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I have a few questions regarding free and bounded (dummy) variables. Let me lay out what I know. (Please tell me if this is incorrect) and then I'll show you what is confusing me.

Let P(x,y) mean that x likes y

P(x,y) x and y are both free variables because they don't have a quantifier.

(forAll(x))P(x,y) x is bound, but y is free.

(forSome(y))P(x,y) x is free, but y is bound.

(forSome(x))(forAll(y))P(x,y) x is bound and y is bound

I don't know if changing the order changes whether something is bound for example:

(forSome(y))(forAll(x))P(x,y) are both x and y still bound even though this doesn't mean that somebody likes everybody. I believe it to mean that For some y, everybody likes x.

Thanks for the help.

1 Answers 1

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I will write $\forall x$ for "forAll(x)", and assume "forSome(x)" means there exists $x$, denoted $\exists x$.

A variable is free if it is not quantified over; the order of quantifiers does not matter. Hence it is correct that in $\exists x \forall y \ P(x,y)$ and $\exists y \forall x \ P(x,y)$ both $x$ and $y$ are bound. The meaning of the formula, however, does change: $\exists x \forall y \ P(x,y)$ means that there is someone who likes everybody; $\exists y \forall x \ P(x,y)$ means that there is someone who is liked by everybody.

Changing the order of quantifiers without changing the variables also leads to different meanings: $\forall y \exists x \ P(x,y)$ means that for every person $y$, there is somebody who likes $y$, and $\forall x \exists y \ P(x,y)$ means that everybody likes someone.