Subsection 5.1.16 Existential Generalization
Existential Generalization P(c/x) P is true for the specific value c
(which appears everywhere a more
general value x might appear).
∴ ∃x (P(x)) There is some value of x for which
P is true.
Restriction: x does not appear free in
This is perhaps the simplest of all of these rules. What it 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.
This is what we have been calling new-rule-to-come-4.
Note that c must be a specific value. It cannot be a function (as described on the previous slide) of some other value.
Now let’s continue the Voting example. We’d like to show that there is someone who can vote. All we need to do is to add line [6]:
[1] x (E(x) V(x)) Premise
[2] x (E(x)) Premise
[3] E(c*) Existential Instantiation [2]
[4] E(c*) V(c*) Universal Instantiation [1]
[5] V(c*) Modus Ponens [3], [4]
[6] x (V(x)) Existential Generalization [5]