If the TV is not on sale, I will not buy it?

Question

If the TV is on sale, I will buy the TV.

The TV is not on sale.
$\therefore$ I will not buy the TV.

$p$: The TV is on sale.

$q$: I will buy the TV.

First statement above: $p\implies q$

Second statement above: $\lnot p$

Third statement: $\lnot q$

Logical statement: $[(p\implies q) \land \lnot p] \implies \lnot q$

Is this a valid argument? Why or why not?

The argument "$p \to q$ and $\lnot p$, therefore $\lnot q$" is not valid and the formula $[(p \to q) \land \lnot p] \to \lnot q$ is not a tautology. — Mauro ALLEGRANZA, Feb 21 '16 at 17:27
To show that the formula is not a tautology, draw a truth-table: the entry for $p$ False and $q$ True will satisfy the antecedent of the conditional but not the consequent. — Mauro ALLEGRANZA, Feb 21 '16 at 17:30

score 2 · Accepted Answer · answered Feb 21 '16 at 17:28

Try drawing a truth table

$$ \begin{array}{c|c|c|c} p&q&p\implies q&\lnot p&\lnot q&(p\implies q) \land \lnot p&((p\implies q) \land \lnot p)\implies\lnot q\\ \hline 0&0&1&1&1&1&1\\ \hline 0&1&1&1&0&1&\color{red}{0}\\ \hline 1&0&0&0&1&0&1\\ \hline 1&1&1&0&0&0&1 \end{array} $$

If the TV is not on sale, I will not buy it?

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