[FOM] Possible Worlds

[A. Mani]

On Friday 22 Dec 2006 23:58, Allen Patterson Hazen wrote:
>
> (((Vocabulary: "Epistemic" logic is often construed broadly, as covering
> knowledge and a variety of related notions.  When knowledge is being
> distinguished from (mere) belief, "doxastic" refers to the belief
> operators.)))

Despite this, belief revision theories often do not adhere to doxastic specs. 
They are used for modelling dynamic reasoning and certain kind of real-time 
processes and sequences of temporal states too.  Dealing with possible-world 
frameworks for mathematics will be simpler from such perspectives (the usual 
real stuff they deal with come with more difficulties).  

Depending on how we want to see it, we can start with a set of possible worlds 
and revise them at later stages. For mathematics this is easier than for say 
modelling fragments of reasoning involved in cooking. 
>
> It is, I think, much harder to model MATHEMATICAL ignorance in a
> possible-worlds framework than it is to give an account of the sense in
> which, e.g., water MIGHT (have turned out) not (to) be H20.  Kripke
> ("Naming and Necessity") pointed out that the latter is an epistemic
> "might" (and not the "metaphysical" possibility he was primarily concerned
> with), but it seems possible to define THIS kind of epistemic possibility:
> roughly
>       It is metaphysically possible for their to be someone
>        whose evidence for a statement of the same logical
>        form as "water is H20" is similar to our evidence
>        (before modern chemistry) for "water is H20" even
>        though their statement is false.
> This won't work for mathematical statements if the "evidence" is allowed
> to include knowledge of the truth of axioms from which the statement
> follows logically.
>
> Many approaches to this problem have been suggested; none to my knowledge
> have been worked out in adequate detail:
You have missed the inconsistency adaptive logics, it is developed somewhat 
but can deal with the issue. 

The soln of the original question by Tim C would be along the following 
lines : 
In the adaptive framework, we will have
A. A logic of possible states corresponding to the initial nice state
B. Abnormalities (that have been mentioned before)
C. Another upper limit logic that will explode inconsistent premise sets    
The proof theory differs in being a marked proof theory with some semantics
thrown in
That is it (see Diderik Batens for example on adaptive logics)
 
>
>     I think the problem is an open one, and perhaps-- given the use of
> epistemic logics in AI-- an important one.  Note that approaches (2) and
> (3) above, and probably even (1) (though Meyer and Routley
> themselves were not Platonists) are consistent with quite extreme forms of
> Platonism.  I would expect a satisfactory solution to be so
> consistent.
In AI they arrive at or use various kind of ad hoc heuristics to solve 
questions of the kind. How would you specify the connection between such 
heuristics and the types of Platonism ? (references ?)


Best

A. Mani
Member, Cal. Math. Soc
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