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    Proofs using Modus Pollens, Modus Tollens

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    Question:
    using the four rules of inference presented (mp,mt,ds and hs), construct a proof for the following valid argument in the answer box below.

    Note:

    A).
    1. w>(pvc)..........premise
    2. ~p.....................premise
    3. w............./c.....premise/conclusion

    B).
    1. H>(D<>A)......................Premise
    2. Mv(R>M).......................Premise
    3. RvH.................................Premise
    4. ~M-------/D<>A.......Premise/conclusion

    C).
    1. ~Dv(L>~F)...................Premise
    2. ~D>~F............................Premise
    3. ~~F......................../~L...........Premise/Conclusion

    D).
    1. (Q>~J)>(M>~D)...................Premise
    2. Q>M.........................................Premise
    3. M>~J........................../Q>~D...... Premise/Conclusion

    E.
    1. ~(~E&~N)>T ............................premise
    2. G>(NvE) .............................premise
    3. (~~Ev~~N)>T ....................premise
    4. (~~Nv~~E)>T ................... Premise
    5. (NvE)> .....................................Premise/conclusion

    F.
    1. (D>C)>(NvW) .......................... premise
    2. D>S...........................................Premise
    3. S>C...........................................Premise
    4. ~N................/ W.........................Premise/conclusion

    G.
    1. ~C>(Cv(J>D)) .............................Premise
    2. C>(C&U) .....................................Premise
    3. ~(C&U)........................................Premise
    4. ~D.................../ ~J......................Premise

    H.
    1. (R>L)>(L>~F)...............................Premise
    2. ~Fv(R>L).......................................Premise
    3. ~~F................/ ~R.........................Premise/Conclusion

    I.
    1. Cv(H>R) ..........................................Premise
    2. Sv(R>E)...........................................Premise
    3. ~ C...................................................Premise
    4. ~ S........................./ H>E.................Premise/conclusion

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    https://brainmass.com/math/logic/proofs-using-modus-pollens-modus-tollens-626379

    Solution Preview

    Format of proof is sequence of "Step [Reason]" statements.

    A).
    1. w>(pvc)..........premise
    2. ~p.....................premise
    3. w............./c.....premise/conclusion

    Proof:
    1. w > (pvc) [Premise]
    2. w [Premise]
    3. (p v c) [Modus ponens on 1 and 2]
    4. ~p [Premise]
    5. c [Disjunctive syllogism of 3 and 4; Conclusion]

    B).
    1. H>(D<>A)......................Premise
    2. Mv(R>M).......................Premise
    3. RvH.................................Premise
    4. ~M-------/D<>A.......Premise/conclusion

    Proof:
    1. M v (R > M) [Premise]
    2. ~M [Premise]
    3. (R > M) [Disjunctive syllogism of 1 and 2]
    4. ~R [Modus tollens of 2 and 3]
    5. R v H [Premise]
    6. H [Disjunctive syllogism on 4 and 5]
    7. H > (D <> A) [Premise]
    8. (D <> A) [Modus ponens on 6 and 7; Conclusion]

    C).
    1. ~Dv(L>~F)...................Premise
    2. ~D>~F............................Premise
    3. ~~F......................../~L...........Premise/Conclusion

    Proof:
    1. ~D > ~F [Premise]
    2. ~~F [Premise]
    3. ...

    Solution Summary

    Stepwise proof for given arguments is provided as a sequence of "Step [Reason]" statements.

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