[Math] Testing whether Argument is valid or not

logic

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

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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)

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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

Best Answer

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

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

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