Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lind

["Rich Cooper" <rich@xxxxxxxxxxxxxxxxxxxxxx>]

John,

 

In your page on Laws, you wrote:

Dunn performed the same substitutions in Kripke's constraints on the accessibility relation. The result is a restatement of the constraints in terms of the laws and facts:

·        System T.  The two axioms p⊃p and p⊃◊p require every world to be accessible from itself. That property follows from Dunn's definition because the laws L of any world are a subset of the facts:  L⊂M.

·        System S4.  System T with axiom S4, p⊃p, requires that R must also be transitive. It imposes the tighter constraint that the laws of the first world must be a subset of the laws of the second world: L₁⊂L₂.

·        System S5.  System S4 with axiom S5, ◊p⊃◊p, requires that R must also be symmetric. It constrains both worlds to have exactly the same laws:  L₁=L₂.

In system T, you’re stating that every necessary p must include at least p, and the laws L for that world Ware “facts” of the same W.  But that doesn’t take into account the conditions where some of the “laws” turn out to be false, or even counter effective.  That might be the case in belief revision of that L|M system based on experiences observed later (future Ms?), or further along the lattice in the specialization direction.  So my question is: why posit that ALL of the L are in M?  Why can’t some L be in ~M, thereby serving to identify a contradiction that can be further explored?

 

Secondly, it looks to me like that system T is a good model of an And-Or tree spanning the LUB and the GLB nodes of the lattice.  The “necessary” and “possible” predicates are suggestive of realistic situations, in specialized environments. 

 

Belief revision is also suggestive of realistic situations, but the study of exceptional cases when revision of the belief base is counter intuitive needs to be explored more deeply.  When a contradiction appears in L or M, or in the combination, that indicates a unsolvability condition.  There has to be practical use in belief revision, with better control of choice making. 

 

-Rich

 

Sincerely,

Rich Cooper

EnglishLogicKernel.com

Rich AT EnglishLogicKernel DOT com

9 4 9 \ 5 2 5 - 5 7 1 2

 

-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Wednesday, January 07, 2015 9:02 PM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf Lindenbaum

 

I also found that the links are intermittent.  I tested them

before I sent the original note, and had no trouble.  When

I read Rich's note, I tried again, and got a time out.

 

John

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