Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Stumped on a logical equivalence proof

  1. Nov 19, 2014 #1
    ~(P<->Q) ⊣ ⊢ (P<->~Q)

    I'm suppose to write the proof for this equivalence but I can't figure it in either direction
    The closest I got was (P->~Q) from ~(P<->Q) but I can't figure anything else out
  2. jcsd
  3. Nov 19, 2014 #2


    User Avatar
    Gold Member

    Do you mean to show that ##\neg (P \leftrightarrow Q) \equiv (P \leftrightarrow \neg Q)##? Why not just use a truth table?
  4. Nov 19, 2014 #3
    I need to do it by formal proof
    this is as far as I got but I cant figure out how to determine the necessay ~Q->P or how to do it in the opposite direction.
    Code (Text):

    ~(P<->Q)                want:P<->~Q
    |P                      want: ~Q
    ||Q                     reductio
    |||P                    want: Q
    |||Q                    reiterate 3
    ||P->Q                  conditional introduction4-5
    |||Q                    want: P
    |||P                    reiterate 2
    ||Q->P                  conditional introduction7-8
    ||P<->Q                 Biconditional definition 6,9
    ||~(P<>Q)               reiterate 1
    |~Q                     indirect proof 3-11
    P->~Q                   conditional introduction2-12
  5. Nov 20, 2014 #4


    User Avatar
    Science Advisor

    To answer, we need to know which axioms and rules of inference that are allowed in this context. This can differ in different textbooks.
  6. Nov 20, 2014 #5
    conjunction introduction
    disjunction introduction
    conjunction elimination
    disjunction elimination
    conditional elimination
    biconditional elimination
    negation introduction/elimination proof
    conditional introduction proof
    bicondional definition

    these are all the rules I have learned
  7. Nov 24, 2014 #6

    Stephen Tashi

    User Avatar
    Science Advisor

    Unfortunately, the rules of logical inference don't all have standardized names. Their titles differ from textbook to textbook. Can you give a link to an article where those rules are written out?
  8. Nov 24, 2014 #7


    User Avatar
    Science Advisor

    I think I know what most of these rules are. But exactly how are negation introduction and elimination defined in your textbook?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook