Variables and Quantifiers

Predicate logic gets more complicated when the subject of a sentence doesn’t name one specific thing. To represent such sentences, we use variables and quantifiers. A variable in predicate logic is a place-holder (or “blank”) into which the name of an individual thing, like ‘Socrates’ or ‘Athens,’ could be inserted. Variables are represented by lowercase letters ‘x,’ ‘y,’ and ‘z.’

A variable can be written to the right of a predicate letter, just like a constant. Unlike constants, however, a variable must be accompanied by a quantifier, which indicates how many individual things the variable represents. The simplest system of predicate logic uses just two types of quantifiers: universal quantifiers and existential quantifiers.

A universal quantifier is represented by the ‘∀’ symbol enclosed in parentheses with a variable, like this: (∀x). The quantifier ‘(∀x)’ indicates that the name of absolutely anything that existsTechnically, the meaning of the quantifiers depends on the domain of quantification, a concept which will be explained later in this chapter. When no domain is specified, it is assumed to include everything that exists. may be inserted in place of the variable x. Similarly, the quantifier ‘(∀y)’ indicates that anything may be named in place of y, and ‘(∀z)’ indicates that anything may be named in place of z.

For example, the proposition that everything is physical (i.e., that there are no non-physical entities like souls or spirits) can be symbolized as follows:

To read the formula ‘(∀x)Px’ aloud, we say “For all x, x is P.”

(∀x)Px

In this example, ‘Px’ means ‘x is physical,’ which is not a complete sentence because it has no subject. The quantifier ‘(∀x)’ means that absolutely anything that exists could be used as the subject of that sentence. Thus, the formula ‘(∀x)Px’ means that any specific thing that exists is a physical thing; or in other words, absolutely everything that exists is physical.

An existential quantifier is represented by the ‘∃’ symbol enclosed in parentheses with a variable, like this: (∃x). The quantifier ‘(∃x)’ indicates that at least one thing that existsTechnically, the meaning of the quantifiers depends on the domain of quantification, a concept which will be explained later in this chapter. When no domain is specified, it is assumed to include everything that exists. may be named in place of the variable x. For example, the proposition that souls exist can be symbolized as follows:

To read the formula ‘(∃x)Sx’ aloud, we we say “There exists an x such that x is S.”

(∃x)Sx

In this example, ‘Sx’ means ‘x is a soul.’ The quantifier ‘(∃x)’ indicates that at least one thing that exists could be used as the subject, in place of x. So, the above formula says that at least one thing that exists is a soul. In other words, there exists at least one soul.

Here are a few more examples involving universal and existential quantifiers.

Everything is awesome is symbolized: (∀x)Ax

Not everything is funny is symbolized: ~(∀x)Fx

Unicorns don’t exist is symbolized: ~(∃x)Ux

If philosophers exist, then thinkers exist is symbolized: ((∃x)Px ⊃ (∃y)Ty)