Relational Predicates

A relational predicate is a predicate that relates two or more objects. Sometimes it is useful to symbolize relational predicates using different constants for each thing that is named in the relation. For example, the sentence Socrates taught Plato could be symbolized ‘Tsp’, where the constant ‘s’ stands for Socrates, ‘p’ stands for Plato, and the relational predicate ‘Txy’ stands for x taught y.

Relational predicates can also be used with quantifiers. Here are a few examples, with the domain of quantification restricted to people:

Constants & predicates	English sentence	Symbolization	How to read the symbolization aloud
m = my name Lxy = x loves y	Somebody loves me.	(∃x)Lxm	There exists an x such that x loves me.
	I love someone.	(∃x)Lmx	There exists an x such that I love x.
	Everybody loves somebody.	(∀x)(∃y)Lxy	For all x, there exists a y such that x loves y.
	Somebody loves everyone.	(∃x)(∀y)Lxy	There exists an x such that for all y, x loves y.

For another example, let’s symbolize the first premise of Anselm’s argument, which was quoted on the previous page:

Something exists in the understanding, at least, than which nothing greater can be conceived.

As mentioned previously, we have to extend the domain of quantification to include not only things that exist, but also things that “exist in the understanding”—i.e., things that are conceptually possible. Using that extended domain of quantification, we can employ the following relational and non-relational predicates:

Ux = x exists in the understanding
Cy = y can be conceived
Gyx = y is greater than x

Now, we can symbolize Anselm’s first premise as follows:

(∃x)(Ux • ~(∃y)(Cy • Gyx))