How to prove the validity of this argument using rules of inference? The premises are:


*

*(P $\rightarrow$ J) $\rightarrow$ ($\lnot$C $\rightarrow$ M)

*$\lnot$J $\rightarrow$ $\space$ $\lnot$P

*($\lnot$ J $\land$ E) $\rightarrow$ $\space$ $\lnot$C

*$\lnot$M $\rightarrow$ $\space$ P


The conclusion is: $\lnot$(J $\land$ $\space$ $\lnot$P) $\rightarrow$ C
You don't necessarily have to answer the question, but I would like to know whether there is such a thing as being too complex for proving with rules of inference. I believe checking the validity would be much easier with a truth tree. 
If it can be done with rules of inference, how would I go about doing it?
Thanks.
 A: The argument is not valid.
If we assume :

$ M := True$
$ E := False$
$ P := False$
$ J := False$
$ C := False$

we will have :

$( \lnot M \rightarrow P )$ is $True$ (because $\lnot M$ is $False$)
$( \lnot J \rightarrow \lnot P )$ i.e. $( P \rightarrow J )$ is $True$ (because $P$ is $False$)
$( (\lnot J \land E) \rightarrow \lnot C)$ is $True$ (because $\lnot C$ is $True$)
$(P \rightarrow J) \rightarrow (\lnot C \rightarrow M)$ is $True$  (because $C$ is $False$ and $M$ is $True$, so that the consequent is $True$).

We have showed that all the four premisses of the argument are $True$.
But we have that the conclusion is $False$, because with $J$ that is $False$ the antecedent of the conclusion is $True$, i.e.

$\lnot (J \land \lnot P)$ is $True$

and $C$ is $False$, so that the conditional

$( \lnot (J \land \lnot P) ) \rightarrow C )$ is $False$.

A: Maybe you should start by noticing that:
$\lnot J \rightarrow P \equiv P \rightarrow J$
and that $(p \rightarrow q) \land (q \rightarrow r) \equiv (p \rightarrow r)$
make use of that in premise 1, use the same argument a few times...
