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After we have explained what Boolean functions and what truth tables are, it is time to use these concepts to introduce important examples of connectives.

## Definition: Negation

Let $$L$$ be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. A negation “$$\neg$$” is a Boolean function

$\neg :=\begin{cases}L & \mapsto L \\ x &\mapsto \neg x \\ \end{cases}$

defined by the following truth table:

Truth Table of the Negation
$[[x]]_I$ $[[\neg x]]_I$
$$1$$ $$0$$
$$0$$ $$1$$

### Notes

• The negation of any proposition changes from one truth value to the other.
• By the axiom of bivalence of truth, the negation of any negated proposition $\neg x$ must be the proposition itself: $\neg(\neg x)=x.$
• Later in the text, it will turn out that negation is one of the most important connectives: We can construct successfully logical systems even if we dispense some connectives, but we cannot do without the negation.

| | | | | created: 2014-06-22 20:51:07 | modified: 2018-05-04 23:32:35 | by: bookofproofs | references: [7838]