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Math Help - free vs bound variable - predicate calculus

  1. #1
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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



    free vs bound variable - predicate calculus-page106.png

    a proof has been given which mentions

    A'= \forallx 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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: free vs bound variable - predicate calculus

    Quote Originally Posted by suzanne View Post
    Hi A'= \forallx 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 ...
    x is free in A_1(x) by hypothesis but not in \forall xA_1(x) . Notice that in the latter case, x always occurs within the scope of the quantifier \forall x .
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    Re: free vs bound variable - predicate calculus

    Quote Originally Posted by suzanne View Post
    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.
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