# Negation of the Rule of Implication proof

tried forever to figure out a solution to this problem. It's based on the rule of Material Implication with a negation in front of both sides. Namely the premise is ~(A>B) with the goal solution being
~(~AvB).

I know how to derive the proof if there were no negations in front, but I am utterly stumped with this one. I'm thinking you want to assume (~AvB) and try to derive a contradiction by deriving (A>B) so you can use ~I as your final step, but I have no clue how this would be done. The negation in front of the premise means it is pretty much useless to work with and I'm always left with some assumptions in the dependencies if I do argue to the goal.

Let me know what any of you guys think of this. Also we are not allowed to use any sequent or theorem introductions like Rule of Implication, DeMorgan's, Law of Excluded Middle, etc. Only the basic operators like assumptions, introduction and elimination of ~,>,<>,v,&, double negation, and stuff like that.

## 1 Answer

As the goal has a negation as its main logical connective, you would need one of the Introduction rules. In particular, Negation Introduction. So, a basic proof skeleton (in Fitch-style) would be: Can you fill in the blanks ?

• no, if we assume ~A and assume B I'm not sure how you would derive A and ~B respectively to contradict each one. Even if you just assume both A and ~B and do this you won't have the premise in the final dependency either, so I'm confused on how the premise is utilized as well. – Ryan Jul 29 '20 at 19:08
• so I figured out you can use the paradoxes of material implication to get from ~A and B to (A>B), but unfortunately I'd have to prove those as well which I'm not sure how to do – Ryan Jul 29 '20 at 19:20
• - @Ryan, are you allowed to use Disjunction Elimination ? This is the rule you need. Tell me if you need further help completing the proof. – F. Zer Jul 29 '20 at 19:58
• Why do you say that we would need to derive A and ~B respectively to contradict each other ? – F. Zer Jul 29 '20 at 21:14
• @Ryan, You do not need to derive a contradiction of each assumption. You only need to derive any contradiction under each assumption. Deriving `A > B` suffices (as it contradicts the premise). In each case, you can derive this by another subproof : a Conditional Proof. – Graham Kemp Aug 20 '20 at 4:28