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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.

| | | | | Contributors: bookofproofs | References: [7838]


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Bibliography (further reading)

[7838] Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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