  A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is
that they often undergo sharp thresholds. For example: if the function $f :
\{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with
$\E_p[f] = \eps$ and $\E_q[f] = 1-\eps$ then $q-p \to 0$ as $n \to \infty$.
Here $\E_p$ denotes the product probability measure on $\{0,1\}^n$ where each
coordinate takes the value $1$ independently with probability $p$. The fact
that symmetric functions undergo sharp thresholds is important in the study of
random graphs and constraint satisfaction problems as well as in social
choice.In this paper we prove sharp thresholds for monotone functions taking
values in an arbitrary finite sets. We also provide examples of applications of
the results to social choice and to random graph problems. Among the
applications is an analog for Condorcet's jury theorem and an indeterminacy
result for a large class of social choice functions.
