# free vs bound variable - predicate calculus

• Jul 28th 2011, 06:50 PM
suzanne
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.
• Jul 28th 2011, 10:43 PM
FernandoRevilla
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\$ .
• Jul 29th 2011, 02:59 AM
emakarov
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.