Help with indirect logic proof please!

  1. Using the five axioms below prove: p→q

    A1: p→~y
    A2: ~r→q
    A3: p→~z
    A4: x→ q or z
    A5: r→x or y

    Do I have to take the contrapositive of some of the axioms to begin this proof?
  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,260
    Staff Emeritus
    Science Advisor

    Yes, that would be the simplest thing to do. The very first "axiom" gives you p-> ~y but there is no "~y-> " so you cannot continue directly. However, you do have "A5: r->x or y which has contrapositive ~(x or y)= (~x) and (~y)->~r and then both "A2: ~r-> q" and "A4: x-> q or z".
  4. Am I on the right track with this?

    Conclusions Justifications
    1. p Given
    2. ~z or ~y All cases
    3. ~z Case 1
    4. ~x A4
    5. ~r A5
    6. q A2
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