The predicates that we have spoken of as two-place predicates are often called relational predicates. In using these, we speak of relations between objects. Relations may have certain important features that, if we learn to recognize them, can help considerably in assessing certain arguments.

Let us think about what this says, taking the predicate "_is a sibling of ." as symbolized by "R_.". (To be a sibling of is to be a brother or sister of.) The symbolic sentence in the box now says that for everyone x and for everyone y, if x is a sibling of y then y is a sibling of x. This seems clearly true. Given that it is true, we will characterize the relation being a sibling as a symmetric relation. Hence we know that, for example, if I am a sibling of Barbara then Barbara is a sibling of me.

Some other relations, such as "being in love with," are not symmetric. Many of us have learned this through painful experience. To say that a relation R is not symmetric is simply to say that it is not the case that (x)(y)(Rxy ® Ryx). Do not confuse this with the following property:

Relations may also be reflexive:

The final property we shall consider is transitivity:

We can most easily grasp the point of this by considering an example:

Many kinship terms are intransitive. Consider the relation of being a father of. If Howard is a father of Max, and Max is a father of me, then Howard is of course not a father of me—a grandfather of me is not a father of me.

Suppose someone presented the following argument:

The properties of relations that we are looking at are themselves related in certain ways. Suppose we have, for example. a relation which is both transitive and symmetric. Does it remain an open question as to whether that relation is reflexive? If you consider this for a bit you should see that the answer is no. Let us briefly see why by means of considering an example. Let us introduce three terms "a" and "b" and a relational predicate "R_." Since by supposition we have transitivity we know that:

But since we have symmetry we also know that:

From these we can conclude that:

This is not, strictly speaking, a proof of our claim, but if you think about it you should see that any relation which is transitive and symmetric is also reflexive.

There are many relations that are transitive and symmetric and thereby, as we just saw, reflexive. Being of the same height is one. But one such relation is typically singled out for special attention. This is the relation of identity. In math this relation is often spoken of as equality. But, with numbers for example, equality is just identity. We shall reserve the letter "I" for identity. Since identity is transitive, symmetric and reflexive we know that all the following are true:

We frequently utilize identity, appealing to these properties, in our reasoning.

Consider the following simple case. I find out that this gun is the murder weapon. I conclude that since I have handled this gun that I have handled the murder weapon. Here I am utilizing the properties of identity. And of course in mathematics we appeal to identity all the time though as noted we typically use an "=" sign. Here is a trivial example:

Again we are appealing to the properties of identity.