I am studying mathematical logic for recreation and I am currently reading Mathematical Logic by Ian Chiswell and Wilfrid Hodges.
I have been stuck for the last couple of days with excercises that deals with arguments using not. And I don't quite seem to get it.
Take for example the first problem (problem 2.6.1, a):
$$ \vdash (\neg (\phi \land (\neg \phi))) $$
So when I first look at this problem and try to understand the question, my mind tells me that $\phi =$"I am a banana" and $\neg\phi$ = "I am not a banana". The next step is then to conclude that those statements cannot be true at the same time, I canno't be and at the same time be a banana. So $(\phi \land (\neg\phi))$ has to be negative. And I get the statement $(\neg(\phi\land (\neg \phi)))$. I really hope you guys say this is one way of interpreting the problem. But I have trouble using the natural deduction rules to prove the statement.
[This is my proof using natural deduction rules]
I have no idea if I am doing this right. Any help greatly appreciated.
Edit 2017-05-10: So I did the deduction according to the hint given to me by Mauro and my first try resulted in this:
My first try after hints from Mauro
The thing that I think I was stuggling with was "how can I discharge the assumption"? Beacuse my mind was telling me that by using $\neg I$ I was proving that $\neg\phi$ was true. But after some thought that is probably wrong, since $\phi$ comes from $(\neg(\phi\land (\neg \phi)))$. So I actually show that the assumption can be discarded, so I should be able to write:
What do you guys think, is this the right derivation?
Your reasoning is sound, by your derivation is not: you have to stay with the rules of Natural Deduction.
We have to start assuming $(\phi ∧ (¬ \phi))$ and "unpack" it with $\land$-E.
From $\phi$ and $¬ \phi$, we apply $\lnot$-E deriving $\bot$, followed by $\lnot$-I, which allows us to conclude with $\lnot (\phi ∧ (¬ \phi))$, discharging the assumption.