Priest (0.1) An Introduction to Non-Classical Logic, ‘Set-theoretic Notation’, summary

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Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

Ch.0 Mathematical Prolegomenon

0.1 Set-theoretic Notation

Brief Summary:

The following are definitions for basic set-theoretical notions that will appear throughout this book. [The following is quotation from pp.xxvii-xxix.]

set

A set, X, is a collection of objects. If the set comprises the objects a₁, ... , a_n, this may be written as {a₁, ... , a_n}. If it is the set of objects satisfying some condition, A(x), then it may be written as {x :A(x)}.

membership

a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X means that a is not a member of X.

singleton

for any a, there is a set whose only member is a, written {a}. {a} is called a singleton (and is not to be confused with a itself).

empty set

There is also a set which has no members, the empty set; this is written as φ.

subset

A set, X, is a subset of a set, Y, if and only if every member of X is a member of Y. This is written as X ⊆ Y. The empty set is a subset of every set (including itself).

proper subset

X ⊂ Y means that X is a proper subset of Y; that is, everything in X is in Y, but there are some things in Y that are not in X. X and Y are identical sets, X = Y, if they have the same members, i.e., if X ⊆ Y and Y ⊆ X. Hence, if X and Y are not identical, X ≠ Y, either there are some members of X that are not in Y, or vice versa (or both).

union

The union of two sets, X, Y, is the set containing just those things that are in X or Y (or both). This is written as X ∪ Y. So a ∈ X ∪ Y if and only if a ∈ X or a ∈ Y.

intersection

The intersection of two sets, X, Y, is the set containing just those things that are in both X and Y. It is written X ∩ Y. So a ∈ X ∩ Y if and only if a ∈ X and a ∈ Y.

relative complement

The relative complement of one set, X, with respect to another, Y, is the set of all things in Y but not in X. It is written Y − X. Thus, a ∈ Y − X if and only if a ∈ Y but a ∉ X.

ordered pair

An ordered pair, ⟨a, b⟩, is a set whose members occur in the order shown, so that we know which is the first and which is the second. Similarly for an ordered triple, ⟨a, b, c⟩, quadruple, ⟨a, b, c, d⟩, and, in general, n-tuple, ⟨x₁, . . . , x_n⟩.

cartesian product

Given n sets X₁, . . . , X_n, their cartesian product, X₁×· · ·×X_n, is the set of all n-tuples, the first member of which is in X₁, the second of which is in X₂, etc. Thus, ⟨x₁, . . . , x_n⟩ ∈ X₁×· · ·×X_n if and only if x₁ ∈ X1 and . . . and x_n ∈ X_n.

subset relation

A relation, R, between X₁×· · ·×X_n is any subset of X₁×· · ·×X_n. | ⟨x₁, . . . , x_n⟩ ∈ R is usually written as Rx₁ . . . x_n.

ternary and binary relations

If n is 3, the relation is a ternary relation. If n is 2, the relation is a binary relation, and Rx₁ x₂ is usually written as x₁Rx₂.

function

A function from X to Y is a binary relation, f , between X and Y, such that for all x ∈ X there is a unique y ∈ Y such that xfy. More usually, in this case, we write: f(x) = y.

(Priest xxvii-xxix)

Summary

Ch.0 Mathematical Prolegomenon

As we will be working in modern logic, we cannot avoid mathematical notations and concepts. Priest will first give three mathematical discussions on: {1} a simple set theoretical notation and its meaning, {2} proof by induction, and {3} equivalence relations and equivalence classes (Priest xxvii).

0.1 Set-theoretic Notation

0.1.1 [intro]

In the following Priest will give a brief explanation of the set-theoretical notation that at times we will encounter later in the text.

0.1.2

[set and membership]

A set [here notated with an italicized capital letter] contains objects [written as lowercase subscripted italicized letters, placed in series between brackets, separated by commas.] The members of the set may be specified either by listing the members explicitly or by stating a condition they all satisfy. When one object is included in the set, it is a member, and it is not a member if it is not included. (See Suppes Intro section 9.2 and Agler Symbolic Logic section 6.4.)

A set, X, is a collection of objects. If the set comprises the objects a₁, ... , a_n, this may be written as {a₁, ... , a_n}. If it is the set of objects satisfying some condition, A(x), then it may be written as {x :A(x)}. a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X means that a is not a member of X.

(Priest xxii)

0.1.3

[examples of sets]

Priest gives these examples of sets. [The first set gives us the set of natural numbers less than 5. The second gives us the set of even natural numbers. The number three is included in the first set but not in the second.]

Examples: The set of (natural) numbers less than 5 is {0, 1, 2, 3, 4}. Call this F. The set of even numbers is {x :x is an even natural number}. Call this E. Then 3 ∈ F, and 5 ∉ E.

(Priest xxvii)

0.1.4

[quantity of set membership]

A set can have any number of members, from none, to one, to infinitely many. If there is one, it is a singleton (see this wikibook entry for singleton, doubleton, etc.). And if there are no members, it is the empty set, φ. (See Suppes Intro section 9.4).

for any a, there is a set whose only member is a, written {a}. {a} is called a singleton (and is not to be confused with a itself). There is also a set which has no members, the empty set; this is written as φ.

(Priest xxvii)

0.1.5

[example of singleton and empty set]

Priest gives the example of the set {3}. It has just one member, the number three. The number three itself is not a set so it has no members. [Nonetheless, it is still something rather than nothing, so the number three is not included in the empty set. I might be wrong in how I interpreted the significance of the final sentence. Let me quote.]

Examples: {3} is the set containing just the number three. It has one member. It is distinct from 3, which is a number, not a set at all, and so has no members. 3 ∉ φ.

(Priest xxviii)

0.1.6

[subset and proper subset]

A set is a subset of another only if all members of the first are members of the second.  We should note that the empty set is actually included in every other set, and it is even included within itself. A set is a proper subset of another only if all the members of the first are included in the second, but not all the second’s are included in the first. Sets are identical when the have exactly the same members. (See Suppes Intro section 9.3.)

A set, X, is a subset of a set, Y, if and only if every member of X is a member of Y. This is written as X ⊆ Y. The empty set is a subset of every set (including itself). X ⊂ Y means that X is a proper subset of Y; that is, everything in X is in Y, but there are some things in Y that are not in X. X and Y are identical sets, X = Y, if they have the same members, i.e., if X ⊆ Y and Y ⊆ X. Hence, if X and Y are not identical, X ≠ Y, either there are some members of X that are not in Y, or vice versa (or both).

(Priest xxviii)

1.1.7

[examples of subsets and proper subsets]

[All even numbers are included in the set of all natural numbers. So the even numbers are a subset of the natural numbers. But since there are natural numbers (namely the odds) which are not included among the evens, that means the even numbers are a proper subset of the natural numbers.]

Examples: Let N be the set of all natural numbers, and E be the set of even numbers. Then φ ⊆ N and E ⊆ N. Also, E ⊂ N, since 5 ∈ N but 5 ∉ E. If X ⊆ N and X ≠ E then either some odd number is in X, or some even number is not in X (or both).

(Priest xxviii)

0.1.8

[union and intersection of sets]

The union of two sets is the set containing whatever is found in either or in both the first two sets. The intersection of two sets is the set of whatever is shared by both the first two. The relative complement of a first set to a second is the set containing whatever is in the second but not in the first. (See Suppes Intro section 9.5 and section 9.6.)

The union of two sets, X, Y, is the set containing just those things that are in X or Y (or both). This is written as X ∪ Y. So a ∈ X ∪ Y if and only if a ∈ X or a ∈ Y. The intersection of two sets, X, Y, is the set containing just those things that are in both X and Y. It is written X ∩ Y. So a ∈ X ∩ Y if and only if a ∈ X and a ∈ Y. The relative complement of one set, X, with respect to another, Y, is the set of all things in Y but not in X. It is written Y − X. Thus, a ∈ Y − X if and only if a ∈ Y but a ∉ X.

(xxviii)

0.1.9

[examples of union, intersection, and complement]

If we unite all even numbers and all odd numbers, we get all natural numbers. If we look for which numbers are both even and odd, we find there are none. And if we wanted to know which numbers are even but not greater or equal to ten, we would get all those below ten.

Examples: Let N, E and O be the set of all numbers, all even numbers, and all odd numbers, respectively. Then E ∪ O = N, E ∩ O = φ. Let T = {x : x ≥ 10}. Then E − T = {0, 2, 4, 6, 8}.

(Priest xxviii)

0.1.10

[Ordered n-tuples, cartesian products, relations, and functions]

A set whose order of its members is determinative of it is an ordered n-tuple, with the n being the number of its members. The members of ordered n-tuples are placed within angle brackets and separated by commas (See Suppes Intro section 10.1 and Agler Symbolic Logic section 6.4.2.) The cartesian product of different sets forms a set of all n-tuples that themselves are formed in such a way that the first member is taken from the first set, the second member from the second, and so on (See Suppes Intro section 10.1.) A relation between members of sets is understood as being some subset of their cartesian product, and it can be written by putting the symbol for the relation first and the terms from each set listed after it (see Suppes Intro section 10.2). When there are two terms it is a binary relation, and three it is ternary. Binary relations of the form Rxy are often written xRy. A function is a binary relation that relates a unique member of a second set to a first set. And we often write functions in the form: f(x)=y. (see Suppes Intro section 11.1.)

An ordered pair, ⟨a, b⟩, is a set whose members occur in the order shown, so that we know which is the first and which is the second. Similarly for an ordered triple, ⟨a, b, c⟩, quadruple, ⟨a, b, c, d⟩, and, in general, n-tuple, ⟨x₁, . . . , x_n⟩. Given n sets X₁, . . . , X_n, their cartesian product, X₁×· · ·×X_n, is the set of all n-tuples, the first member of which is in X₁, the second of which is in X₂, etc. Thus, ⟨x₁, . . . , x_n⟩ ∈ X₁×· · ·×X_n if and only if x₁ ∈ X1 and . . . and x_n ∈ X_n. A relation, R, between X₁×· · ·×X_n is any subset of X₁×· · ·×X_n. | ⟨x₁, . . . , x_n⟩ ∈ R is usually written as Rx₁ . . . x_n. If n is 3, the relation is a ternary relation. If n is 2, the relation is a binary relation, and Rx₁ x₂ is usually written as x₁Rx₂. A function from X to Y is a binary relation, f , between X and Y, such that for all x ∈ X there is a unique y ∈ Y such that xfy. More usually, in this case, we write: f(x) = y.

(Priest xxviii-xxix)

0.1.11

[Examples of ordered n-tuples, cartesian products, relations, and functions]

Priest first shows the determining role of order in ordered n-tuples by giving the example: ⟨2, 3⟩ ≠ ⟨3, 2⟩. He then has N = all numbers. So now we think of a relation that takes a member from N and relates it to another member in N. This double application of the domain is written N × N. We consider the first member being related as n and the second as m. The relation places them into the ordered couple ⟨n,m⟩. We take as our example the relation R consisting of the set {⟨2, 3⟩, ⟨3, 2⟩}. Since the first term of each couple is from N and the second is also from N, we can write: R ⊆ N × N. We can also say it is a binary relation between N and itself. Now we suppose that we have a binary relation that is a function, f. Its domain is N. It forms ordered couples where the first term is a member of N and the second is that same number squared. We could write this as f(n) = n².

Examples: ⟨2, 3⟩ ≠ ⟨3, 2⟩, since these sets have the same members, but in a different order. Let N be the set of numbers. Then N × N is the set of all pairs of the form ⟨n,m⟩, where n and m are in N. If R = {⟨2, 3⟩, ⟨3, 2⟩} then R ⊆ N × N and is a binary relation between N and itself. If f = {⟨n, n²⟩ : n ∈ N}, then f is a function from numbers to numbers, and f(n) = n².

(Priest xxix)

Priest, Graham. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University, 2001, 2008.