Let $T=\{\tau_1,\cdots, \tau_k\}$ be the set of truth assignments. Consider the tree of the truth assignments $2^\omega$. 

Consider the formula 
$\Gamma_T = \{ \underset{\tau \in T}\lor \tau_{|n} \mid n \in \omega\}$
where $\tau_{|n}$ is the formula that expresses $\tau$ up to atom $p_n$, i.e. $\underset{i\leq n}\wedge l_i(\tau)$
where 
$l_i(\tau) = \begin{cases} 
p_i & \tau(p_i)=\top \\ 
\lnot p_i & \tau(p_i)= \bot
\end{cases}$.

It is easy to show that every $\tau \in T$ we have $\tau \vDash \Gamma_T$. 

For any $\tau \notin T$, there is some $n\in \omega$ such that $\tau_{|n}$ is different from those in $T$ and therefore $\tau$ does not satisfy $\underset{\tau \in T}\lor \tau_{|n}$ and therefore $\tau\nvDash \Gamma_T$.

You can generalized this construction to find a topological characterization of the set of truth assignments that can be defined. You may be interested in [Stone duality][1] and [type (model theory)](http://en.wikipedia.org/wiki/Type_%28model_theory%29).


  [1]: http://en.wikipedia.org/wiki/Stone_duality