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.


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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