Chapter 35 Properties of relations

Symbolization keys allow us to assign two-place predicate symbols like ‘R(x,y)’ to English predicates with two gaps, e.g., ‘blankx is older than blanky’. This allows us to symbolize arguments such as:

  • Rudolf is older than Kurt.

  • Kurt is older than Julia.

  • Rudolf is older than Julia.

This argument is valid, but its symbolization in FOL,

  • R(r,k)

  • R(k,j)

  • R(r,j)

is not. That’s because the validity of the argument depends on a property of the relation ‘x is older than y’, namely that if some person x is older than a person y, and y is also older than z, then x must be older than z. This property of ‘older than’ is called transitivity . We can symbolize this property itself in FOL as:


Whenever the validity of an argument only depends on a property of a relation involved, and this property can be symbolized in FOL, we can add this symbolization to the argument and obtain an argument that is in fact valid in FOL:

  • xyz((R(x,y)R(y,z))R(x,z))

  • R(r,k)

  • R(k,j)

  • R(r,j)

Properties of relations such as transitivity are important concepts especially in applications of logic in the sciences. For instance, order relations between numbers such as < and (and also > and ) are transitive. The identity relation = is also transitive.

There are other properties of relations that are important enough to have names, and often show up in applications. Here are some:

  • A relation R is transitive if⁠f it is the case that whenever R(x,y) and R(y,z) then also R(x,z).

  • A relation R is reflexive if⁠f for any x, R(x,x) holds.

  • A relation R is symmetric if⁠f whenever R(x,y) holds, so does R(y,x).

  • A relation R is anti-symmetric if⁠f for no two different x and y, R(x,y) and R(y,x) both hold.

We have already seen that transitivity can be symbolized in FOL. The others can, too:

  • R is reflexive: xR(x,x)

  • R is symmetric:

  • R is anti-symmetric:


    or, equivalently,


Relations expressing an equivalence in some respect are reflexive, e.g., ‘is the same age as’, ‘is as tall as’, and the most stringent equivalence of them all, identity =. They are also symmetric: e.g., whenever x is as tall as y, then y is as tall as x. In fact, relations that are reflexive, symmetric, and transitive are called equivalence relations .

Equivalences aren’t the only symmetric relations. For instance, ‘is a sibling of’ is symmetric, but it is not reflexive (as no one is their own sibling).

Relations that are reflexive, transitive, and anti-symmetric are called partial orders . For instance, and are partial orders. The relation ‘is no older than’, by contrast, is reflexive and transitive, but not anti-symmetric. (Two different people can be of the same age, and so neither is older than the other.)

Practice exercises

A. Give examples of relations with the following properties:

  1. 1.

    Reflexive but not symmetric

  2. 2.

    Symmetric but not transitive

  3. 3.

    Transitive, symmetric, but not reflexive

B. Show that a relation can be both symmetric and anti-symmetric by giving an interpretation that makes both of the following sentences true: