1. ## Sentential Derivation (logic) help please

I really can't figure this derivation out.

Assumptions:
1. $\neg (F \vee G)$ iff $(\neg F \supset \neg F)$
2. $\neg G \supset F$

I need to derive contradicting sentences to show that it is inconsistent, I can tell it is since $\negF\supset\negF$ is always true, which means ~(F v G) is true, so F and G both have to be false individually, which would contradict the $\neg G \supset F$ since ~G would be true and F is false.

However, I don't know how to formulate this in SD (going by The Logic Book 5th edition by Bergmann). Can anyone help?

2. So, you're asked to produce a contradiction out of the following assumptions:

1. $\neg(F\vee G)\iff(\neg F\to\neg F)$
2. $\neg G\to F.$

Forgive me for changing notation, but I hate the horseshoe for implication. I can never remember which way it's supposed to go!

I would go about it this way:

1. First, assume $F$.
2. Then, show that $\neg F\to\neg F$.
3. Second, assume $\neg F$.
4. Then, show that $\neg F\to\neg F$.
5. The law of the excluded middle (which in natural deduction you can derive from scratch!) tells you that $F\vee\neg F$.
6. Therefore, you can conclude that $\neg F\to \neg F$.

Alternatively, you can go like this:

1. $F\vee\neg F$. Law of excluded middle.
2. $\neg F\to \neg F$ by equivalence with step 1.

I'm just following the outline of the (valid) proof you've already provided. I think once you get started, you'll be able to continue.

So, how would you write this up so far in a two-column proof? And how would you continue?

3. Originally Posted by swtdelicaterose
I really can't figure this derivation out.

Assumptions:
1. $\neg (F \vee G)$ iff $(\neg F \supset \neg F)$
2. $\neg G \supset F$

I need to derive contradicting sentences to show that it is inconsistent, I can tell it is since $\negF\supset\negF$ is always true, which means ~(F v G) is true, so F and G both have to be false individually, which would contradict the $\neg G \supset F$ since ~G would be true and F is false.

However, I don't know how to formulate this in SD (going by The Logic Book 5th edition by Bergmann). Can anyone help?