[Math] Using a truth table to determine if valid or invalid

logicpropositional-calculus

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!enter image description here

Best Answer

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.

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:

enter image description here

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

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