# How to prove the validity of this argument using rules of inference?

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.

• When you say 'The argument is', do you mean 'The premises are'? What are your inference rules? – Git Gud Jan 27 '14 at 1:53
• Yes, those are the premises. The inference rules are here: homepages.ius.edu/rwisman/C251/html/ch01/t01_5_001.jpg – user124071 Jan 27 '14 at 1:56
• Please, are you sure that the formulae are written correctly ? – Mauro ALLEGRANZA Jan 27 '14 at 9:24

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 $False$.

• This is the correct answer, however, wouldn't this just be guesswork? Keep assuming truth values for the propositions until the premises end up all true and the conclusion false? Unless there's a method to go about doing this. Thanks. – user124071 Jan 27 '14 at 12:49
• I used tableaux method (that is a proof system) to check whether $(1+2+3+4) \rightarrow [\lnot (J \land \rightarrow P) \rightarrow C)]$ is a tautology. The procedure gives two answer: (i) it is not; so, because in propositional logic : $(A \vDash B)$ iff $(\vDash A \rightarrow B)$, we can conclude that the last formula in your question is not a logical consequence of $(1+2+3+4)$. (ii) The method gives us a counter-example, in case that the formula is not a tautology: this is how I've found the boolean valuation that satisfy all the premises but not the conclusion. – Mauro ALLEGRANZA Jan 27 '14 at 13:12

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...