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a unless b but only if c.

glee66glee66 Free Trial Member
edited October 2014 in Logical Reasoning 4 karma
For instance, a -> (b -> c) simplifies to a + -b -> -c. What about (-a -> b) -> c. How can I simplify it?
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  • chrijani7chrijani7 Alum Member
    827 karma
    I believe the second solution is the proper approach.

    (-a->b)->c

    because you have a conditional within a conditional. View the lessons on Demorgans laws & mastery to see a similar issue that JY solves.

    But unless=negate sufficient, so negate a and sufficient, then only if= necessary

    so -a->b is its only conditional, but it is a conditional that has a conditional that applies to it as well, so (-a->b) servers as the entire sufficient, and then only if ties that it together.

    Hope that makes sense, these are difficult to explain, if your having issues like I said watch the videos on Demorgans law.
  • jeffersonairplane14jeffersonairplane14 Alum Member
    8 karma
    To simplify: (-a -> b) -> c, think of the expression (-a -> b) as y. That is, y=(-a -> b). Then, we get

    y -> c,
    or, -c -> -y (taking contrapositive)
    or, -c -> (-a -> b)
    or, -c AND -a -> b
  • shinny117shinny117 Alum Member
    69 karma
    Would someone help me with this one? (I'm reviewing JY's explanation, but just making sure I'm understanding this concept...)

    this I get:
    not (A -> B) would be A some -B

    this I'm not so sure:
    not (-A -> B)


    Any help is much appreciated!
  • iiiSpooniiiSpoon Alum Inactive ⭐
    277 karma
    @shinny117 another way to think of not(A>B) would be "you can have A and not B." So not(-A>B) would be "you can have not A and not B." (-A>B) is just an "or relation" in which "one must always be in." Saying nope to this would be "you don't need to always have one in," which is "you can have both out (not A and not B). I hope this helped.
  • lglglglglglglglg Free Trial Member
    19 karma
    So,
    (A -> B) -> C = not C -> A and not B
    (not A -> B) -> C = not C -> not A and not B
    A -> (B->C) = A and not B -> C

    ?
  • iiiSpooniiiSpoon Alum Inactive ⭐
    277 karma
    @leeginnyy, the first two are spot on, but for the third conditional statement A -> (B->C) = A and not B -> C, the contrapositive should be (B and not C) > not A since your embedded necessary condition is (B > C).

    If A exists then the relation B > C kicks in. If the relation B > C cannot kick in, then A cannot exist.
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