Testing whether Argument is valid or not

Question

I am to determine if argument is valid by making truth table

ATTEMPT

Let

W= Warning lights will come on
P= Pressure is high
R=Relief valve is clogged

Then i have premises as

W $\leftrightarrows$ P AND R ,where the symbol indicates bi conditional statement (1st Premise)

Negatition R (2nd Premise)

Therefore W $\leftrightarrows$ P (Conclusion)

Now i made truth table as usual with 8 rows and written all other stuff. In first row where W,P,R are false , the premises and conclusion is coming out to be true which makes argument valid. But textbook states that it is invalid. I would like to know where i am going wrong

Thanks

As $R$ is false. "$P$ and $R$" is false too. And you get that $W$ is always false. — quid, Jul 31 '15 at 15:33
Is there any reason why you are trying to use a truth table here? It doesn't seem at all necessary. — Daniel W. Farlow, Jul 31 '15 at 15:40

score 1 · Answer 1 · answered Jul 31 '15 at 15:33

1

The claimed conclusion is not $W\leftrightarrow R$ as you wrote but rather it is $W\leftrightarrow P$. From the given $\neg R$ we can conclude that $P\land R$ is false and hence the equivalent $W$ is also false. If the claim were correct,we could infer $\neg P$, but we certainly can't.

Incidentally, your mistake in encoding the claim as $W\leftrightarrow R$ does give a valid conclusion: As seen above, $W$ is false and as both $W$ and $R$ are false, $W\leftrightarrow$ is true.

answered Jul 31 '15 at 15:33

Hagen von Eitzen

374,180

no no it was a typo. i meant P and not R – Taylor Ted Jul 31 '15 at 15:35
I have uploaded my work, where is mistake? – Taylor Ted Jul 31 '15 at 15:40
The premises i have ticked are the ones should be consistent with each other, aren't they? – Taylor Ted Jul 31 '15 at 15:48
In the third line the premises $\neg R$ amd $ are both true, but the conlusion is false – Hagen von Eitzen Jul 31 '15 at 15:55
But i was talking about first line – Taylor Ted Jul 31 '15 at 15:56

Aymane Fihadi · Accepted Answer · 2015-11-08T12:04:51.227

I don't see why you want to solve your problem using the truth table.

Firstly, here is an intuitive approach that I believe the first thing to do when you have such problems.

$$ ``\mbox{Warning lights is on}" \Longleftrightarrow ``\mbox{Pressure is high}" \wedge \ `` \mbox{Relief valve is clogged }" $$

is equivalent to :

$$ ``\mbox{Warning lights is } \color{#C00}{off} " \Longleftrightarrow `` \mbox{Pressure is } \color{#C00}{not} \mbox{ high}" \vee \ `` \mbox{Relief valve is } \color{#C00}{not} \mbox{ clogged }" \tag{P} $$

Now, we consider the statements $(P)$ to be true and that $``\mbox{the Relief valve is not clogged}"$, that implies that Warning lights is off ( regardless of the pressure). So the conclusion is obviously invalid (because the pressure can be too hight and the warning still off).

Now if the table is required by the question, you have to add an arrow in your table that contain the truth values of

$$ \left( (W \Longleftrightarrow R \wedge P) \wedge \lnot R \right) \Longrightarrow \left( W \Longleftrightarrow P \right) $$

Testing whether Argument is valid or not

2 Answers2