Suppes (9.1-9.2) Introduction to Logic, “Introduction”, "Membership", summary

by Corry Shores

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[The following is summary. My commentary is in brackets. Boldface is mine.]

 

Summary of

 

Patrick Suppes

 

Introduction to Logic

 

Ch. 9 Sets

 

§9.1 Introduction
and
§9.2 Membership

 

 

Brief summary:
A set is any kind of a collection of entities of any sort, and the members are said to “belong to” the set. This membership relation is symbolized ∈, and the set members are listed between braces, {}. If two sets have the same members, then those two sets are identical. This is the principle of extensionality for sets. There is an empty set, which has no members, and it is here symbolized Λ. Set {{1,2}} is not identical with set {1,2}, because the first set has one member, namely, {1,2}, and the second has these two members: 1,2. Most sets do not include themselves as members. And the membership relation is not symmetric, so from A ∈ B it does not follow that B ∈ A. The order of the members does not matter, so {1,3,5}={1,5,3}. And we do not count an element of a set twice, so {1,1,3,5}={1,3,5}. Also, the relation of set membership is not transitive. So from A∈B and B∈C, it does not follow that A∈C. For example, 2∈{1,2} and {1,2}∈{{1,2},{3,4}}. However, 2∉{{1,2},{3,4}}. We can understand properties in terms of set membership by saying that a thing has a given property if and only if it belongs to the set of things having this property. And finally, we can express the principle of the identity of indiscernibles in terms of set membership by saying that if y belongs to every set to which x belongs, then y=x.

 

 

 

Summary

 

§9.1 Introduction

 

Here Suppes will introduce set theory intuitively rather than axiomatically (177).

 

 

§9.2 Membership

 

A set is “any kind of a collection of entities of any sort” (177). Other terms for “set” are “class,” “collection,” and “aggregate”, which may be used interchangeably (177).

 

A set’s members belong to the set, and we use a modified Greek letter epsilon ∈ to mean “belongs to”. So instead of

Elizabeth II belongs to the class of women

We could write

Elizabeth II ∈ the class of women
(177d)

In ordinary language this sentence would be formulated

Elizabeth II is a woman.
(178)

“Thus the verb ‘to be’ often has the meaning of set membership” (178).

 

“[A] set is completely determined when its members are given” (178). So “A and B are sets which have exactly the same members, then A = B” (178). [If a triangle has sides all of the same length, then all its angles will each be 60 degrees.] Thus:

The set of equilateral triangles = the set of equiangular triangles,

for something belongs to the first set if and only if it belongs to the second, since a triangle is equilateral if and only if it is equiangular.
(178)

So we see then there there is a principle of identity for sets when they have the same members. It is normally called the principle of extensionality for sets. We may formulate it symbolically as:

A = B ↔ (x)(x ∈ A ↔ x ∈ B)
(Suppes 178)

[So set A is identical with set B if and only if for all x, x belongs to A if and only if x belongs to B. (or, if and only if all members of A are members of B and vice versa).]

 

We will want to speak of sets that might not even have any members, which are called empty sets (178).

 

No consider this formula:

(x)(x ∈ A → x ∈ B)
(Suppes 178)

Suppose that A is an empty set. That means the antecedent is false, and thus the whole statement is true (178). Suppose now that B is empty. That means this sentence is true too:

(x)(x ∈ B → x ∈ A)
(178)

Recall the principle of extensionality for sets:

A = B ↔ (x)(x ∈ A ↔ x ∈ B)

So if

(x)(x ∈ A → x ∈ B)

and

(x)(x ∈ B → x ∈ A)

are true for empty sets A and B, then they must be identical, in accordance with the principle of extensionality for sets. This means that there can only be one empty set, which we notate with the Greek capital letter lambda Λ. Since it is the empty set, it is the one “set such that for every x, x does not belong to Λ”, which can be symbolically rendered:

(x)–(x ∈ Λ).

The part

–(x ∈ Λ)

We can abbreviate as

x ∉ Λ

So we can write the formulation for the empty set as:

(x)(x ∉ Λ).

We often use the symbol ∉ to mean that something does not belong to some set (179).

 

We can “describe a set by writing down names of its members, separated by commas, and enclosing the whole with braces” (179). Thus we can write:

{Roosevelt, Parker}

for “the set consisting of the two major candidates in the 1904 American Presidential election” (179). Or for “the set consisting of the first three odd positive integers” we can write:

{1, 3, 5}

[Recall that sets are equal when they have the same members, which was the principle of extensionality.] We can see that because they have the same members:

{1, 3, 5} = {1, 5, 3}
(Suppes 179)

Also, “we do not count an element of a set twice” (179), thus:

{1, 1, 3, 5} = {1, 3, 5}

 

We also note that the “members of a set can themselves be sets. Thus a political party can be conceived as a certain set of people, and it may be convenient to speak of the set of political parties in a given country” (179). We can also have a set made of other sets that are comprised of integers, as in:

{ {1, 2}, {3, 4}, {5, 6} }
(Suppes 179)

This set has just three members, namely:

{1, 2}
{3, 4}
{5, 6}

And the set:

{ {1, 2}, {2, 3} }

has the two members:

{1, 2}
{2, 3}

 

Now,

A set having just one member is not to be considered identical with that member. Thus the set

{ {1, 2} }

is not identical with the set

{1, 2};

this is clear from the fact that {1, 2} has two members, whereas

{ {1, 2} }

has just | one member (namely, {1, 2}
(179-180).

For the same reason,

{Elizabeth II} ≠ Elizabeth II

“for Elizabeth II is a woman, while {Elizabeth II} is a set” (180).

 

In most cases, sets are not members of themselves. So “a set of chairs is not a member of the set of chairs: i.e., the set of chairs is not itself a chair” (180). Here we see an important difference between identity and set membership:

A = A

is always true. However,

A ∈ A

is in most cases false (180).

 

[Identity is symmetric, perhaps because A = B is the same as B = A.] Unlike the identity relation, the relation of set membership is not symmetric: “from A ∈ B it does not follow that B ∈ A. So while:

2 ∈ {1, 2}

it does not follow from this that {1, 2} is included in 2, so:

{1, 2} ∉ 2

Another interesting [and perhaps surprising] thing to note is that

the relation of membership is not transitive: from A ∈ B and B ∈ C it does not follow that A ∈ C. Thus, for example, we have:

2 ∈ {1, 2}

and:

{1, 2} ∈ { {1, 2} , {3, 4} }

but:

2 ∉ { {1, 2} , {3, 4} }

for the only members of { {1, 2} , {3, 4} } are {1, 2} and {3, 4}, and neither of these sets is identical with 2.
(Suppes 180)

 

Suppes further explains this idea by noting the following. We suppose that {a, b} is “any set with two members” (180). This means that

for every x, x ∈ {a, b} if and only if either x = a or x = b, that is symbolically:

(x)(x ∈ {a, b} ↔ (x = a ∨ x = b)).

Similarly, if {a, b, c} is a set with three members, then x ∈ {a, b, c} if and only if either x = a or x = b or x = c. It is for this reason that we just said that  2 ∉ { {1, 2} , {3, 4} } ; for if x ∈ { {1, 2} , {3, 4} }, then either x = {1, 2} or x = {3, 4} ; and since 2 ≠ {1, 2} and 2 ≠ {3, 4} , it follows that  2 ∉ { {1, 2} , {3, 4} }.
(Suppes 180)

 

Suppes then notes that “there is a close relationship between saying that something has a property and saying that it belongs to a set” (180d). For, “a thing | has a given property if and only if it belongs to the set of things having the property” (180-181). He gives an illustration. When we say that “6 has the property of being an even number,” that  “amounts to saying that 6 belongs to the set of even numbers” (181). And in fact, “Since we can always in this way express things in terms of membership in sets instead of in terms of the possession of properties, we do not find it necessary to give any more detailed discussion of properties” (181).

 

Previously in §5.1, Suppes discussed the principle of the identity of indiscernibles in terms of properties (181). Now he expresses the principle of the identity of indiscernibles in terms of set membership:

If y belongs to every set to which x belongs, then y = x. Put in this form, the principle has perhaps a more obvious character than it has when put in terms of properties. For x ∈ {x} (i.e., x belongs to the set whose only member is x), and hence, if y belongs to every set to which x belongs, we conclude that y ∈ {x}, so that y = x.
(181)

 

 

 

 

Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.

 

 

 

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