# propositional Modal logic filtration definition

Hello I have a slightly unusual question which relates to a definition of filtration structure. The following is my current state of the definition:

$\mathcal{M} = (W, R, L)$, W is a set of worlds, $R$ is a binary relation on the worlds, $R \subseteq W \times W$, and $L$ is a labelling function $L : W \rightarrow\ \mathcal{P} (PropAtoms)$, here $PropAtoms$ is the set of atomic formulas.

Let $\mathcal{M} = (W, R, L)$ be a Kripke model and let $\Gamma$ be a set of formulas that is closed under subformulas. Now, for each world $w \in W$ we define the following: $\Gamma_{w} = \{ \psi \in \Gamma : (\mathcal{M},w) \vDash \psi \}$,

that is, $\Gamma_{w}$ contains formulas from $\Gamma$ that are true at $w$ and consequently in model $\mathcal{M}$. Next we define an equivalence relation $\sim_{\Gamma}$ on worlds $w, w' \in W$, $\sim_{\Gamma} w w'$ if and only if $\Gamma_{w} \sqsubset \Gamma_{w'}$, that is, worlds $w$ and $w'$ are equivalent if and only if they satisfy the same formulas in $\Gamma$. We also have equivalence classes $[w]$ of worlds $w$ with respect to $\sim_{\Gamma}$.

Lets now present the following quotient structure $\mathcal{M}_{\Gamma} = (W_{\Gamma}, R_{\Gamma}, L_{\Gamma})$ which is in fact filtration of $\mathcal{M}$ with respect to $\Gamma$. $\mathcal{M}_{\Gamma}$ is defined as follows: .........

Here is the bit I need help with, I am struggling to define my filtration model. This is mainly because the definitions online don't use the same labelling function but rather use a valuation function which is of the form $v(p)$. And such function takes an atomic formula whearas my labelling function takes in a specific world written $L(x)$. Could someone please help me our here and try providing correct definition which will match my model $M$ definition?

You are probably looking for something like: $L_\Gamma([w]) = L(w) \cap \Gamma$. This will falsify any propositional letter not in $\Gamma$.

(Filtration is (afaik) usually defined as a property that a model may have. There could be several (or none) concrete models satisfying this property.)

You got quite close to the final model.

Keeping your $$\mathcal{M} = (W, R, L)$$, $$\Gamma$$, $$\sim_{\Gamma}$$ on worlds $$w, w' \in W$$, and $$[w]$$ of worlds $$w$$ with respect to $$\sim_{\Gamma}$$. Define further $$W_\Gamma = \{ [w] \mid w \in W \}$$.

Suppose that $$\mathcal{M}^f = (W^f, R^f, L^f)$$ is any model satisfying

• $$W^f = W_\Gamma$$,
• If $$Rwv$$ then $$R^f[w][v]$$
• If $$R^f[w][v]$$ then for all $$\Diamond \phi \in \Gamma$$, if $$\mathcal{M}, v \Vdash \phi$$ then $$\mathcal{M}, w \Vdash \Diamond \phi$$
• $$V^f(p) = \{ [w] \mid\mathcal{M}, v \Vdash p \}$$ for all $$p$$ in PropAtoms.

Then $$\mathcal{M}^f$$ is called a filtration of $$\mathcal{M}$$ through $$\Gamma$$.

Now, for your question on construction. The above restrictions only impose restrictions on $$R^f$$:

Let

1. $$R^s[w][v]$$ if and only if $$\exists w' \in [w], \exists v' \in [v] . Rw'v'$$
2. $$R^l[w][v]$$ if and only if for all formulae $$\Diamond \phi$$ in $$\Gamma$$, $$\mathcal{M}, v \Vdash \phi$$ implies $$\mathcal{M}, w \Vdash \phi$$

These two give rise to the largest and smallest filtrations, $$(W_\Gamma, R^s, L^f)$$ and $$(W_\Gamma, R^l, L^f)$$.

Based on P Blackburn, M De Rijke, and Y Venema. "Modal Logic (Cambridge Tracts in Theoretical Computer Science)." (2002), pp 77-80.