Using a truth table to determine if valid or invalid

Question

I have some questions like if $P$ then $Q, P$ therefor $Q$ for example, how can you tell from writing your truth table if therefor $Q$ is valid or invalid? I mean I know its true because Modus Ponens tells me it is but that doesn't really help on more complex issues like;

p∨q
r
r → ¬q
−−−−−−
therefore p

I can make a table but what am I looking for in it to show me therefore p is valid or invalid.

Thanks

As per conversation with amwhy is this an accurate reflection of what you are trying to explain? I can see that the column with all true R is also true. Therefore its valid!

Answer 1

You need to check the following:

The argument is valid if and only if whenever you have a row in which (all) entries under the following columns evaluate to true,

$p\lor q$

$r$

$r\rightarrow \lnot q$

Then we must also have $p$ true.

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This is equivalent to checking whether the statement $$[(p \lor q) \land r\land (r\rightarrow \lnot q)]\rightarrow p$$ is a tautology (i.e., whether the statement evaluates to true for every possible truth-value assignment given to $p, q, r$. If it is a tautology, then the argument is valid:

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Can you see why the two approaches listed above are equivalent?

Answer 2

This isn't correct. A row on which the premises and the conclusion are all true only shows that the premises and conclusion could be all true, that is, that they are consistent. This doesn't make the argument valid, as you could have an invalid argument with such a row. What you should check for is the PRESENCE or ABSENCE of a row in which the premises are true while the conclusion is false. As it happens, the argument you asked about is valid, but your truth table is wrong so there such a row.