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Single Idea 13373
[filed under theme 5. Theory of Logic / L. Paradox / 1. Paradox
]
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.
Gist of Idea
Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong
Source
Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §5)
Book Ref
-: 'Mind' [-], p.32
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.
Related Idea
Idea 13334
Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey]
The
37 ideas
from Graham Priest
9123
|
Someone standing in a doorway seems to be both in and not-in the room
[Priest,G, by Sorensen]
|
9672
|
Free logic is one of the few first-order non-classical logics
[Priest,G]
|
9697
|
X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets
[Priest,G]
|
9685
|
<a,b&62; is a set whose members occur in the order shown
[Priest,G]
|
9695
|
An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order
[Priest,G]
|
9696
|
A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets
[Priest,G]
|
9686
|
A 'set' is a collection of objects
[Priest,G]
|
9687
|
A 'member' of a set is one of the objects in the set
[Priest,G]
|
9675
|
a ∈ X says a is an object in set X; a ∉ X says a is not in X
[Priest,G]
|
9674
|
{x; A(x)} is a set of objects satisfying the condition A(x)
[Priest,G]
|
9673
|
{a1, a2, ...an} indicates that a set comprising just those objects
[Priest,G]
|
9677
|
Φ indicates the empty set, which has no members
[Priest,G]
|
9676
|
{a} is the 'singleton' set of a (not the object a itself)
[Priest,G]
|
9688
|
A 'singleton' is a set with only one member
[Priest,G]
|
9689
|
The 'empty set' or 'null set' has no members
[Priest,G]
|
9690
|
A set is a 'subset' of another set if all of its members are in that set
[Priest,G]
|
9691
|
A 'proper subset' is smaller than the containing set
[Priest,G]
|
9680
|
The empty set Φ is a subset of every set (including itself)
[Priest,G]
|
9679
|
X⊂Y means set X is a 'proper subset' of set Y
[Priest,G]
|
9678
|
X⊆Y means set X is a 'subset' of set Y
[Priest,G]
|
9681
|
X = Y means the set X equals the set Y
[Priest,G]
|
9683
|
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets
[Priest,G]
|
9682
|
X∪Y indicates the 'union' of all the things in sets X and Y
[Priest,G]
|
9684
|
Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X
[Priest,G]
|
9694
|
The 'relative complement' is things in the second set not in the first
[Priest,G]
|
9693
|
The 'intersection' of two sets is a set of the things that are in both sets
[Priest,G]
|
9692
|
The 'union' of two sets is a set containing all the things in either of the sets
[Priest,G]
|
9698
|
The 'induction clause' says complex formulas retain the properties of their basic formulas
[Priest,G]
|
13366
|
The least ordinal greater than the set of all ordinals is both one of them and not one of them
[Priest,G]
|
13367
|
The next set up in the hierarchy of sets seems to be both a member and not a member of it
[Priest,G]
|
13368
|
The 'least indefinable ordinal' is defined by that very phrase
[Priest,G]
|
13370
|
'x is a natural number definable in less than 19 words' leads to contradiction
[Priest,G]
|
13369
|
By diagonalization we can define a real number that isn't in the definable set of reals
[Priest,G]
|
13371
|
If you know that a sentence is not one of the known sentences, you know its truth
[Priest,G]
|
13372
|
There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar
[Priest,G]
|
13373
|
Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong
[Priest,G]
|
8720
|
A logic is 'relevant' if premise and conclusion are connected, and 'paraconsistent' allows contradictions
[Priest,G, by Friend]
|