Ideas from 'Intro to Non-Classical Logic (1st ed)' by Graham Priest [2001], by Theme Structure

[found in 'Introduction to Non-Classical Logic' by Priest,Graham [CUP 2001,0-521-79434-x]].

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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
Φ indicates the empty set, which has no members
{a} is the 'singleton' set of a (not the object a itself)
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,} indicates that a set comprising just those objects
{x; A(x)} is a set of objects satisfying the condition A(x)
a ∈ X says a is an object in set X; a ∉ X says a is not in X
X⊆Y means set X is a 'subset' of set Y
X = Y means the set X equals the set Y
X⊂Y means set X is a 'proper subset' of 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 'union' of all the things in sets X and Y
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets
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
The 'empty set' or 'null set' has no members
A 'singleton' is a set with only one member
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 'relative complement' is things in the second set not in the first
The 'union' of two sets is a set containing all the things in either of the sets
The 'intersection' of two sets is a set of the things that are in both sets
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)