 words', we can generate a number which is and is not one of these numbers. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: [not enough space to spell this one out in full] |
| 13369 | By diagonalization we can define a real number that isn't in the definable set of reals [Priest,G] |
| Full Idea: Richard: φ(x) is 'x is a definable real number between 0 and 1' and ψ(x) is 'x is definable'. We can define a real by diagonalization so that it is not in x. It is and isn't in the set of reals. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: [this isn't fully clear here because it is compressed] |
| 13366 | The least ordinal greater than the set of all ordinals is both one of them and not one of them [Priest,G] |
| Full Idea: Burali-Forti: φ(x) is 'x is an ordinal', and so w is the set of all ordinals, On; δ(x) is the least ordinal greater than every member of x (abbreviation: log(x)). The contradiction is that log(On)∈On and log(On)∉On. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
| 13367 | The next set up in the hierarchy of sets seems to be both a member and not a member of it [Priest,G] |
| Full Idea: Mirimanoff: φ(x) is 'x is well founded', so that w is the cumulative hierarchy of sets, V; &delta(x) is just the power set of x, P(x). If x⊆V, then V∈V and V∉V, since δ(V) is just V itself. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
| 13371 | If you know that a sentence is not one of the known sentences, you know its truth [Priest,G] |
| Full Idea: In the family of the Liar is the Knower Paradox, where φ(x) is 'x is known to be true', and there is a set of known things, Kn. By knowing a sentence is not in the known sentences, you know its truth. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
| A reaction: [mostly my wording] |
| 13372 | There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar [Priest,G] |
| Full Idea: There are liar chains which fit the pattern of Transcendence and Closure, as can be seen with the simplest case of the Liar Pair. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
| A reaction: [Priest gives full details] Priest's idea is that Closure is when a set is announced as complete, and Transcendence is when the set is forced to expand. He claims that the two keep coming into conflict. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 21332 | We don't get a love of 'order' from nature - which is thoroughly chaotic [Mill] |
| Full Idea: Even the love of 'order' which is thought to be a following of the ways of nature is in fact a contradiction of them. All which people are accustomed to deprecate as 'disorder' is precisely a counterpart of nature's ways. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: The Greeks elevated the idea that the cosmos was orderly, but almost entirely based on the regular movement of the planets. They turned a blind eye to the messy bits of nature. As you magnify nature, order and chaos seem to alternate. |
| 21333 | Evil comes from good just as often as good comes from evil [Mill] |
| Full Idea: If good frequently comes out of evil, the converse fact, evil coming out of good, is equally common. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.117) | |
| A reaction: Mill surmises that on the whole good comes from good, and evil from evil, but the point is that the evidence doesn't favour the production of increased good. |
| 21335 | Belief that an afterlife is required for justice is an admission that this life is very unjust [Mill] |
| Full Idea: The necessity of redressing the balance [of injustice] is deemed one of the strongest arguments for another life after death, which amounts to an admission that the order of things in this life is often an example of injustice, not justice. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874]) | |
| A reaction: It certainly seems that an omnipotent God could administer swift justice in this life. If the whole point is that we need freedom of will, then why is justice administered at a much later date? The freedom seems to be illusory. |
| 21334 | No necessity ties an omnipotent Creator, so he evidently wills human misery [Mill] |
| Full Idea: If a Creator is assumed to be omnipotent, if he bends to a supposed necessity, he himself makes the necessity which he bends to. If the maker of the world can all that he will, he wills misery, and there is no escape from the conclusion. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.119) | |
| A reaction: If you add that the Creator is supposed to be perfectly benevolent, you arrive at the paradox which Mackie spells out. Is the correct conclusion that God exists, and is malevolent? Mill doesn't take that option seriously. |
| 21329 | Nature dispenses cruelty with no concern for either mercy or justice [Mill] |
| Full Idea: All of this [cruel killing] nature does with the most supercilious disregard both of mercy and of justice, emptying her shafts upon the best and noblest indifferently with the meanest and worst | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: The existence of an afterlife at least offers an opportunity to rectify any injustice, but that hardly meets the question of why there was injustice in the first place. It would be odd if it actually is justice, but none of us can see why that is so. |
| 21328 | Killing is a human crime, but nature kills everyone, and often with great tortures [Mill] |
| Full Idea: Killing, the most criminal act recognised by human laws, nature does once to every being that lives, and frequently after protracted tortures such as the greatest know monsters purposely inflicted on their living fellow creatures | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: We certainly don't condemn lions for savaging gazelles, but the concept of a supreme mind controlling nature forces the question. Theology needs consistency between human and divine morality, and the supposed derivation of the former from the latter. |
| 21330 | Nature makes childbirth a miserable experience, often leading to the death of the mother [Mill] |
| Full Idea: In the clumsy provision which nature has made for the perpetual renewal of animal life, ...no human being ever comes into the world but another human being is literally stretched on the rack for hours or day, not unfrequently issuing in death. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: This is a very powerful example, which is rarely cited in modern discussions. |
| 21331 | Hurricanes, locusts, floods and blight can starve a million people to death [Mill] |
| Full Idea: Nature often takes the means by which we live. A single hurricane, a flight of locusts, or an inundation, or a trifling chemical change in an edible root, starve a million people. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: [second sentence compressed] The 'edible root' is an obvious reference to the Irish potato famine. Some desertification had human causes, but these are telling examples. |