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Mathematical Logic

  1. Sep 29, 2009 #1
    This from Alonzo Church's Mathematical Logic, been stuck on it for a week =(.

    1. The problem statement, all variables and given/known data
    14.3 Present a Formal Proof: p [tex]\Rightarrow[/tex] (q [tex]\Rightarrow[/tex] r) [tex]\Rightarrow[/tex] ((p [tex]\Rightarrow[/tex] q) [tex]\Rightarrow[/tex] r)


    2. Relevant equations
    3. The attempt at a solution

    A truth table has shown that the previous implication is a tautology therefore we should be able to prove it. The first half is easily obtained from modus ponens... p [tex]\Rightarrow[/tex] (q [tex]\Rightarrow[/tex] r) however I have not been able to get ((p [tex]\Rightarrow[/tex] q) [tex]\Rightarrow[/tex] r) any suggestions or guidance would be appreciated.
     
    Last edited: Sep 30, 2009
  2. jcsd
  3. Sep 30, 2009 #2
    I assume you are asked to show [itex]p \rightarrow (q \rightarrow r) \Rightarrow (p \rightarrow q) \rightarrow r[/itex].

    Proofs involving conclusions of the form "if A then B" are usually best proven by assuming the premises of the claim and A and then showing B is a consequence.

    Basically:

    Given [itex]p \rightarrow (q \rightarrow r), (p \rightarrow q)[/itex].

    Show [itex]r[/itex].

    As a hint, I'd suggest assuming p as a first step in the proof.

    --Elucidus

    P.S.: This method is valid due to the equivalence [itex](A \wedge B) \rightarrow C \equiv A \rightarrow (B \rightarrow C)[/itex]
     
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