1. ## 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)

2. ## Re: proving logic

Originally Posted by Miafuller
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.

3. ## 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. .

4. ## Re: proving logic

Originally Posted by Miafuller
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) $\begin{gathered} P \wedge \neg Q \hfill \\ \neg (\neg P \vee Q) \hfill \\ \neg (P \to Q) \hfill \\ \end{gathered}$

5. ## Re: proving logic

Originally Posted by Miafuller
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.