[extropy-chat] Possible Worlds Semantics

[Ian Goddard]

--- Hal Finney <hal at finney.org> wrote:

> Ian writes about Kripkean "possible worlds" 
> semantics.  I've been trying to learn a little 
> about this because Robin has used it in some of 
> his papers, and it is a widely used framework for
> economic and game theory analysis of certain issues.


 Last year Technotranscendence recommended the modal
logic text "Possibilities and Paradox." [1] I found it
to be wonderful and amazingly rapid overview of modal
logics. Highly recommended! But be sure to review its
errata page. [2]

[1] http://www.amazon.com/gp/product/0199259879
[2]
http://www.princeton.edu/~fraassen/Possib&paradERRATA.htm

More rigorous and even more recommended is:
http://www.amazon.com/gp/product/0226280888
But it also covers a lot more than modal logic.


> the first son looks inside 
> his envelope and finds $10,000. Disappointed at
> the meager amount, he calculates that the odds are
> fifty-fifty that he has the smaller amount in his
> envelope.  Since the other envelope contains either 
> $1,000 or $100,000 with equal probability, the 
> first son realizes that the expected amount in the 
> other envelope is $50,500.  Unbeknownst to him the 
> second son has seen that there is only $1,000 in 
> his envelope.


 Not sure if that last sentence means the first son
may expect the second NEVER looked in his envelope. Or
does "unbeknownst" apply only to the first son knowing
WHAT the second saw in his envelope? I'll assume the
latter, and thus that both brothers know the other
knows what's in their own envelope, but each does not
know the content of the other's envelope.

 Assuming that, I'll dare to answer one question:


> Suppose instead that the sons were not permitted 
> to look at each other, but instead they had to 
> write their confirmation of the deal on separate
> pieces of paper and hand them to their father?  
> What should they write?"

Given:

First son = A
Second son = B

A's envelope has $10^4
B's envelope has $10^3

Answer:

B should write: "If A wants to know if I say 'yes' to
the switch, then I agree to switch."

A should write: "If B wants to know if I want to know
if he says 'yes' to the switch, then I do NOT agree to
switch." 

Trying to explain my answer runs into describing a
thicket of nested possible worlds where each brother
needs to know what the other needs to know about the
other in order to know/infer if the other has a higher
or lower valued envelope. The critical parameters are
the max and min values and knowing that someone with
the max will necessarily say "NO switch," and someone
with the min will necessarily say "switch." Given the
indicated amounts, neither can assume the other has
the max or min. However, the effect of those necessary
answers flows through the nested set of "other minds"
in possible worlds each builds in trying to sense what
the other got. ~Ian




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