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Single Idea 13371

[filed under theme 5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox ]

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.

Gist of Idea

If you know that a sentence is not one of the known sentences, you know its truth

Source

Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4)

Book Ref

-: 'Mind' [-], p.30

A Reaction

[mostly my wording]

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]