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.
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$\begingroup$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.
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