## Subsection 4.4.4 Counterexamples

We are about to present a collection of techniques for showing that a claim must be true in *all* interpretations. (Or, similarly, that it is false in *all* interpretations.) If there are infinitely many such interpretations, we need something other than “try them all”.

But suppose that we want to show that some claim is *not* valid. Or that it is *not* a contradiction. Then it suffices to find a single interpretation that makes a liar out of it. We’ll call such an interpretation a *counterexample*. Universally quantified statements are typically the easiest to refute in this way.

Also note that we can refute the claim that an expression is a contradiction by showing a single set of values that make the expression true.

Let the universe be the rational numbers. Now consider:

\(\forall \)x, y (\(\exists \)z (z = x/y))We can show that this claim is not a tautology by finding a single (x, y) pair of values such that the required z does not exist.

That is easy. Let y = 0. (It doesn’t matter what x is.)

### Exercises Exercises

#### Exercise Group.

1. Consider the following set of claims:

[1] ∀*x* (*Cat*(*x*) → *Loves*(*x*, *Catnip*))

[2] ∀*x* (*Loves*(*x*, *Catnip*) → *Cat*(*x*))

[3] ∀*x* (*Loves*(*x*, *Chocolate*) → *Loves*(*x*, *IceCream*))

[4] ∀*x* ((*Cat*(*x*) ∧ *Calico*(*x*)) → *Loves*(*x*, *Chocolate*))

[5] ∀*x* (*Cat*(*x*) → ¬*Dog*(*x*))

Let *ModelOfCatsAndDogs* be the conjunction of these five claims.

For each of the following statements, indicate whether or not it is a counterexample to *ModelOfCatsAndDogs*:

##### 1.

(Part 1) *Cat*(*Lucy*) ∧ *Calico*(*Lucy*) ∧ ¬*Loves*(*Lucy*, *Icecream*)

##### 2.

(Part 2) *Cat*(*Agnes*) ∧ *Loves*(*Agnes*, *Paper*)

##### 3.

(Part 3) *Dog*(*Scarlet*) ∧ ∀*x* (*Loves*(*Scarlet*, *x*))

##### 4.

(Part 4) Dog(*Lloyd*) ∧ *Loves*(*Lloyd*, *Chocolate*))

#### Exercise Group.

2. *Zamzows and Tenockritus* For the purpose of this question, assume that the zamzow is a newly discovered species and tenockritus is an ailment that afflicts species like the zamzow. For each of the following claims, indicate an assertion that, if true, would *falsify* the claim:

##### 1.

(Part 1) ∀*x* (*Zamzow*(*x*) → *HasTenockritus*(*x*)) All zamzows have tenockritus.

Alfred and Reggie are zamzows that have tenockritus.

There are no zamzows.

Clora isn’t a zamzow but has tenockritus.

Peapod is a zamzow and, by the way, no one has tenockritus.

No one has tenockritus.

##### 2.

(Part 2) ∃*x* (*Zamzow*(*x*) ∧ *HasTenockritus*(*x*)) Some zamzows have tenockritus.

Alfred and Reggie are zamzows that have tenockritus.

Peapod is a zamzow who does not have tenockritus.

Chlora isn’t a zamzow but has tenockritus.

Tiddlywinks is the only zamzow and does not have tenockritus.

Only bears have tenockritus.

##### 3.

(Part 3) ¬(∃*x* (*Zamzow*(*x*) ∧ *HasTenockritus*(*x*)) ) No zamzow has tenockritus.

Alfred and Reggie are zamzows that have tenockritus.

Peapod is a zamzow who does not have tenockritus.

Chlora isn’t a zamzow but has tenockritus.

Tiddlywinks is the only zamzow and does not have tenockritus.

Only bears have tenockritus.