# Thread: Help with conditional proof (natural deduction)

1. ## Help with conditional proof (natural deduction)

Hey guys, I've been banging my head against the wall trying to figure this one out. It's the first problem in my problem set (so it should be one of the easier ones), but I have taken many dead end routes trying to figure this out. The problem is as follows:

1. $(\sim U \rightarrow Z) \rightarrow (Y \wedge \sim B)$

2. $\sim U \rightarrow (H \vee Z)$

3. $H \rightarrow Z$

/ $\sim B$

So far I've been trying to use a conditional to prove $\sim U \rightarrow Z$ so that I can use MP to get $Y \wedge \sim B$, and by simplification $\sim B$ , but everything I've tried thus far is a dead end. Any and all help would be greatly appreciated! Thanks!

2. ## Re: Help with conditional proof (natural deduction)

Are these three separate statements that you are supposed to combine to prove $\sim B$?

3. ## Re: Help with conditional proof (natural deduction)

Originally Posted by phys251
Are these three separate statements that you are supposed to combine to prove $\sim B$?
Yes, using the 18 rules of natural deduction

4. ## Re: Help with conditional proof (natural deduction)

I feel like I'm getting a bit closer. When assuming $\sim U$ , I'm able to create the statement $\sim H \rightarrow Z$, giving me both $\sim H \rightarrow Z$ and $H \rightarrow Z$. Abstractly, I can see how this would lead to being able to prove $\sim U \rightarrow Z$, since $H$ and $\sim H$ both imply $Z$, but I'm unable to put it down concretely in terms of the rules of natural deduction.

5. ## Re: Help with conditional proof (natural deduction)

If you are completely lost, you could try a truth table. Note, however, that since you have five variables, you would have to have $2^5 = 32$ entry rows in your table.

Alternatively, don't forget properties such as contrapositives.

6. ## Re: Help with conditional proof (natural deduction)

Originally Posted by phys251
If you are completely lost, you could try a truth table. Note, however, that since you have five variables, you would have to have $2^5 = 32$ entry rows in your table.

Alternatively, don't forget properties such as contrapositives.
Unfortunately, I don't think a truth table will do much here. The nature of this proof is that it is a tautology, since regardless of the input truth values, we will always be able to deduce $\sim B$.

7. ## Re: Help with conditional proof (natural deduction)

Ok guys, I decided to scrap this one and do another problem of similar form. It seems like I wasn't creating a construction dilemma. For anyone wondering, this was the proof: (note that the symbols used for implication and conjunction are different. I don't know why, but they are)

8. ## Re: Help with conditional proof (natural deduction)

Originally Posted by nateneal
Ok guys, I decided to scrap this one and do another problem of similar form. It seems like I wasn't creating a construction dilemma. For anyone wondering, this was the proof: (note that the symbols used for implication and conjunction are different. I don't know why, but they are)

There is a long tradition in logic of people find new and more reasonable ways to teach the subject. Natural Deduction is just one more example. In fact, it is one of the more obscure efforts. Thus you will be lucky to find someone here willing to take this up, not I.

9. ## Re: Help with conditional proof (natural deduction)

Originally Posted by Plato
There is a long tradition in logic of people find new and more reasonable ways to teach the subject. Natural Deduction is just one more example. In fact, it is one of the more obscure efforts. Thus you will be lucky to find someone here willing to take this up, not I.
I've actually taken discrete math courses before and I agree. I'm taking this class as more of an easy A and the emphasis on ND was definitely a surprise. I've never taken a class so picky when it comes to these kinds of proofs

10. ## Re: Help with conditional proof (natural deduction)

Originally Posted by nateneal
I've never taken a class so picky when it comes to these kinds of proofs
I have been out of active research for c10 years. But before that, your complaint about being 'picky' was the most frequent that I heard. Some traditional things just always work. But the need to publish drives the need to find the new.

11. ## Re: Help with conditional proof (natural deduction)

Here is a way to create the conditional : $\sim U \rightarrow Z$

1. $(\sim U \rightarrow Z) \rightarrow (Y \wedge \sim B)$

2. $\sim U \rightarrow (H \vee Z)$

3. $H \rightarrow Z$

4. $\sim U$.................................................. .................................................. ...ACP

5. $\sim Z$.................................................. ........................................Assumption for contradiction

6. $\sim Z\rightarrow\sim H$.................................................. .......................3 CONTRAPOSITIVE

7. $\sim H$.................................................. .................................................5 6 MP

8. $\sim H\wedge \sim Z$.................................................. ................................5 7 Conjunction

9. $\sim(H\vee Z)$.................................................. ..........................................8 D.Morgan

10.. $(H\vee Z)$.................................................. ..........................................2 4 MP

11. $\sim(H\vee Z)\wedge(H\vee Z)$.................................................. .......................9 10 Conjunction

12. $Z$.................................................. .................................................. ..........5 TO 11 CONTRADICTION

13. $\sim U \rightarrow Z$.................................................. .................................4 to 12 Conditional proof

12. ## Re: Help with conditional proof (natural deduction)

Originally Posted by nateneal
Ok guys, I decided to scrap this one and do another problem of similar form. It seems like I wasn't creating a construction dilemma. For anyone wondering, this was the proof: (note that the symbols used for implication and conjunction are different. I don't know why, but they are)

Step 11 is totally wrong