Predicate Logic
Existential Instantiation
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. |