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I am desperately trying to figure out the formal proof for this argument.

$$\begin{array}{r} A\lor B\\ A\lor C\\ \hline A\lor (B \land C) \end{array}$$

I am trying to apply the backwards method here. I am trying to infer A, in order to use vIntro in the last step and introduce the final disjunction. But I got stuck finding sufficient proof for A.

Any hint will be greatly appreciated. Thank you!

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    from $A\vee B$, $A\vee C$, you get $(A\vee B)\land(A\vee C)$ and from this you get $A\lor(B\land C)$. your approach cant work because if $A$ is false and both $B,C$ are true, the prop holds2011-12-13
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    Well, it isn't really true that you can infer A from what you're given, is it? There is a situation where the assumptions hold, but A is still false, right? What situation is that? What happens then?2011-12-13
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    Without knowing what formal proof system you are using, it is impossible to answer. The details of the formal proof will vary immensely from one system to another.2011-12-13
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    For example, in one system I am used to, every (substitution instance of a) tautology is an axiom. So you can use $(A \lor B) \to (A \lor C) \to (A \lor (B \land C))$ as an axiom, apply modus ponens twice to cut $A \lor B$ and $A \lor C$ and you will be left with the conclusion you want.2011-12-13
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    If your not familiar with a term like "formal proof system", I'll ask *exactly* what rules of inference can you use? Exactly, what axioms (if any... you might not have any) can you use?2011-12-13
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    As Doug hints, we don't know what formal proof system you are supposed to use. The course instructor DOES. Why not seek help there?2011-12-13
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    Such problems show up here now and then; but it is never defined what is meant by a "formal proof". Usually everyone is happy when the obvious has been regurgitated in the form of ten consecutive sentences in everyday language.2011-12-13
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    @Christian, "never" is a very strong term. [Here](http://math.stackexchange.com/questions/90340/proving-p-to-q-to-r-to-p-to-q-to-p-to-r) is a counterexample.2011-12-13
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    the OP keeps asking these kinds of (very basic) questions, as has been observed in another comment elsewhere. The problem is that he refuses to provide further information on his deduction method or book reference. He does not even bother to edit his own questions to provide extra information as requested by other members. It certainly looks like a lazy student asking us to do his homework. Prove me wrong (by the "backwards method").2011-12-16

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