Question

Which one of these first -order logic formulae is valid?

• A. $\forall x\left(P\right(x)\Rightarrow Q(x\left)\right)\Rightarrow (\left(\forall xP\right(x\left)\right)\Rightarrow \left(\forall xQ\right(x\left)\right))$
• B. $\exists x\left(P\right(x)\vee Q(x\left)\right)\Rightarrow (\left(\exists xP\right(x\left)\right)\Rightarrow \left(\exists xQ\right(x\left)\right))$
• C. $\exists x\left(P\right(x)\wedge Q(x\left)\right)\iff (\left(\exists xP\right(x\left)\right)\wedge \left(\exists xQ\right(x\left)\right))$
• D. $\forall x\exists yP(x,y)\Rightarrow \exists y\forall xP(x,y)$

Answer 1

The correct option is A $\forall x\left(P\right(x)\Rightarrow Q(x\left)\right)\Rightarrow (\left(\forall xP\right(x\left)\right)\Rightarrow \left(\forall xQ\right(x\left)\right))$

Option (a) is a standard oneway distributive property of predicates.