Question

Which of the following is a valid first order formula? (Here $\alpha$ and $\beta$ are first order formulae with x as their only free variable)

• A. $(\left(\forall x\right)\left[\alpha \right]\Rightarrow \left(\forall x\right)\left[\beta \right]\Rightarrow \left(\forall x\right)[\alpha \Rightarrow \beta ])$
• B. $\left(\forall x\right)\left[\alpha \right]\Rightarrow \left(\exists x\right)[\alpha \wedge \beta ]$
• C. $\left(\forall x\right)[\alpha \vee \beta ]\Rightarrow \left(\exists x\right)\left[\alpha \right]\Rightarrow \left(\forall x\right)\left[\alpha \right]$
• D. $\left(\forall x\right)[\alpha \Rightarrow \beta ]\Rightarrow (\left(\forall x\right)\left[\alpha \right]\Rightarrow \left(\forall x\right)\left[\beta \right])$

Answer 1

The correct option is D $\left(\forall x\right)[\alpha \Rightarrow \beta ]\Rightarrow (\left(\forall x\right)\left[\alpha \right]\Rightarrow \left(\forall x\right)\left[\beta \right])$

$\left(\forall x\right)[\alpha \Rightarrow \beta ]\Rightarrow (\left(\forall x\right)\left[\alpha \right]\Rightarrow \forall \left(x\right)\left[\beta \right])$
is a logical equivalence and therefore , a valid first order formula.