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Thread: proving logic

  1. #1
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    proving logic

    Hi everyone, does anyone know what the solutions to these questions would be? I am really struggling with this particular piece of homework, so would appreciate any help or explanations. I have to prove that the following statements are valid in proven logic. Thank you
    = not → = if/ then
    (P & Q) : (Q & P) (6)
    P → Q: Q → P (6)
    : (P →Q) → (Q →P) (5) Principle of transposition
    Q → R : (Q →P) →(P →R) (9)
    (P & Q) →R : R →(P →Q) (11)
    P: [((Q → R) →P)] →[( R →Q)] (9)
    P, Q: (P →Q) (6)
    P, P : Q (8)
    : P → (P →Q) (10) Law of Dun Scotus
    P → P : P (11)
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  2. #2
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    Re: proving logic

    Quote Originally Posted by Miafuller View Post
    Hi everyone, does anyone know what the solutions to these questions would be? I am really struggling with this particular piece of homework, so would appreciate any help or explanations. I have to prove that the following statements are valid in proven logic. Thank you
    = not → = if/ then
    (P & Q) : (Q & P) (6)
    P → Q: Q → P (6)
    : (P →Q) → (Q →P) (5) Principle of transposition
    Q → R : (Q →P) →(P →R) (9)
    (P & Q) →R : R →(P →Q) (11)
    P: [((Q → R) →P)] →[( R →Q)] (9)
    P, Q: (P →Q) (6)
    P, P : Q (8)
    : P → (P →Q) (10) Law of Dun Scotus
    P → P : P (11)
    First most of the notation is completely standard.
    But I have not idea how or what the " : " for/means.
    What do the (#) mean? Do they refer to numbered statements? If so what are that the propositions?

    Moreover, I don't see that actual question.
    Please state it clearly.
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  3. #3
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    Re: proving logic

    Sorry I should have said, the numbers in the brackets represent how many lines each answer should be. The exercise is taken from Paul Tomassi's book 'Logic'. I have attached a picture of the exercise which will hopefully make it clearer. proving logic-img_4055.png. proving logic-img_4054.png
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    Re: proving logic

    Quote Originally Posted by Miafuller View Post
    Sorry I should have said, the numbers in the brackets represent how many lines each answer should be. The exercise is taken from Paul Tomassi's book 'Logic'. I have attached a picture of the exercise which will hopefully make it clearer.
    The best I can do is show #s 5, 7. You have to fill in the reasons your text uses.
    I gather that $A:B$ means to assume $A$ and prove $B$ from that.

    5) $\begin{gathered} (P \wedge Q) \to \neg R \hfill \\ R \to \neg (P \wedge Q) \hfill \\ R \to (\neg P \vee \neg Q) \hfill \\ R \to (P \to \neg Q) \hfill \\ \end{gathered}$

    7) $\begin{gathered} P \wedge \neg Q \hfill \\ \neg (\neg P \vee Q) \hfill \\ \neg (P \to Q) \hfill \\ \end{gathered}$
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    Re: proving logic

    Quote Originally Posted by Miafuller View Post
    Hi everyone, does anyone know what the solutions to these questions would be? I am really struggling with this particular piece of homework, so would appreciate any help or explanations. I have to prove that the following statements are valid in proven logic. Thank you
    = not → = if/ then
    (P & Q) : (Q & P) (6)
    P → Q: Q → P (6)
    : (P →Q) → (Q →P) (5) Principle of transposition
    Q → R : (Q →P) →(P →R) (9)
    (P & Q) →R : R →(P →Q) (11)
    P: [((Q → R) →P)] →[( R →Q)] (9)
    P, Q: (P →Q) (6)
    P, P : Q (8)
    : P → (P →Q) (10) Law of Dun Scotus
    P → P : P (11)
    1. ~~(P & Q)
    2. Show ~~(Q & P)
    3. P & Q 1, DN
    4. P 3, S
    5. Q 3, S
    6. Q & P 4, 5, Adj.
    7. ~~(Q & P) 6, DN
    Q.E.D.

    1. ~P → ~Q
    2. Show Q → P
    3. Q A
    4. P 1, 3, MT
    5. Q → P 3-4, CONDITIONAL INTRODUCTION
    Q.E.D.

    ...and so on and on.
    They are all easy.
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