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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.

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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 $x\neq 2$.

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