Ideas of Graham Priest, by Theme

[British, fl. 2000, At Queensland University, then Professor at the University of Melbourne, and St Andrew's University.]

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2. Reason / B. Laws of Thought / 3. Non-Contradiction
Someone standing in a doorway seems to be both in and not-in the room [Sorensen]
4. Formal Logic / E. Nonclassical Logics / 5. Relevant Logic
A logic is 'relevant' if premise and conclusion are connected, and 'paraconsistent' allows contradictions [Friend]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic is one of the few first-order non-classical logics
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
{a} is the 'singleton' set of a (not the object a itself)
a ∈ X says a is an object in set X; a ∉ X says a is not in X
{x; A(x)} is a set of objects satisfying the condition A(x)
Φ indicates the empty set, which has no members
X⊂Y means set X is a 'proper subset' of set Y
X⊆Y means set X is a 'subset' of set Y
X = Y means the set X equals the set Y
Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets
X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets
<a,b&62; is a set whose members occur in the order shown
{a1, a2, ...an} indicates that a set comprising just those objects
X∪Y indicates the 'union' of all the things in sets X and Y
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order
A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets
A 'member' of a set is one of the objects in the set
A 'set' is a collection of objects
A 'singleton' is a set with only one member
The 'empty set' or 'null set' has no members
A set is a 'subset' of another set if all of its members are in that set
A 'proper subset' is smaller than the containing set
The 'intersection' of two sets is a set of the things that are in both sets
The 'union' of two sets is a set containing all the things in either of the sets
The 'relative complement' is things in the second set not in the first
The 'induction clause' says complex formulas retain the properties of their basic formulas
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
The empty set Φ is a subset of every set (including itself)
5. Theory of Logic / L. Paradox / 1. Paradox
Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / b. König's paradox
The 'least indefinable ordinal' is defined by that very phrase
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
'x is a natural number definable in less than 19 words' leads to contradiction
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / d. Richard's paradox
By diagonalization we can define a real number that isn't in the definable set of reals
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
The least ordinal greater than the set of all ordinals is both one of them and not one of them
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
The next set up in the hierarchy of sets seems to be both a member and not a member of it
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
If you know that a sentence is not one of the known sentences, you know its truth
There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar