Duality (was "NEAR DEATH LOGIC")


Le 20-juil.-05, à 14:50, I wrote:



It would help me to proceed if you tell me, you Stathis or any reader of 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 2 then 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/