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Stumped on a logical equivalence proof

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  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

    ZetaOfThree

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    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

    Erland

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    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
    reiteration

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

    Stephen Tashi

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    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

    Erland

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    Science Advisor

    I think I know what most of these rules are. But exactly how are negation introduction and elimination defined in your textbook?
     
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