## 7929**Lemma**: Affirming the Consequent of an Implication

Similar to mixing-up the necessary and sufficient condition of an implication is a fallacy known as the **affirming the consequent**. It is often used to manipulate the opinion of the audience about a proposed action. Given two propositions $p$ and $q$, it takes the following form:

$$\begin{array}{rll}

p\Rightarrow q&\text{major premise}&\text{e.g. If we want to succeed, then we have to take the risk.}\\

q&\text{minor premise}&\text{e.g. I tell you, we have to take the risk.}\\

\hline

p&\text{conclusion}&\text{e.g. Therefore, we will succeed.}\\

\end{array}

$$

Another, more mathematical example of this fallacy is

$$\begin{array}{rll}

p\Rightarrow q&\text{major premise}&\text{e.g. If a number $n\neq 2$ is a prime number, then it is odd.}\\

q&\text{minor premise}&\text{e.g. The number $n$ is odd.}\\

\hline

p&\text{conclusion}&\text{e.g. Therefore, $n\neq 2$ and $n$ is a prime number.}\\

\end{array}

$$

| | | | | Contributors: *bookofproofs* | References: [6823]

## 79331.**Proof**: *(related to "Affirming the Consequent of an Implication")*

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[6823] **Kane, Jonathan**: “Writing Proofs in Analysis”, Springer, 2016

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