10 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) |
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. |
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] |
21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
18006 | Chomsky's 'interpretative semantics' says syntax comes first, and is then interpreted [Chomsky, by Magidor] |
Full Idea: Chomsky and his followers (whose position was labelled 'interpretative semantics') claimed that a sentence is first assigned a syntactic structure by an autonomous syntactic module, and this structure is then provided as input for semantic interpretation. | |
From: report of Noam Chomsky (Aspects of the Theory of Syntax [1965]) by Ofra Magidor - Category Mistakes 1.3 | |
A reaction: This certainly doesn't fit the experience of introspecting speech, but then I suppose good pianists focus entirely on the music, and overlook the finger movements which have obvious priority. But I don't know the syntax of the sentence when I begin it. |