Let \(L\) be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. A disjunction “\(\vee\)” is a Boolean function
$$\vee :=\begin{cases}
L \times L & \mapsto L \\
(x,y) &\mapsto x \vee y.
\end{cases}$$
defined by the following truth table:
$[[x]]_I$ | $[[y]]_I$ | $[[x\vee y]]_I$ |
---|---|---|
\(1\) | \(1\) | \(1\) |
\(0\) | \(1\) | \(1\) |
\(1\) | \(0\) | \(1\) |
\(0\) | \(0\) | \(0\) |
We read the disjunction $x\vee y$
“$x$ or $y$”.
$x=$“The month is January”, $y=$“The month is February”
The logical disjunction $x\vee y$ is true if one of $x,y$ (or both) are true, but in English, only one can be true, but never both.
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| created: 2014-06-22 20:32:11 | modified: 2018-05-23 23:08:18 | by: bookofproofs