After learning direct rules of inference, it is tempting to treat every conditional pattern as a valid deduction. Invalid conditional reasoning studies two common mistakes: affirming the consequent and denying the antecedent. These errors look similar to Modus Ponens and Modus Tollens, but their logical forms are different. In this lesson, invalid conditional reasoning will be identified by form, by examples, and by counterexamples. :::definition[Converse Error] Let $p$ and $q$ be statements. The argument form $$\begin{equation*} p\to q,\ q\ \therefore\ p \end{equation*}$$ is called the converse error or the fallacy of affirming the consequent. This is not a valid rule of inference. ::: :::definition[Invalid Converse Form] The implication $$\begin{equation*} [(p\to q)\land q]\to p \end{equation*}$$ is not a tautology. Thus, $$\begin{equation*} p\to q,\ q\ \therefore\ p \end{equation*}$$ is invalid. ::: :::example[Converse Fallacy] Let $p$ mean “Margaret Thatcher is president of the United States,” and let $q$ mean “Margaret Thatcher is at least $35$ years old.” The argument $$\begin{equation*} p\to q,\ q\ \therefore\ p \end{equation*}$$ has the converse form. Although $p\to q$ may be true and $q$ may be true, it does not follow that $p$ is true. Hence, the argument commits the converse error. ::: :::definition[Inverse Error] Let $p$ and $q$ be statements. The argument form $$\begin{equation*} p\to q,\ \neg p\ \therefore\ \neg q \end{equation*}$$ is called the inverse error or the fallacy of denying the antecedent. This is not a valid rule of inference. ::: :::definition[Invalid Inverse Form] The implication $$\begin{equation*} [(p\to q)\land\neg p]\to\neg q \end{equation*}$$ is not a tautology. Thus, $$\begin{equation*} p\to q,\ \neg p\ \therefore\ \neg q \end{equation*}$$ is invalid. ::: :::example[Inverse Fallacy] Let $p$ mean “$2+3=6$,” and let $q$ mean “$2+4=6$.” The argument $$\begin{equation*} p\to q,\ \neg p\ \therefore\ \neg q \end{equation*}$$ has the inverse form. Here $\neg p$ is true because $2+3\neq6$, but $q$ is also true because $2+4=6$. Therefore $\neg q$ is false. Hence, the argument is invalid. ::: The valid conditional forms are $$\begin{equation*} p,\ p\to q\ \therefore\ q \end{equation*}$$ and $$\begin{equation*} p\to q,\ \neg q\ \therefore\ \neg p. \end{equation*}$$ The invalid conditional forms are $$\begin{equation*} p\to q,\ q\ \therefore\ p \end{equation*}$$ and $$\begin{equation*} p\to q,\ \neg p\ \therefore\ \neg q. \end{equation*}$$ Students often confuse these because all four involve the same conditional $p\to q$. The safe method is to identify which part of the conditional is being affirmed or denied. The same conditional statement can appear in both valid and invalid argument forms. Set truth values for $p$ and $q$, then compare Modus Ponens, Modus Tollens, the converse error, and the inverse error. A form fails when its premises are true while its conclusion is false. Try $p=0$ and $q=1$ to see why the two fallacies are not valid rules. :::scientific-preview[Conditional Form Comparator]