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

Question

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.

Answer 1

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:

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

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