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I need to show that this formula $(\forall x(A \to B) \to (\forall x A\to \forall x B))$ is true for all interpretation. Could you help me please?

Thank you!

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    @Doug: The question *as phrased* is asking for a model-theoretic argument. It’s a question about semantics, not synta$x$.2011-09-16

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If $\forall x(Ax \rightarrow Bx)$ is false, formula is true by (Tarski's inductive) definition of truth.

If $\forall x(Ax \rightarrow Bx)$ is true, then by d.o.t., for any element $d \in D$ (where $D$ is your model) it's true that $Ad \rightarrow Bd$.

The consequent says $\forall x Ax \rightarrow \forall x Bx$. By d.o.t., if $\forall x Ax$ is false, the consequent is true, and thus the formula is true.

If however $\forall x Ax$ is true, by d.o.t. we have that for all $d \in D$, $Ad$. So by combining this with $Ad \rightarrow Bd$, by d.o.t. we have that for all $d \in D$, $Bd$, and this is defined to be same as $\forall x Bx$.

Since regardless of model (interpretation) the formula holds, it holds for all interpretations.

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    That's what is desribed (implicitly), by not referrin$g$ to any specific interpretation, it was proven for any (arbitrary) interpretation, and thus for all.2014-12-08