##### Predicate Logic

Existential Generalization

### Subjects to be Learned

• existential generalization rule

### Contents

Rule:

P(c)
----------
x P(x)

where c is an element of the universe.

Restrictions:

x must not appear free in P(c).

Explanation:

What this rule says is that if there is some element c in the universe that has the property P, then we can say that there exists something in the universe that has the property P.

Example:

For example the statement "if everyone is happy then someone is happy" can be proven correct using this existential generalization rule.

To prove it, first let the universe be the set of all people and let H(x) mean that x is happy.

Then the argument is
x H(x) x H(x)

The proof is
 1. x H(x) Hypothesis 2. H(c) Universal instantiation 4. x H(x) Existential generalization.

Note:

Without the restriction that x must not appear free in P(c), one may produce an incorrect formula by existential generalization. For example, x Q(x, x) may be derived from Q(x,c) by existential generalization. But Q(x,c) x Q(x, x) is not valid, as you can see if Q(x,y) means "x is not equal to y", or "x > y", for example.