[Math] Using a truth table to show that an argument form $(p\rightarrow q) \land q \rightarrow p$ is invalid.

discrete mathematicslogic

Use a truth table to show that the following argument form is invalid.

p$\rightarrow$q

q

∴p

My attempt:
I made a truth table as below, but in what column and row can I find the argument form is invalid?

$$\begin{array}{c|c|c|c}
\space\space\space\space(p \rightarrow q)&\land &q& \rightarrow& p
\\\hline
T\space T\space T&\space T&\space T&T&T
\\TFF&F&F&T&T\\F\color {red} {T}T&T&\color {red} {T}&F&\color {red} {F}\\FTF&F&F&T&F
\\\hline
step 1\space2\space 1&\space3&\space1&\space4&\space1
\end{array}$$

[EDIT] I think it's found invalid in the row where I wrote in red since it shows the conclusion p is false when all the premises (p$\rightarrow$q) and $q$ are true.

FYI
""A row of the truth table in which all the premises are true is called a critical row. …You can show that an argument is invalid by constructing a truth table for the argument form and finding at least one critical row in which all the premises are true but the conclusion is false."

Source: Discrete Mathematics with Applications by Susanna Epp

Best Answer

Note that the question is not about the formula $$(p\to q)\land q \rightarrow p $$ but about the proposed inference $$ (p\to q), q \vdash p $$ This is important because $\vdash$ is not a connective and shouldn't have a truth value.

In order to use a truth table, as requested by the question, you should have "input" columns $p$ and $q$, and "output" columns $p\to q$ as well as $q$ and $p$:

$$ \begin{array}{cc|ccc} p & q & p\to q & q & p \\ \hline F & F & T & F & F \\ T & F & F & F & T \\ F & T & \cdots \\ T & T & & \cdots \end{array} $$

If you can find a row where the $p \to q$ and $q$ outputs are both T and the $p$ output is $F$, then your truth table shows that the inference $(p\to q), q\vdash p$ is not valid.