# Prove falsity of argument schemas in predicate logic

I have to give a counterexample for the following argument schema:

∃x (Px ∧ Qx) ⊨ ∀x (Px ∨ Qx)

by definig its domain and the interpretation function, which is where I have some slight problems. On a technical level, I think I understood, what's the problem with the schema, namely, that the conclusion is false because of the universal quantifier.

But I don't really understand how to express this in an interpretational function, that proves that the first statement is true, where as the second is false.

## 1 Answer

You can set the domain to be $$\mathbb{R}$$, so your quantifiers range over that set. And you can define the interpretation $$\mathcal{I}$$ such that,

\begin{align*} \mathcal{I}(P)(x)&\ \triangleq\ x^2 = 4\\ \mathcal{I}(Q)(x)&\ \triangleq\ x+2=4 \end{align*}

Then your schema becomes,

$$\exists x\in\mathbb{R}. (x^2=4) \land (x+2=4)\vDash \forall x\in\mathbb{R}. (x^2=4) \lor (x+2=4)$$

And it is trivial to show that the first is true taking $$x=2$$, and that the second is false choosing any $$x\neq 2$$.