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What is the definition of the negation of a statement? I know that if a statement is true then its negation is not true and also that either the statement is true or its negation is true but they cannot both be true. However, I do not see how this constitutes a definition.

user213066
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    There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. – Elias Costa Feb 04 '15 at 22:13
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    Negation is simply the incorporation of the NOT logical operator before the statement taken as a whole. – David G. Stork Feb 04 '15 at 22:14
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    I see two close votes as 'too broad'. How's this too broad? As far as I see any possible answer will go something like what David said above. – Git Gud Feb 04 '15 at 22:20

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One does this explicitly by parts. You got the first thing correct — if a statement is true, its negation is defined to be false. But what you forgot is the second thing: If a statement is false, its negation is defined to be true. To conclude:

Let $A$ be a statement. We define $$\lnot A :\equiv \begin{cases} \text{false}& A \:\text{is true} \\ \text{true} & A \:\text{is false} \end{cases} $$ This definition is valid, because for any statement $A: A \:\text{is true}\dot\lor A\:\text{is false}$.

What you said afterwards is a direct consequence of this definition:

Assume $A$ is true. Then, $A\lor\lnot A$is true as well. Assume $A$ is false. Then, $\lnot A$ is true, and thus is $A \lor\lnot A$. From that, we can conclude that For all statements $A: A\lor\lnot A \:\text{is true}$.

Your second assumption, that for all statements $A: A\land\lnot A\:\text{is false}$, can be proved the same way.

Lukas Juhrich
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