Question regarding the meaning of bound variables in First Order Logic.

I have two questions, which are probably easy explained but I am quite confused.

1- If I write: ∀xDx → ∃xAx, then the x in ∀xDx and the x in ∃xAx can be different things in an interpretation? Should they mean the same thing?

2- Is ∀yDy → ∃xAx = ∀y∃x(Dy → Ax) ?If it is, then what happens with ∀xDx → ∃xAx = ∀x∃x(Dx → Ax)? Wouldn't it be confusing?

Thanks.

• 1) $\forall x Dx$ means "every thing is $D$". $\exists x Ax$ means "some thing is $A$". Thus, they do not "mean the same thing". Dec 14, 2017 at 13:36
• My question with the first one is whether the x could be a black cat and the x could be a Siamese dog (assume there are Siamese dogs). I understand that the same thing can have different predicates, my question is whether the same variable can be different "things" in the same sentence. Dec 14, 2017 at 13:46
• Correct me if I am wrong, but then in question 2 the first example is correct and the second is not? Or neither of them are correct? Dec 14, 2017 at 13:48

The equivalences with nested quntifiers are a little bit tricky...

In general, we have that: $(\alpha \to \exists x \ \beta) \equiv \exists x \ (\alpha \to \beta)$, provided that $x$ is not free in $\alpha$.

Thus, $(∀yDy \to ∃xAx) \equiv ∃x(∀yDy \to Ax)$, because $x$ is not free in $∀yDy$.

Nothing change if we write $∀xDx$.

But (alas!) we have that: $(\forall x \alpha \to \beta) \equiv \exists x \ (\alpha \to \beta)$, provided that $x$ is not free in $\beta$.

So, if we apply the rule to the formula above, what we get is:

$(∀yDy \to ∃xAx) \equiv ∃x(∀yDy \to Ax) \equiv ∃x∃y(Dy \to Ax)$.

Finally, we have that: $(∀xDx \to ∃xAx) \equiv ∃x(Dx \to Ax)$.

• Wow, any place where I can read further about this? My introductory book does not take such depth. Dec 14, 2017 at 13:59
• @César Dalí Vázquez Navarro - Many textbooks... Kleene, Mathematical Logic or Simpson, Mathematical Logic. Dec 14, 2017 at 14:05
• Grazie mille :) Dec 14, 2017 at 14:07