Example 1 Let
$$
\begin{array}{l}
A=\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{d}{|d|}-\frac{a b}{|a b|}-\frac{a c}{|a c|}- \\
\frac{a d}{|a d|}-\frac{b c}{|b c|}-\frac{b d}{|b d|}-\frac{c d}{|c d|}+\frac{a b c}{|a b c|}+ \\
\frac{a b d}{|a b d|}+\frac{a c d}{|a c d|}+\frac{b c d}{|b c d|}-\frac{a b c d}{|a b c d|} .
\end{array}
$$

Find the value of $A$.
According to Proposition 2, we have
$$
\begin{aligned}
A & =1-\left(1-\frac{a}{|a|}\right)\left(1-\frac{b}{|b|}\right)\left(1-\frac{c}{|c|}\right)\left(1-\frac{d}{|d|}\right) \\
& =\left\{\begin{array}{ll}
1, & \text{if at least one of } a, b, c, d \text{ is positive; } \\
-15, & \text{if all of } a, b, c, d \text{ are negative. }
\end{array}\right.
\end{aligned}
$$