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free vs bound variable - predicate calculus

Hi

I am a new member. As per my understanding, if a variable in a formula does not fall under a quantifer, it is free and if it does, it is bound.

Could you tell me if my understanding correct?

I am currently reading the book "mathmatical logic for computer science" page 106. Please see the attached image or visit

Mathematical logic for computer science - Google Books

Attachment 21910

a proof has been given which mentions

A'= $\displaystyle \forall$x A1(x)

and then the text: but x is the only free variable in A1 ...........

My question is: is x free? It falls under for all quantifier ...

Sorry for the long question and thank you for your help.

Suzzane.

Re: free vs bound variable - predicate calculus

Quote:

Originally Posted by

**suzanne** Hi A'= $\displaystyle \forall$x A1(x) and then the text: but x is the only free variable in A1 ........... My question is: is x free? It falls under for all quantifier ...

$\displaystyle x$ is free in $\displaystyle A_1(x)$ by hypothesis but not in $\displaystyle \forall xA_1(x)$ . Notice that in the latter case, $\displaystyle x$ always occurs within the scope of the quantifier $\displaystyle \forall x$ .

Re: free vs bound variable - predicate calculus

Quote:

Originally Posted by

**suzanne** I am a new member. As per my understanding, if a variable in a formula does not fall under a quantifer, it is free and if it does, it is bound.

Could you tell me if my understanding correct?

This is correct; however, to be more precise, "free" or "bound" pertain to *occurrences* of variables. A variable can have both free and bound occurrences in a single formula. In this case, though, bound occurrences should be renamed for clarity.

See also this post for links about free and bound variables.