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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?

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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.)

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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.

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