Is $ (\forall x)(A \rightarrow B) \vdash (\forall x)A \rightarrow (\forall x)B $ an absolute theorem schema ?
- If you think yes, then give a proof.
- If you think no, construct a counter model or prove the invalid strong generalization from it.
How can I do this one ?
The inference is valid; here is a proof with Natural Deduction :
1) $(∀x)(A \rightarrow B)$ --- premise
2) $(∀x)A$ --- assumed [a]
3) $A \rightarrow B$ --- from 1) by $\forall$-elimination
4) $A$ --- from 2) by $\forall$-elimination
5) $B$ --- from 4) and 3) by $\rightarrow$-elimination
6) $\forall x B$ --- from 5) by $\forall$-introduction
7) $(∀x)A \rightarrow (∀x)B$ --- from 2) and 6) by $\rightarrow$-introduction, discharging assumption [a]
Thus, from 1) and 7) we have :