more on this theme
|
more from this text
Single Idea 9698
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
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.
Gist of Idea
The 'induction clause' says complex formulas retain the properties of their basic formulas
Source
Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.2)
Book Ref
Priest,Graham: 'Introduction to Non-Classical Logic' [CUP 2001], p.-5
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]
|