[Math] How to prove the validity of this argument using rules of inference

discrete mathematicslogic

The premises are:

  1. (P $\rightarrow$ J) $\rightarrow$ ($\lnot$C $\rightarrow$ M)
  2. $\lnot$J $\rightarrow$ $\space$ $\lnot$P
  3. ($\lnot$ J $\land$ E) $\rightarrow$ $\space$ $\lnot$C
  4. $\lnot$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.

Best Answer

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 $False$.

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