##### Predicate Logic

Existential Instantiation

### Subjects to be Learned

• existential instantiation rule

### Contents

Rule:

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

where c is some element of the universe of discourse. It is not arbitrary but must be one for which P(c) is true.

Restrictions:

c must be a new name or constant symbol.

Explanation:

What this rule says is that if P holds for some element of the universe, then we can give that element a name such as c (or x, y, a etc). When selecting symbols, one must select them one at a time and must not use a symbol that has already been selected within the same reasoning/proof.

Example:

For example, if x P(x) x Q(x) is true, then select a name for P, say c, then for Q, say d. One must NOT select c for Q as well as for P.

Consider the following argument: If you get 95 on the fianl exam for CS 398, then you get an A for the course. Someone, call him/her say c, gets 95 on the final exam. Therefore c gets an A for CS398. This argument uses Existential Instantiation as well as a couple of others as can be seen below.

Let the universe be the set of all people in the world, let N(x) mean that x gets 95 on the final exam of CS398, and let A(x) represent that x gets an A for CS398.

Then the proof proceeds as follows:
 1. x [ N(x) A(x) ] Hypothesis 2. x N(x) Hypothesis 3. N(c) Existential instantiation on 3. 4. N(c) A(c) Universal instantiation on 1. 5. A(c) Modus ponens on 3 and 4.

In step 3 above, a specific person with property N was given the name c. For that same person c, the statement [ N(c) A(c) ] holds by the universal instantiation.
Note that the order of steps 3 and 4 can not be reversed.