I am having some difficulties understanding logical arguments. I was taught that the notion of a valid argument is formalized as follows:
"An argument $P_1, P_2,\cdots , P_n ⊢ Q $ is said to be valid if $Q$ is true whenever all the premises $P_1, P_2,\cdots , P_n$ are true. An argument which does not satisfy this condition is a fallacy."
But (according to my book) the argument $$p \rightarrow q,\quad q\;⊢ \;p\; $$ is a fallacy, but I don't see why. If we construct a truth table
$p$ | $q\;$ | $p \rightarrow q$ |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
we see that when $p$ and $q$ are true, $p \rightarrow q$ is also true (line 1). So then, why isn't this valid? I do know that for an argument to be valid, with premises $P_1, P_2,\cdots , P_n$, the proposition $$(P_1 \land P_2 \land \cdots \land P_n) \rightarrow Q$$ should be a tautology, but that doesn't dispel my confusion.
What am I missing?
Best Answer
In the third line of the truth table, $q$ and $p \to q$ are both true and $p$ is false. So $q, p\to q \vdash p$ is not a valid argument.