Establish the validity by means of a truth table

Question

The following argument is valid, establish the validity by means of a truth table. Determine which rows of the table are crucial for assessing the validity of the argument.

$[p \wedge (p \rightarrow q) \wedge r] \rightarrow [(p \vee q) \rightarrow r]$

how do I know which row of the truth table is valid?

Answer 1

An implication $A \rightarrow B$ can only produce $F$, if $A= T$ and $B=F$.

So, the "crucial" rows in the truth table would be those where

• $A = p \wedge (p \rightarrow q) \wedge r = T$

There, the expression $B=(p \vee q) \rightarrow r$ must be $T$ as well for the argument to be valid.

Answer 2

In my eyes it makes little sense to say that some rows of the truth-table would be more 'crucial' than others.

It makes even less sense to say that some specific row(s) make it valid.

And completely incorrect is to say, as you do, that you would be looking for 'valid rows'

Finally, what you have here is not an argument, but rather a single statement; a conditional with an antecedent ('if') and a consequent ('then').

But fine, I'll play along and consider this to be an 'argument', and where we are asked to see if the 'then' part logically follows from the 'if' part. This would be true if it is not the case that there is a row where the 'if' part is true, but the 'then' part is false.

As such, one could consider the row(s) where the 'if' part is true to be the 'crucial' row(s), as it is in only those rows where we need to make sure the 'then' part is also true ... any rows where the 'if' part is false is not a row we are interested in. I assume that this is what the question has in mind, because this is how many people think about this.

However, note that one can likewise say that the rows where the 'then' part is false that are 'crucial', for in that case we need to make sure that the 'if' part is also false, lest we are dealing with an invalid 'argument' ... in our search as to whether or not there is a row with a true 'if' part and a false 'then' part, we can ignore any rows where the 'then' part is true.

So this is what I meant by saying that it doesn't make much sense to say that some rows are more 'crucial' than others.