where c is some element of the universe of discourse. It is not
but must be one for which P(c) is true.
c must be a new name or constant symbol.
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.
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.|