FIGURE 1. Three inverted lollipops
(i.e., the submodel of $\mathcal{M}$ that consists of all nodes that can be reached from $w$ by a forward path via the relation $R$).
If $(W, R)$ is a frame, a modal assertion is valid for $(W, R)$ if it is true at all worlds of all Kripke models having $(W, R)$ as a frame. If $\mathcal{C}$ is a class of frames, a modal theory is sound with respect to $\mathcal{C}$ if every assertion in the theory is valid for every frame in $\mathcal{C}$. A modal theory is complete with respect to $\mathcal{C}$ if every assertion valid for every frame in $\mathcal{C}$ is in the theory. Finally, a modal theory is characterized by $\mathcal{C}$ if it is both sound and complete with respect to $\mathcal{C}$ [15, p. 40].
We remind the reader of the technique called filtration (cf. [8, pp. 267–268]): if $\mathcal{M} = (W, R, V)$ is a Kripke model and $\Gamma$ is a subformula-closed set of formulas, we can define an equivalence relation $\sim_{\Gamma}$ on $W$ by saying that $u \sim_{\Gamma} v$ if and only if $u$ and $v$ agree on the truth values of all formulas in $\Gamma$. We define $W_{\Gamma} := W/\sim_{\Gamma}$ and
and say that a Kripke model $\mathcal{M}' = (W_{\Gamma}, R', V_{\Gamma})$ is a filtration of $\mathcal{M}$ with respect to $\Gamma$ if for all $\psi \in \Gamma$ and $w \in W$, we have $\mathcal{M}, w \models \psi$ if and only if $\mathcal{M}', [w]{\sim{\Gamma}} \models \psi$. Note that in general, there can be more than one filtration with respect to a given $\Gamma$.
THEOREM 1 (Filtration Theorem). For every Kripke model $\mathcal{M} = (W, R, V)$ and every subformula-closed set of formulas $\Gamma$, there is a filtration $\mathcal{M}' = (W_{\Gamma}, R', V')$ of $\mathcal{M}$ with respect to $\Gamma$. One such filtration is the minimal filtration defined by $UR'V$ if and only if there are $u \in U$ and $v \in V$ such that $uRv$. Furthermore, if $\Gamma$ was finite, then so is $W_{\Gamma}$.
We also remind the reader of the technique of canonical models (cf. [1, Chapter 4.2]): If $\Lambda$ is a normal modal theory, then we can construct the canonical model $(W^\Lambda, R^\Lambda, V^\Lambda)$ satisfying all of the formulas in $\Lambda$ where $W^\Lambda$ is the set of maximal $\Lambda$-consistent sets of formulas, $R^\Lambda$ is the canonical accessibility relation, and $V^\Lambda$ is the canonical valuation defined by $V^\Lambda(\varphi, w) := 1$ if and only if $\varphi \in w$.
THEOREM 2 (Canonical Model Theorem). Any normal modal theory $\Lambda$ is complete with respect to its canonical model. That is, for any $\varphi \notin \Lambda$, there is a node $w \in W^\Lambda$ such that $\neg\varphi \in w$.