# Is it possible to provide a proof of the following arguments?

1. G & ¬ H
2. ¬ H ⇒ H

Therefore,

1. S & I

Is it possible to prove the validity of this argument without having any information of S or I in the premises?

1. H

Therefore,

1. S ⇒ (B ⇒ H)

Similarly, is it possible to prove the validity of this argument without any information of S or B in the premises?

I'm wondering if there is an error in the problems. Any help would be appreciated!

• Welcome to Philosophy! Sharing a little bit about the context of your question can help improve the likelihood of getting a great answer. Maybe you could share what you have found out so far, or indicate where you are running into trouble? – Joseph Weissman Oct 29 '14 at 2:25

## 2 Answers

It's absolutely possible to prove both. I don't know your set of inference rules, so I'm going to make it easy for me by including material implication to solve the second one and to just let things follow directly from a contradiction for the first:

``````1. G & ¬ H   A
2. ¬ H ⇒ H   A
3. ¬ H       &E1
4. H         MP2,3
5. H & ¬ H   &I 3,4
6. S & I     From contradiction at 5.
``````

If you aren't allowed to just draw any conclusion from a contradiction, then you bracket this by assuming : ¬ ( S & I) as a subproof as step 3, drawing the contradiction (now 3-6), then proof by contradiction using the subproof to get S & I

``````1. H        A
2. ¬ B v H  vI 1
3. B ⇒ H    Material Implication 2
4. ¬ S v ( B ⇒ H ) vI 3
5. S ⇒ ( B ⇒ H ) Material Implication 4
``````
• Thanks for the help! I really wanted to know if it was possible more than anything. I've got some reading to do to figure my way through this. – Brandon Oct 29 '14 at 2:46

Yes. For the first, observe that ~H -> H is simply H v H, which is simply H. That contradicts the right conjunct of (1), viz. ~H. So you can conclude anything you want. This is known as the explosion principle.

For the second, observe that S -> (B -> H) is simply ~S v (B -> H), which is simply ~S v (~B v H), which is equivalent to ~S v ~B v H, which follows from H by disjunction-introduction.

• Beaten by 5 seconds by the Hunan logic monster. – virmaior Oct 29 '14 at 2:28