Text 2 the logic of relations

The part of logic which we shall mention here in addition to the algebra of classes is the logic of relations. The idea of "relation" is an easy one to grasp.

What is meant by a relation is best explained with a few illustrations. For exam­ple, "being taller than", "being father of', "being equal to", "being greater than" are relations in which things of various kinds may stand to one another. The first two relations mentioned above may hold between people, while the last two may hold between numbers.

If we say that "George is older than Paul", that "3 is equal to ⁶/₂, that "New York is east of Chicago" and that " 12 is a multiple of 3", we are expressing rela­tions between people, cities and numbers.

Some propositions which contain two or more proper names (or individuals) are correctly interpreted as truth-functional compounds of singular propositions having different subject terms. For example, the proposition "Lincoln and Grant were presidents" is properly interpreted as the conjunction of the two singular propositions "Lincoln was a president and Grant was a president". Relations which hold between two individuals are called "binary" or "dyadic". Other rela­tions may relate three or more individuals or objects. If we say that "Chicago lies between Denver and New York" we are expressing a relation between three things and it is said to be a "ternary" or triadic relation, while "quaternary" or tetradic relations are expressed by the proposition "Nick, Pete, Charlie and John played bridge together".

There are a number of interesting properties that relations themselves may possess. We shall consider only a few of more familiar ones, and our discussion will be confined to properties of binary or dyadic relations.

Symbolically, we shall represent relations with capital letters such as R, S, and T, and if R relates two objects a and b (in that order), we shall express this by writing aRb. For example: "Harry is the brother of John" can be written as aRb, where a stands for Harry, b for John, and R for "is the brother of ”. Binary rela­tions may be characterized as symmetric and asymmetric or non-symmetric. Var­ious symmetric relations are designated by the phrases: "is next to", "is married to", "has the same weight as".

Consider an arbitrary relation R and let x, y, z etc. represent things which may stand in their relations to each other. If x has the relation R to y, we may write xRy. It may happen that whenever x has the relation R to x, then y must also have the relation R to x. If this is the case, the relation R is called symmetric. Thus, "is equal to" for number is a symmetric relation since if a=b, then b=a.

Similarly, "is the cousin of, "is as tall as", "is a co-worker of, "lives in the same house of are symmetric relations, as, if "Henry is a cousin of George", it follows necessarily that George is also a cousin of Henry.

On the other hand, "is older than", "is the brother of, "is smarter than" are asymmetric relations, as, if "Bob is older than John", then John is younger (and not older) than Bob, and "Jim is the brother of Mary" does not imply that "Mary is the brother of Jim". Hence, "is the brother of” is not symmetric. A relation is symmetric only if aRb implies bRa for all a and b to which the relation applies.

An asymmetric relation is one such that if one individual has that relation to a second individual, then the latter cannot have that relation to the former. Var­ious asymmetric relations are designated by the phrases: "is north of, "is parent of, "weighs more than".

A relation R is said to be reflexive if for every a to which R applies it is true that aRa, namely, that a has relation R with itself.

For instance, "is as intelligent as" is reflexive since every person (or animal) is as intelligent as himself. Other reflexive relations are "is as old as", "is as tall as", "is equal to", "is as rich as" etc. Among the many relations that are not reflexive we find, for example, "is heavier than", "is the father of” and "is smarter than".

It may happen that if x has the relation R to y, and y has the relation R to z, then x must have the relation R to z. If this is the case, the relation R is called transitive. Thus, a relation R is said to be transitive if for every a, b, and c to which R applies, aRb and bRc implies aRc. For example, "weighs as much as" is a tran­sitive relation. If Henry weighs as much as Charles and Charles weighs as much as Fred, then Henry weighs as much as Fred.

Similarly, if a and b are natural numbers, then the relation "is a multiple of is transitive. If a is a multiple of b and b is a multiple of c, then a must neces­sarily be a multiple of c.

Among the relations that are not transitive we find "is the father of” and "is a friend of”. If a is the father of b and b is the father of c, then a is the grandfa­ther (and not the father) of c. Also, if Mary is a friend of Jean, and Jean is a friend of Betty, it does not necessarily follow that Mary is also a friend of Betty.

Some relations may possess two or all these three properties. Consider, for example, the relation "implies" which may hold between propositions p, q, r etc. This relation is clearly reflexive since p implies p; that is if p is true, then p is true. It is not symmetric, since "p” implies “g" may be correct, while “q” implies “p" is not; that is, a proposition may be correct while its converse is not. It is transi­tive since if p implies q and q implies r, then p implies r, in fact, this is an important principle of logic which we use continually in our proofs without the slightest hesitation.

Consider next the relation "was born in the same town as" which may hold among people. This is clearly reflexive, symmetric and transitive.

If we now consider some relations that are reflexive, symmetric and transitive, we will find that they all express some kind of equality. "Is as tall as" ex­presses an equality of height, "is as rich as" expresses an equality of wealth, "is congruent to" expresses an equality of the sides and angles of triangles, and "is as old as" expresses an equality of age. It is for this reason that relations which are reflexive, symmetric and transitive are referred to as equivalence relations. For example, equality for numbers is an equivalence relation.

Had we studied relations in the early parts of this textbook, we would have found it less difficult to explain the meaning of the equality sign. We could simply have said that "=" is an equivalence relation between the various numbers. Thus, equalities of differences of natural numbers, fractions and complex num­bers are, indeed, reflexive, symmetric, and transitive.

EXERCISES