--- ## Definition > [!tldr] Definition > The [[Negation|negation]] of the [[Conditional statements|conditional statement]] $P \rightarrow Q$ is the [[Conjunction|conjunction]] $P \wedge (\neg Q)$. **Notes**: - In English, the negation of "If $P$ then $Qquot; is "$P$ and not $Qquot;. - In pure notation, $\neg (P \rightarrow Q)$ is $P \wedge (\neg Q)$. > [!IMPORTANT] > **The negation of a conditional statement is not another conditional statement**. In particular, the negation of $P \rightarrow Q$ is not any of the following: > > - $P \rightarrow (\neg Q)$ > - $(\neg P) \rightarrow Q$ > - $(\neg P) \rightarrow (\neg Q)$ > > For example, the negation of "If it is raining then I will carry an umbrella" is not, "If it is not raining then I will not carry an umbrella". > > The negation of a conditional statement must be a complete sentence as well. For example the negation of "If it is raining then I will carry an umbrella" is not: *It is raining I am not carrying an umbrella* -- without the word "and", this sentence is semantically meaningless and could have multiple interpretations. ## Examples and Non-Examples - The negation of "If it is raining outside then I will carry an umbrella" is "It is raining and I am not carrying an umbrella". - The word "but" can be used in place of "and": "It is raining *but* I am not carrying an umbrella". ## Resources