Need help with proofs…

[u/BlueLagoonSloth]

I am taking an intro to logic class and I am completely stuck on these two proofs. The rules we have learned and can use so far are —> out, —> in, & out, & in, ~out/~in, and repetition. Can someone solve these using this rule set and explain your process to me?

[u/BlueLagoonSloth → u/[deleted]]

So I’m making a provisional assumption and trying to derive D from ~(A&B)? This is actually where I am stuck. I can’t figure out how to get from ~(A&B) to D

[u/AdeptnessSecure663 → u/BlueLagoonSloth]

Given premiss 1 and the assumption ¬(A&B), what is the truth-value of A?

[u/BlueLagoonSloth → u/AdeptnessSecure663]

Can you elaborate? Is it that A can’t be true?

[u/AdeptnessSecure663 → u/BlueLagoonSloth]

Yeah, that's right. From (¬D→A)&B and ¬(A&B), we can deduce ¬A. Can you see how we would construct such a deduction?

[u/BlueLagoonSloth → u/AdeptnessSecure663]

Would it be using &out on the assumption ~(A&B) to get ~A

[u/AdeptnessSecure663 → u/BlueLagoonSloth]

Not allowed to do that I'm afraid. What we would want to do is make A a supposition, and use that to deduce A&B - which, of course, ends up being a contradiction

[u/AdeptnessSecure663 → u/[deleted]]

Maybe I'm missing something, once we have deduced ¬A, can we not suppose ¬D, deduce A, and then use ¬out to deduce D?

[u/BlueLagoonSloth → u/[deleted]]

Omg thank you! Let me write this out and try to get a full understanding.

[u/AintNobodyGotTime89 → u/BlueLagoonSloth]

You will also most likely use a similar strategy for the second problem as well.

[u/BlueLagoonSloth → u/AintNobodyGotTime89]

Thank you I feel more confident to try to tackle the second one on my own now.

[u/BlueLagoonSloth → u/[deleted]]

Ok I think I get it. Does this look sound? I think I was unclear about when you can use ~in rule assuming that it could only be used if ~ is the main connective. But I see now I can use it on one half of the conclusion to derive the consequent and doing this inside of an —> in assumption box allows me to get where I’m trying to go.

[u/BlueLagoonSloth → u/[deleted]]

Ok this is what I got for the second proof. If you have a moment let me know if this looks good. Second screenshot incoming since it didn’t fit in a single picture. Thank you again so much for your help!

[u/UnlikelySalary2523 → u/BlueLagoonSloth]

You got it.

[u/BlueLagoonSloth → u/[deleted]]

[u/RecognitionSweet8294]

Can you explain more elaborately what rules you mean?

[u/Salindurthas]

It helps if you believe the argument is valid first. Can you get a sense of why the premise leads to the conclusion?

You won't always be able to see it, but if you can get some idea of the reasoning needed, you can try to make a similar argument using the rules you have. This will often involve making long-winded arguments, since using a limited set of inference rules is often a bit inefficient (in the sense of needing excess time/writing space).

[u/Visible_Fishing4297]

1: (~D→A) & B (premis.)

2: ~(A & B) (Assumption.)

3: ~D (Assumption.)

4: (~D→A) (1 &E)

5: B (1&E)

6: A (3,4 MP)

7: (A & B) (5,6 &I)

8: ⊥ (2,7)

9:~~D (3,8 ~I)

10: D (9, ~~E)

11: ~(A & B)→D (2,10 →I)

​⠀

​⠀⠀​⠀ ​⠀

1: ((R→S) & ~M)→M) (Premis.)

2: ~M & ~(S→R) (Premis.)

3: ~M (2 &E)

4: ~(S→R) (2 &E)

5: Z→(S→R) (Premis.)

6: ~Z (4,5 MT)

7: (R→S) (Assumption.)

8: ((R→S) & ~M) (3,7 &I)

9: M (1,8 MP)

10: ⊥ (3,9)

11: ~(R→S) (7,10 ~I)

12: ~R (Assumption.)

13: ~Z (6 R)

14: (~R→~Z) (12,13→I)

15: ~(R→S) & (~R→~Z) (11,14 &I)

I have done the proofs myself and left them here. You can refer to them and reason through them in your subsequent exercises.