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Valid and Invalid Arguments

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data

    "Watson, I have uncovered the following facts:

    * If Mrs Smith is lying then Moriarty has not escaped
    * Either Moriarty is dead or he is really Jones
    *If Moriarty is really Jones then he has escaped.
    *I am convinced Mrs Smith is lying.

    "Good Lord Holmes," replied Dr Watson, "what can you make of all this?"
    "Elementary my dear Watson, Moriarty is dead."

    Is Holmes correct? Justify your answer.



    2. Relevant equations



    3. The attempt at a solution

    So I started by

    Let

    A = Mrs. Smith is lying.
    B = Moriarty has escaped.
    C = Moriarty is really Jones.
    D = Moriarty is dead.

    so then we have

    A→~B
    D→~C
    C→B
    A
    _______
    ∴ D

    Is this the correct way of setting out the Argument.... I was a little confused as how to approach the either in the second line.
    And the last line before the conclusion should that just be A?

    After this am I supposed to use a contradiction to prove it??
    ie do I assume that D is false??
    This is really confusing me!

    Help Please..
     
  2. jcsd
  3. Apr 28, 2012 #2
    That looks alright to me, except if you say "A" therefore "D" you might want to elaborate out a bit more.

    So if you say A, then you go back and look at what you've constructed:

    A

    A->~B [Mr. Smith is lying, therefore Moriarty has not escaped]
    ~B->~C [Moriarty has not escaped, therefore Moriarty is not Jones]
    ~C->D, [Moriarty is not Jones, therefore he is dead]

    There are several ways to do this, this is essentially propositional logic.
     
  4. Apr 28, 2012 #3
    So is that all you have to say...
    Therefore Holmes is correct.
    ??
     
  5. Apr 28, 2012 #4

    I like Serena

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    Homework Helper

    The "either-or" is a so called exclusive or:
    http://en.wikipedia.org/wiki/Xor


    Either D and not C, or not D and C.

    Symbollically: D.~C + ~D.C


    Luckily the result is the same.
     
  6. Apr 28, 2012 #5
    Ok thanks!
     
  7. May 1, 2012 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Here is how I would do this.

    A * If Mrs Smith is lying then Moriarty has not escaped
    B * Either Moriarty is dead or he is really Jones
    C *If Moriarty is really Jones then he has escaped.
    D *I am convinced Mrs Smith is lying.

    Since D, Mrs Smith is lying. So from A, Moriarty has not escaped. C is equivalent to its "contrapositive" which is "If Moriarty has not escaped, then he is not really Jones. Since "he is really Jones" is false, the other possibility in B, "Moriarty is dead" must be true.
     
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