## 9 Apr 2016

### Suppes (9.6) Introduction to Logic, “Domains and Individuals”, summary

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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

§9.6 Domains and Individuals

Brief summary:
A domain of individuals (also called a domain of discourse) is a specific set. For example, in sociology, if we speak of the set of albinos, we implicitly mean only those albinos within the specific set (within the domain) of humans. We use the symbol “V” to denote a domain. Suppose we have a domain V and a set A. The complement of A relative to the domain are all those items in the domain that are not in A. We symbolize it either as V∼A or just ∼A.

Summary

Suppose we are working in the field of sociology. And suppose further that we speak of “the set of albinos”. Now, since we are dealing in sociology, we of course do not mean to refer to the set of all albino monkeys, of all albino mice, and of all other albino animals. Rather, we just mean the set of all albino humans.

Often one is interested, not in all possible sets, but merely in all the subsets of some fixed set. Thus in sociology, for instance, it might be natural to be talking mostly about sets of human beings; and to speak with the understanding that when a set was mentioned it was to be taken to be a set of people, unless an explicit statement to the contrary was made.
(187)

Or in geometry when we say “set” we might really more specifically mean “set of points” (187).

[So we are limiting our set to a particular limited one, which is our domain, since we are excluding things outside it.]

When a fixed set D is taken as given in this way, and one confines himself to the discussion of subsets of D, we shall call D the domain of individ­uals, or sometimes the domain of discourse. Thus the domain of individuals of the sociological discussion mentioned above is the set of all human beings.
(187)

We will use the symbol ‘V’ to stand for the domain of individuals. Its contents are different depending on how it is defined in its particular context (187).

[When we are working with domains, we might want to know what is in that domain V which is not in some other domain A. We call this set the complement of A.]

When dealing with a fixed domain of individuals V, it is convenient to introduce a special symbol for the difference of V and a set A:

A = V∼A.

We call ∼A the complement of A. More generally, the difference BA of B and A is called the complement of A relative to B; so the complement of a set is simply its complement relative to the given domain of individuals.
(188)

[We can also find complements of sets that we have already performed other operations upon.]

As should be expected, we may find the complement of a set which itself results from operations on other sets. For example, let

V = {1, 2, 3}
A = {1, 2}
B = {2, 3}.

Then

A = V∼A = {1, 2, 3}∼{1, 2} = {3},

and correspondingly,

∼(A B) = V∼(A B) = {1, 2, 3}∼({1, 2} ∪ {2, 3})
= {1, 2, 3}∼{1, 2, 3} = Λ.
(Suppes 188)

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

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