Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 5.1.10 Working with Existentially Quantified Statements: “The One”

Now we need a similar process for working with existentially quantified statements. But there is a crucial difference. An existential quantifier guarantees the existence of (possibly only) one element that satisfies its predicate. So we can’t pick an arbitrary element and assume that it’s “the one”. What we can do is to give a name to “the one”. We’ll soon describe exactly how to do this with new-rule-to-come-3.

As before, the key is that we must pick a name that doesn’t match any other elements that we’re reasoning about. We know nothing special about “the one” except that it exists and satisfies one particular predicate. So, in particular, if we are working with multiple existentially quantified expressions, we must not assume that “the one” for one of them is necessarily the same as “the one” for any others.

Suppose that we have:

[1] x (Likes(x, Rootbeer))

[2] x (Likes(x, Beets))

Assuming that we have new-rule-to-come-3, which will let us name “the one”, we could write:

[1a] Likes(d*, Rootbeer)

[2a] Likes(e*, Beets)

What’s critical is that we not use the same symbol in both [1a] and [2a]. It’s possible that d* = e* (in other words that it’s the same person who likes both rootbeer and beets), but we have no basis for concluding that.

And now for the key difference between the arbitrary elements that arise when we’re working with universally quantified expressions and the existential elements (“the ones”) that arise from existentially quantified expressions:

• Dealing with arbitrary elements: We saw that if we have proved a claim about an arbitrary element, then we can (using new-rule-to-come-2) generalize that claim so that we create a new universally quantified statement.

• Dealing with existential elements: After proving a claim about “the one”, all we know is that our claim is true of that one and, thus, of at least one element. So we can generalize our claim but only to an existentially quantified statement.

So the final rule that we’re about to describe (for now we’ll call it new-rule-to-come-4) will let us assert that existentially quantified statement.

As a bookkeeping matter: When we’re writing a proof, we must clearly distinguish between arbitrary elements (about which we’ve made no assumptions) and “the ones” (about which we’ve made the specific assumption that they are “the ones”). To make it easy to keep these two kinds of objects distinct, every time we introduce a name for “the one”, we’ll end the name with an asterisk (as we’ve just done).

While it’s critical that we not confuse arbitrary elements and “the ones”, it’s worth pointing out that people do sometimes fail to make that distinction. When that happens, logical nonsense results.

Suppose I say, “Some kittens are fluffy”:

[1] x (Kitten(x)  Fluffy(x))

Since this is an existentially quantified statement, we are allowed to write:

[1a] Kitten(k*)  Fluffy(k*)

In this case, k* is “the one”, about whom we know absolutely nothing except that it’s a fluffy kitten.

Now suppose that someone else says, “My kitten Lucy isn’t fluffy” (as if that were a counterexample to my claim). We’d write that specific statement as:

[2] Kitten(Lucy)  Fluffy(Lucy)

Now compare [1a] and [2]. It is possible that Lucy and k* are aliases for the same individual. But nothing requires that they be. So we can’t conclude any contradiction between them. Our guy k* may be a big ball of fluff while his third cousin Lucy could be as sleek as can be.

So I should respond, “I did not make a universal statement about all kittens. I said there is at least one fluffy kitten. Lucy might not be that one fluffy kitten”.