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Thread: Can i change the order of quantifiers in this case?

  1. #1
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    Can i change the order of quantifiers in this case?

    In this sentence:

    "No hero is cowardly and some soldiers are cowards."

    Assuming h(x) = x is a hero

    s(x) = x is a soldier

    c(x) = x is a coward.

    So the sentence is like this i think:

    (∀x (h(x)->C(x))∧(∃y (s(y)∧C(y))
    In this case, are prenex formulas bellow the same thing?

    ∀x ∃y (h(x)∨C(x))∧((s(y)∧C(y))
    ∃y ∀x (h(x)∨C(x))∧((s(y)∧C(y))
    Last edited by Lyunth; Nov 24th 2015 at 07:25 AM.
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  2. #2
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    Re: Can i change the order of quantifiers in this case?

    Your translation is correct. Just personal preference, I prefer to translate "no hero is cowardly" as
    $$\neg((\exists x)(h(x)\wedge c(x)))$$

    Yes, you can rewrite your symbolic statements in either of the two ways; I'm not sure why you want to do this. I think the original is much clearer. You can formally prove this by going through the business of universal and existential instantiation, some logical equivalences and then existential and universal generalization.
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  3. #3
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    Re: Can i change the order of quantifiers in this case?

    Quote Originally Posted by Lyunth View Post
    In this sentence:
    "No hero is cowardly and some soldiers are cowards."
    Assuming h(x) = x is a hero
    s(x) = x is a soldier
    c(x) = x is a coward.
    So the sentence is like this i think:
    (∀x (h(x)->C(x))∧(∃y (s(y)∧C(y))
    In this case, are prenex formulas bellow the same thing?
    ∀x ∃y (h(x)∨C(x))∧((s(y)∧C(y))
    ∃y ∀x (h(x)∨C(x))∧((s(y)∧C(y))
    The problem of quantifier order usually occurs when a predicate applies to more than one variable such as a relationship (is a class mate of).
    Here you have two singular predicates. Isn't true that $\displaystyle \left( {\forall x} \right)\left[ {h(x) \to \neg C(x)} \right] \wedge \left( {\exists y} \right)\left[ {s(y) \wedge C(y)} \right]$ is the same as $\displaystyle \left( {\exists y} \right)\left[ {s(y) \wedge C(y)} \right] \wedge \left( {\forall x} \right)\left[ {h(x) \to \neg C(x)~?} \right]$
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