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.

Branches: 1
Corollaries: 2 3
Definitions: 4 5 6 7
Examples: 8
Explanations: 9
Lemmas: 10 11 12 13 14 15 16
Parts: 17
Proofs: 18 19 20 21 22 23 24 25 26

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References

Bibliography

Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016

$[[x]]_I$	$[[\neg x]]_I$
\(1\)	\(0\)
\(0\)	\(1\)

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Definition: Negation

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References

Bibliography