| 1994 | The Structure of Paradoxes of Self-Reference |
| §2 | p.27 | 13366 | The least ordinal greater than the set of all ordinals is both one of them and not one of them |
| 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) |
| §2 | p.27 | 13367 | The next set up in the hierarchy of sets seems to be both a member and not a member of it |
| 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) |
| §3 | p.28 | 13368 | The 'least indefinable ordinal' is defined by that very phrase |
| 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. |
| §3 | p.29 | 13370 | 'x is a natural number definable in less than 19 words' leads to contradiction |
| 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] |
| §3 | p.29 | 13369 | By diagonalization we can define a real number that isn't in the definable set of reals |
| 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] |
| §4 | p.30 | 13371 | If you know that a sentence is not one of the known sentences, you know its truth |
| 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] |
| §4 | p.30 | 13372 | There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar |
| 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. |
| §5 | p.32 | 13373 | Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong |
| 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. |
| 1998 | What is so bad about Contradictions? |
| p.73 | 9123 | Someone standing in a doorway seems to be both in and not-in the room | |
| Full Idea: Priest says there is room for contradictions. He gives the example of someone in a doorway; is he in or out of the room. Given that in and out are mutually exclusive and exhaustive, and neither is the default, he seems to be both in and not in. | |||
| From: report of Graham Priest (What is so bad about Contradictions? [1998]) by Roy Sorensen - Vagueness and Contradiction 4.3 | |||
| A reaction: Priest is a clever lad, but I don't think I can go with this. It just seems to be an equivocation on the word 'in' when applied to rooms. First tell me the criteria for being 'in' a room. What is the proposition expressed in 'he is in the room'? |
| 1998 | works |
| p.160 | 8720 | A logic is 'relevant' if premise and conclusion are connected, and 'paraconsistent' allows contradictions | |
| Full Idea: Priest and Routley have developed paraconsistent relevant logic. 'Relevant' logics insist on there being some sort of connection between the premises and the conclusion of an argument. 'Paraconsistent' logics allow contradictions. | |||
| From: report of Graham Priest (works [1998]) by Michčle Friend - Introducing the Philosophy of Mathematics 6.8 | |||
| A reaction: Relevance blocks the move of saying that a falsehood implies everything, which sounds good. The offer of paraconsistency is very wicked indeed, and they are very naughty boys for even suggesting it. |
| 2001 | Intro to Non-Classical Logic (1st ed) |
| Pref | p.-9 | 9672 | Free logic is one of the few first-order non-classical logics |
| Full Idea: Free logic is an unusual example of a non-classical logic which is first-order. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], Pref) |
| 0.1.0 | p.-5 | 9697 | X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets |
| Full Idea: X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets, the set of all the n-tuples with its first member in X1, its second in X2, and so on. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.0) |
| 0.1.10 | p.-6 | 9685 | <a,b&62; is a set whose members occur in the order shown |
| Full Idea: <a,b> is a set whose members occur in the order shown; <x1,x2,x3, ..xn> is an 'n-tuple' ordered set. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10) |
| 0.1.10 | p.-6 | 9695 | An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order |
| Full Idea: An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10) |
| 0.1.10 | p.-5 | 9696 | A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets |
| Full Idea: A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10) |
| 0.1.2 | p.-7 | 9686 | A 'set' is a collection of objects |
| Full Idea: A 'set' is a collection of objects. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2) |
| 0.1.2 | p.-7 | 9687 | A 'member' of a set is one of the objects in the set |
| Full Idea: A 'member' of a set is one of the objects in the set. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2) |
| 0.1.2 | p.-7 | 9674 | {x; A(x)} is a set of objects satisfying the condition A(x) |
| Full Idea: {x; A(x)} indicates a set of objects which satisfy the condition A(x). | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2) |
| 0.1.2 | p.-7 | 9673 | {a1, a2, ...an} indicates that a set comprising just those objects |
| Full Idea: {a1, a2, ...an} indicates that the set comprises of just those objects. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2) |
| 0.1.2 | p.-7 | 9675 | a ∈ X says a is an object in set X; a ∉ X says a is not in X |
| Full Idea: a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X indicates that a is not in X. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2) |
| 0.1.4 | p.-7 | 9677 | Φ indicates the empty set, which has no members |
| Full Idea: Φ indicates the empty set, which has no members | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4) |
| 0.1.4 | p.-7 | 9676 | {a} is the 'singleton' set of a (not the object a itself) |
| Full Idea: {a} is the 'singleton' set of a, not to be confused with the object a itself. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4) |
| 0.1.4 | p.-7 | 9688 | A 'singleton' is a set with only one member |
| Full Idea: A 'singleton' is a set with only one member. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4) |
| 0.1.4 | p.-7 | 9689 | The 'empty set' or 'null set' has no members |
| Full Idea: The 'empty set' or 'null set' is a set with no members. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4) |
| 0.1.6 | p.-6 | 9690 | A set is a 'subset' of another set if all of its members are in that set |
| Full Idea: A set is a 'subset' of another set if all of its members are in that set. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.6 | p.-6 | 9691 | A 'proper subset' is smaller than the containing set |
| Full Idea: A set is a 'proper subset' of another set if some things in the large set are not in the smaller set | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.6 | p.-6 | 9678 | X⊆Y means set X is a 'subset' of set Y |
| Full Idea: X⊆Y means set X is a 'subset' of set Y (if and only if all of its members are members of Y). | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.6 | p.-6 | 9679 | X⊂Y means set X is a 'proper subset' of set Y |
| Full Idea: X⊂Y means set X is a 'proper subset' of set Y (if and only if all of its members are members of Y, but some things in Y are not in X) | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.6 | p.-6 | 9681 | X = Y means the set X equals the set Y |
| Full Idea: X = Y means the set X equals the set Y, which means they have the same members (i.e. X⊆Y and Y⊆X). | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.6 | p.-6 | 9680 | The empty set Φ is a subset of every set (including itself) |
| Full Idea: The empty set Φ is a subset of every set (including itself). | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |
| 0.1.8 | p.-6 | 9683 | X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets |
| Full Idea: X ∩ Y indicates the 'intersection' of sets X and Y, which is a set containing just those things that are in both X and Y. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.1.8 | p.-6 | 9684 | Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X |
| Full Idea: Y - X indicates the 'relative complement' of X with respect to Y, that is, all the things in Y that are not in X. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.1.8 | p.-6 | 9682 | X∪Y indicates the 'union' of all the things in sets X and Y |
| Full Idea: X ∪ Y indicates the 'union' of sets X and Y, which is a set containing just those things that are in X or Y (or both). | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.1.8 | p.-6 | 9692 | The 'union' of two sets is a set containing all the things in either of the sets |
| Full Idea: The 'union' of two sets is a set containing all the things in either of the sets | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.1.8 | p.-6 | 9693 | The 'intersection' of two sets is a set of the things that are in both sets |
| Full Idea: The 'intersection' of two sets is a set containing the things that are in both sets. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.1.8 | p.-6 | 9694 | The 'relative complement' is things in the second set not in the first |
| Full Idea: The 'relative complement' of one set with respect to another is the things in the second set that aren't in the first. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8) |
| 0.2 | p.-5 | 9698 | The 'induction clause' says complex formulas retain the properties of their basic formulas |
| Full Idea: The 'induction clause' says that whenever one constructs more complex formulas out of formulas that have the property P, the resulting formulas will also have that property. | |||
| From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.2) |