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Math Help - confusion between "and" and "implies"

  1. #1
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    confusion between "and" and "implies"

    So lets say P(x) is "x is a lion"
    Q(x) is "x is fierce"
    R(X) is "x drinks coffee"

    According to textbook ∀x(P(x) -> q(x)) = "All lions are fierce"
    ∃x((P(x)^ČR(x))= "Some lions do not drink coffee"

    It also says it cannot be ∃x((P(x)->ČR(x)). Why?

    P(x)=x is a hummingbird
    Q(x)=x is large
    R(x)=x lives on honey
    S(x)=x is richly colored

    Textbook says

    ∀x(P(x) -> S(x))=All hummingbirds are richly colored
    Why is this statement implied and not "and". but the previous statement is "and" and cannot be implied? Whats the difference?

    Thanks
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  2. #2
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    Hallo bfpri,
    your text book should explain it, but if not think about it like this:
    (\forall x)[P(x) \rightarrow Q(x)]
    "for all x, if x is a Lion, then it is fierce," which means, "every lion is fierce."

    (\forall x)[P(x) \land Q(x)]
    would mean that "everything is a lion and everything is fierce."

    And the problem with
    (\exists x)[P(x) \rightarrow \neg R(x)]
    is that it means "if there are no lions then coffee is not drunk." So, if the antecedent is false ("there exist no lions") then the conclusion is true. "If some lions exist, then nothing drinks coffee (or coffee isn't drunk)."
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  3. #3
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    Quote Originally Posted by bmp05 View Post
    Hallo bfpri,
    your text book should explain it, but if not think about it like this:
    (\forall x)[P(x) \rightarrow Q(x)]
    "for all x, if x is a Lion, then it is fierce," which means, "every lion is fierce."

    (\forall x)[P(x) \land Q(x)]
    would mean that "everything is a lion and everything is fierce."
    I'm with you up to this part
    And the problem with
    (\exists x)[P(x) \rightarrow \neg R(x)]
    is that it means "if there are no lions then coffee is not drunk."
    I'm confused. I thought it meant for some X, if X is a lion then x does not drink coffee..Where did the "no lions" come from?

    thanks for the reply
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  4. #4
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    because it's an implication, what happens if the antecedent is false, then the conclusion is true. The antecedent is false if there aren't any lions, right? So, if you're universe is your university or somewhere, where there are no lions, then the implication is that "coffee is not drunk."
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  5. #5
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    Quote Originally Posted by bmp05 View Post
    because it's an implication, what happens if the antecedent is false, then the conclusion is true. The antecedent is false if there aren't any lions, right? So, if you're universe is your university or somewhere, where there are no lions, then the implication is that "coffee is not drunk."
    Ah, Ok. What if the domain is "all creatures". Wouldn't that mean for some creatures, if x is not a lion, x does not drink coffee? Sorry, I'm a bit slow on this logic stuff. =/
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  6. #6
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    Have you studied the distinction between universal statements over against existential statements?
    With due respect to bmp05, that distinction is the heart of your question.

    The statements, “All P is Q” and “No P is Q” are universal statements.
    Universal statements are implications, that is hypotheticals.
    “All Marians are green.” translates as if X is a Marian then X is green.
    “No dogs are green.” translates as if X is a dog then X is not green.
    Each of those is hypothetical.

    Existential statements are statements of fact.
    "Some P is Q" and "Some P is not Q", for example.
    “Some pigs can fly” translates, as there is a pig that flies. I can show you one such.
    “Some cats are not kind” translates, as there is a cat that is not kind. I can show you one such.

    NOTE that the negation of a universal statement is a existential statement.
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  7. #7
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    Thanks!
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