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I know this should be quite easy but I can't figure out how I have to write down a model as a counterexample for this:

$\forall x (P(x) \rightarrow Q(x)), \exists x(P(x)) \vdash \forall xQ(x)$

Let's say

$ P = \{ \text{students who attended exam E.123} \}$

and

$ Q = \{ \text{students who passed exam E.123} \}$

How do I define my universe of values and how do I give the definitions of the functions and predicates? How to show a model $\mathcal{M}$ as counterexample?

Is it sufficient to define $\mathcal{A}_\mathcal{M} = \{ a, b\}$ and say

$ \begin{align} P_\mathcal{M}(a) &= T \\ P_\mathcal{M}(b) &= T \\ Q_\mathcal{M}(a) &= F \\ Q_\mathcal{M}(b) &= F \end{align} $

?

Stefan Falk
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1 Answers1

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1) $\forall x ((x > 0) \rightarrow (x > 0))$ --- is the generalization of a tautology; thus, it is valid

2) $\exists x (x > 0)$ --- it is true in $\mathbb N$

3) $\forall x(x > 0)$ --- it is false in $\mathbb N$, because not $0 > 0$.

Of course, we can "shrinck" it to a model $\mathcal M$ with $|\mathcal M |= \{ 0,1 \}$.

Mauro ALLEGRANZA
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