Le 20-juil.-05, à 14:50, I wrote:

## Advertising

It would help me to proceed if you tell me, you Stathis or any readerof the list, if you have understand (or not) that the fact that [Bf,Bt, actually any B<something> hold in the dead-end world] is as"obvious" as the fact that [for ALL numbers x, if x is bigger than 2then x is bigger than 1.]

`And I was expecting some answer or comment, but I can perhaps help a`

`little bit more.`

`OK, I will give you a completely different explanation why, for any`

`proposition p, Bp is true in any cul-de-sac world. But first I explain`

`a fundamental duality in modal logics.`

`In modal logic there is an important duality, which has been discovered`

`by Aristotle (the founder of modal logic), and which is encapsulated in`

`the Aristotelian Square:`

`Read Bp as "p is true in all (accessible) world". What I will say holds`

`both for the Leibniz multiverse (= a collection of observer-moments or`

`worlds), and Kripke multiverse (the same but with a accessibility`

`relation among observer-moments).`

`To make it concrete read B by "everywhere in Belgium", and p by "it`

`rains".`

`For someone (in Belgium or not) we can distinguish the following four`

`nuances, which I put in the Aristotelian square:`

1. Bp 2. ~Bp 3. B~p 4. ~B~p

`"1." (Bp) means "p is true in all (accessible) worlds" (here: it rains`

`everywhere in Belgium)`

`"2." (~Bp) means "it is false that p is true in all (acc.) worlds`

`(here: it is false that it rains everywhere in Belgium). That means`

`that there is a (acc.) world where p is not true. Here:there is a place`

`in Belgium where it does NOT rain.`

`"3." (B~p) means "~p is true in all (accessible) worlds", which is the`

`same as "p is false in all (accessible) world", or "there is no (acc.)`

`worlds in which p is true. Here: "there is no place in Belgium where it`

`rains", or "nowhere does it rain in Belgium", or "good wether`

`everywhere in Belgium"`

`"4." (~B~p) means "it is false that ~p is true in all accessible`

`worlds". This is the same as saying that there is an accessible world`

`where p is true. In our example: it is false that [it does not rain]`

`everywhere in Belgium, or "there is a place in Belgium where it rains.`

~B~p is called the dual of Bp.

`Bp means p true in all accessible world, more precisely (Kripke) Bp is`

`true in some world \alpha if p is true in all world \beta such that`

`\alpha reaches \beta (which we could write \a R \b).`

`Thus, ~B~p true in \a, means "it is false that ~p is true in all world`

`accessible from \a", but this means (by classical logic again) that`

`there is a world accessible from \a where p is true.`

`Let us write Dp for ~B~p. D can be seen as a new modality defined from`

`B.`

`Not that ~D~p, the dual of D, is the same as (by definition)`

`~(~(B~(~p))), but that is Bp (in classical logic p is equivalent with`

`~~p). So B and D are duals of each other. If you read the B as a Box,`

`read D as a diamond. B = [], and D = <> (with the previous notation)`

`Now, the "pedagogical" problem, linked to the truth table of the`

`implication, has reappeared just above, when I mention "by classical`

`logic again".`

`I want explain you how I could have manage to hide it (which could be,`

`or not, a good pedagogy).`

The idea is to take at once the D modality as primitive.

`In that case it is enough to define the meaning of Dp in a world \a by`

`saying that Dp is true in \a when there exists an accessible world`

`(from \a) where p is true. It is the equivalent of the "KRIPKE`

`IMPORTANT LINE" from the preceding posts.`

`Now take, in some multiverse, a dead-end world (terminal world,`

`cul-de-sac world, etc.), i.e. a world from which you can access NO`

`worlds (not even itself).`

`In that case it is obvious that, in such dead-end world, any`

`proposition of the form ~D~p are true, because if ~D~p was false then`

`D~p would be true (because all worlds obey classical logic, so all`

`propositions are either true or false). But then, D~p being true, it`

`means there would be an accessible world where ~p is true. In`

`particular there would be an accessible world, but that cannot be the`

`case given that we have consider a dead-end world.`

`So it is obvious (it should be obvious) that in any dead-end world, the`

`proposition of the form ~D<whatever> are always true. They literally`

`assert the non existence of an accessible world.`

`Now we can define Bp by its dual ~D~p, and this entails, without going`

`trough the logical difficulty (of the implication) that Bp, whatever p`

`is, is always true is dead-end world.`

`If you read Bp as "necessary p", and "Dp" as "possible p", you see that`

`dead-end world are not particularly funny: in dead-end world everything`

`is necessary and nothing is possible! Caution!`

Here is the Aristotelian Square in term of the D modality: 1. ~D~p 2. D~p 3. ~Dp 4. Dp You should see that ~Dp is the same as B~p, and ~Bp is the same as D~p. I give you the duality for well known modalities: B = obligation; D = permission B = everywhere; D = somewhere B = always; D = once B = necessary; D = possible B = for all; D = it exists B = provable; D = consistent

`Please verify. For example if Bp = "p is obligatory" Dp = ~B~p = it is`

`not obligatory that ~p = p is permitted. Also, if ~p is not permitted,`

`it means p is obligatory.`

`To sum up, if you have a difficulty to grasp that Bp is true in`

`dead-end world, just take the diamond D as primitive modality, remember`

`its Kripke meaning, i.e. Dp is true in \a if there is a world \b with`

`both \a R \b and p true in \b.`

`And then define Bp by ~D~p. D<wathever> is always false in dead end,`

`and so B<whatever> is always true in dead-end.`

`Obviously the proposition (Bp -> p) is false in dead-end. If it where`

`true, then we would have Bf and Bf->f, so by the rule of modus ponens`

`(worlds obey classical logic) we would have f in dead-end, but f is`

`false in all world!`

`Note that a characteristic formula of the "near death" multiverse was`

`"(~Bf -> ~B(~Bf))", with the D modality, you can write it Dt -> DBf,`

`which you can read "if I am alive then I can die" (or: if I am in a`

`transient world then I can reach a dead-end world).`

Hope that helps, Bruno http://iridia.ulb.ac.be/~marchal/