We have a few loose ends to gather up from last time.

So let us briefly recall filtrations.

Did you call closure of alpha capital sigma or did you just call it closure of alpha?

Only closure of alpha.

Only closure of alpha.

So let us recall the definition.

We take a formula alpha and M a model.

How do we call valuations?

Pi or?

Pi.

Pi.

Yes.

A model with these data set to W of worlds, transition relation R, valuation pi.

And there was a definition of what it meant to filter M by alpha and the filtration of

M by alpha would not be uniquely defined but it would be a relation, an equivalence relation

on the set of worlds and well, the equivalent menstruation on the set of worlds is defined

by alpha but the relation that we get on the corresponding quotient model is only exhumatized

within a certain range.

So we have this, did you give the equivalence relation a name?

Induced by alpha?

I just wrote subscript alpha so I don't have to, no I didn't name equivalence relation.

Okay so let's say that V is alpha equivalent to W if and only if for all formulas phi in

the closure of alpha if and only if V satisfies phi if and only if W satisfies phi so two

states in the model are equivalent if and only if indistinguishable by subformulas of

alpha.

And then a relation on the quotient at set, quotient by this indistinguishability relation

under alpha is a filtration if and only if the following hold.

So let me remember there's two conditions basically, I shouldn't call this R as well,

I should call this R bar.

So first of all it has at least a minimal prescribed size, this relation, namely whenever

I have WRV upstairs in the original model then I also want it downstairs where, well

I'm omitting the triple equality sign on the index here so by writing index alpha here

I mean equivalence classes under this equivalence relation here.

So whenever I have a transition upstairs it survives downstairs so that's the minimal

relation that I have basically.

And second, there's basically a condition that bounds the size of R bar from above and

it says that whenever I put two equivalence classes in relation R bar then this must be

compatible with the satisfaction of modal formulas when restricted to the closure of

alpha.

So that means, well let's write this in terms of boxes, for all formulas no not sigma, phi,

box phi in the closure of alpha.

So yeah we do strictly mean here that the box should be there as well so the whole formula

box phi should be a sub-formula of alpha.

If W satisfies, so the original W in the original model, if W satisfies box phi then V satisfies

phi.

So this definition was there last time so Esther Deish noticed afterwards, well not

that he could have done anything against it because the time was up, there was three things

missing.

First of all we have to show that we really do have a minimal and a maximal filtration

where we define our bar by precisely either the first condition or the second condition