Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 5.1.14 Existential Instantiation

Existential Instantiation ∃x (P(x))

∴ P(c*/x)

Restrictions:

• c* must be a symbol that has not previously been used.

• If the quantified expression ∃x (P(x)) originally occurred inside the scope of one or more universal quantifiers that have already been instantiated then:

If any of those universals have been instantiated to arbitrary elements, then we must describe c* as depending on the values of those arbitrary elements. (See example below to make this clearer.)

This rule says that if P holds for some element of the universe, then we can give that element a name such as c*. When selecting symbols, we must be careful: the symbols must be selected one at a time and must not duplicate any symbol that has already been selected within the same proof.

Thus, if ∃x (P(x)) ∧ ∃y (Q(y)) is true, then, to use this rule, we select a name, let’s say c*, for the object of which P must be true. Then we select a different name, let’s say d*, for the object of which Q, must be true.

This is what we have been calling new-rule-to-come-3.

To simplify our bookkeeping and to guarantee that we don’t confuse the names we choose here for “the ones” with the names we choose for arbitrary elements when we apply Universal Instantiation, we’ll always choose names (like c* and d*) that end with an asterisk.

Voters

Consider the following argument: If you are 18 years old then you can vote. Someone, call the person c, is 18 years old. Therefore c can vote.

Define:

E(x): True if x is 18 years old.

V(x): True if x can vote.

Then the proof proceeds as follows:

[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]

In step 3 above, a specific person with property E was given the name c*. For that same person c*, the statement E(c*)  V(c*) holds by Universal Instantiation. Note that the order of steps 3 and 4 cannot be reversed. We can do Existential Instantiation first and give the name c* to “the one”. Then we can use Universal Instantiation since, when we apply it, we may make the claim about any element and thus, in particular c*. But, if we do Universal Instantiation first and choose c* as the name of some arbitrary element, then that name has already been used. The best we’d be able to do, if we apply Existential Instantiation at that point, would be to choose a second name for “the one”. But then we’d not be able to conclude anything about either c* or (say) d* because they’d be different.

Notice, in this example, that we’ve ended by concluding something (i. e., can vote) of some object c* about which we know nothing else. But we do in fact know one more thing. We now know that there exists someone who can vote. We need one more rule, to be described next. It will let us take that step.