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7929Lemma: 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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Bibliography (further reading)

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

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