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Math Help - Universal senteces & bound variables

  1. #1
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    Universal senteces & bound variables

    Hi,
    Does the fact that a sentence is an universal sentence (i.e. x+y=y+z) automatically make all variables bound?
    Let me give you an example. In the expression "for any x, x-y=x+(-y)" is only an x variable bound or both x & y are bound variables?

    MY doubt is due to the convention that in many universal sentences the quantifiers are customarily dropped.

    Thanks
    BTW the above question is from A Tarski "Intro to logic..."
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  2. #2
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    Re: Universal senteces & bound variables

    Quote Originally Posted by mathphile View Post
    In the expression "for any x, x-y=x+(-y)" is only an x variable bound or both x & y are bound variables?
    All instances of x are bound and no instances of y are bound.

    Quote Originally Posted by mathphile View Post
    the convention that in many universal sentences the quantifiers are customarily dropped.
    In many contexts, we often leave off the universal quantifiers. If you wish to know whether an instance of a variable is bound or not in a formula, then just specify whether you mean the formula as it is actually displayed or as it is, in some context, understood to stand for the universal closure of the displayed formula.

    If "for all x, x-y=x+(-y)" is to be regarded as actually displayed, then no instances of y are bound in it.

    If "for all x, x-y=x+(-y)" is to be regarded as standing for its universal closure "for all y, for all x, x-y=x+(-y)" then all instances of y are bound in it.

    Note that the key theorem of logic for this matter is:

    For all formulas F we have |- F if and only if |- AxF. But it is not the case that for all formulas F we have |- F <-> AxF.
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  3. #3
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    Re: Universal senteces & bound variables

    Thanks,

    Quote Originally Posted by MoeBlee View Post

    Note that the key theorem of logic for this matter is:

    For all formulas F we have |- F if and only if |- AxF. But it is not the case that for all formulas F we have |- F <-> AxF.
    Sorry, I'm pretty new to math & logic -- can you tell me how to read the theorem (symbols).
    Thanks
    mathphile
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  4. #4
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    Re: Universal senteces & bound variables

    '|-' stands for 'is a theorem'.

    For all formulas F, we have that F is a theorem if and only if AxF is a theorem.

    But it it not the case that for all formulas F, we have that F <-> AxF is a theorem. ('<->' is the if and only if symbol).
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  5. #5
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    Red face Re: Universal senteces & bound variables

    Many thanks,
    I must be pain in the neck, but could you explain what AxF mean?
    Last edited by mathphile; July 13th 2011 at 05:40 AM.
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  6. #6
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    Re: Universal senteces & bound variables

    You're not a pain. I was using 'A' for the universal quantifier "for all".
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  7. #7
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    Red face Re: Universal senteces & bound variables

    Thanks. I've learned something.
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