1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Logical Equivalences

  1. Sep 15, 2012 #1

    Bashyboy

    User Avatar

    Say that I have p represent, "Swimming at the NJ shore is allowed;" and say that q represents, "Sharks have been spotted near the shore."

    If I have the compound proposition [itex]\neg p \rightarrow \neg q[/itex], then it is equivalent to writing [itex]q \rightarrow p[/itex]

    Yet, that would suggest that the two propositions in English are equivalent. To be precise, "If swimming at the NJ shore is not allowed, then the sharks have not been spotted near the shore." and "If sharks have been spotted near the shore, then swimming at the NJ shore is allowed." Personally, I do not see how these to compound propositions, in English, are remotely equivalent.
     
    Bashyboy, Sep 15, 2012
  2. jcsd
    Phys.org - latest science and technology news stories on Phys.org
  3. Sep 15, 2012 #2

    christoff

    User Avatar

    Well, in English, those two propositions are silly. But they are still equivalent. I'm just going to alter the phrasing a bit to simplify the explanation...
    p="I will swim"
    q="There are sharks"

    So ¬p→¬q means "If there are no sharks, then I will not swim".
    and q→p means "If I will swim, then there are sharks".

    "If there are no sharks, then I will not swim"... If I will swim (p), then there must be sharks. Why? Because "If there are no sharks, then I will not swim". Since I cannot both swim and not swim, there must be sharks. Therefore, "If I will swim, then there are sharks".

    "If I swim, then there are sharks"... If there are no sharks (¬q), then I cannot possibly be swimming, since "If I swim, then there are sharks". Since there cannot be both sharks and no sharks, I must not be swimming. Therefore, "If there are no sharks, then I will not swim".

    The equivalence follows from p∧(¬p) being a fallacy for propositions p.
     
    christoff, Sep 15, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook