Reasoning: An Introduction to Logic, Sets, and Functions

Subsubsection 4.2.1 What If There Aren’t Any?

It’s tempting to think of ∀ as making a stronger claim than ∃ does. And often it does.

For example, consider: Existential Bears with Tails

[1] \(\forall \) x (Bear(x) \(\rightarrow \) HasTail(x)) All bears have tails.

[2] \(\exists \) x (Bear(x) \(\wedge \) HasTail(x)) There exists a bear with a tail.

[1] feels substantially stronger than [2] does. And, in fact, it says more about the real world

But it is possible to write a universal claim that actually says less than the corresponding existential claim does. This happens when, in fact, there are no objects to which the universal claim applies. Sometimes, when this happens, we’ll say that the universal claim is, “trivially true”, in the sense that it can’t be false because it doesn’t apply to anything.

Let Lucy be my calico cat. Now consider two things I might say:

[3] Lucy has won every game of solitaire she’s ever played.

\(\forall \) x (SolitaireGame(x) \(\wedge \) Played(Lucy, x) \(\rightarrow \) Won(Lucy, x))

[4] There is a solitaire game, played by Lucy, that she’s won.

\(\exists \)x (SolitaireGame(x) \(\wedge \) Played(Lucy, x) \(\wedge \) Won(Lucy, x))

Not unsurprisingly, there are no solitaire games played and lost by Lucy. So [3] is trivially true. Yet [4], the seemingly weaker existential claim, is false since Lucy has (to my knowledge) never played solitaire

By the way, you’ll notice, in both of the examples we just showed, that while the universal claim contains an implication, the existential one does not. It has an and instead. Why? Suppose that we had written:

[5] ∃x (Bear(x) → HasTail(x))

It’s hard to understand what this even means in this form. Recall that an implication is true unless the antecedent is true and the consequent is false. So we can make it true simply by making the antecedent false. (We did this formally in Boolean logic with the Conditional Disjunction rule that tells us that (p → q) ≡ (¬p ∨ q). We’ll soon see how to use this rule in quantified statements.) Thus we can read [5] as, “There exists something that is not a bear or that has a tail.” Of course, the universe contains both many nonbears (including you) and many tailed objects (including Lucy, above). So this statement is true. But it says nothing about whether there must exist at least one actual bear that has a tail. Since that’s what we were trying to talk about, we used ∧ instead of →.

We’ll keep returning to this idea. It’s very rare that what you want is an implication inside an existentially quantified statement.