On 22 Nov 2012, at 19:26, Stephen P. King wrote:
In my research for my earlier post (Re: Nothing happens in the
Universe of the Everett Interpretation) I found the following:
Hierarchy of beliefs
"Construction by Jean-François Mertens and Zamir implementing with
John Harsanyi's proposal to model games with incomplete information
by supposing that each player is characterized by a privately known
type that describes his feasible strategies and payoffs as well as a
probability distribution over other players' types.
Such probability distribution at the first level can be interpreted
as a low level belief of a player. One level up the probability on
the belief of other players is interpreted as beliefs on beliefs. A
recursive universal construct is built—in which player have beliefs
on their beliefs at different level—this construct is called the
hierarchy of beliefs.
The result is a universal space of types in which, subject to
specified consistency conditions, each type corresponds to the
infinite hierarchy of his probabilistic beliefs about others'
probabilistic beliefs. They also showed that any subspace can be
approximated arbitrarily closely by a finite subspace.
Another popular example of the usage of the construction is the
Prisoners and hats puzzle. And so is Robert Aumann's construction of
Common knowledge (logic)."
I think that we can identify the concept of a "privately known
type" with the concept of 1p as discussed in the UDA. My concept of
a "reality" as that which is incontrovertible for some finite
collection of 1p can also we seen as synonymous with Aumann's Common
Knowledge which was mentioned above:
"Common knowledge is a special kind of knowledge for a group of
agents. There is common knowledge of p in a group of agents G when
all the agents in G know p, they all know that they know p, they all
know that they all know that they know p, and so on ad infinitum."
Bruno seems to use a radically flattened version of this idea -
as do most logicians - where the 'players' have complete knowledge
(ala Bp&p) of each other.
Bp & p is not complete knowledge. Dt is true, but BDt & Dt is false,
I see this as the natural reaction to the implications of the idea
of incomplete knowledge. We see a nice discussion of this here: http://plato.stanford.edu/entries/nonwellfounded-set-theory/harsanyi-type-spaces.html
"Type spaces are mathematical structures used in theoretical
parts of economics and game theory. The are used to model settings
where agents are described by their types, and these types give us
“beliefs about the world”, “beliefs about each other's beliefs about
the world”, “beliefs about each other's beliefs about each other's
beliefs about the world”, etc. That is, the formal concept of a type
space is intended to capture in one structure an unfolding infinite
hierarchy related to interactive belief.
John C. Harsanyi (1967) makes a related point as follows:
It seems to me that the basic reason why the theory of games with
incomplete information has made so little progress so far lies in
the fact that these games give rise, or at least appear to give
rise, to an infinite regress in reciprocal expectations on
the part of the players."
My claim is that we can avoid the problems of infinite regress
by both taking the possibility of regress seriously and
understanding that beliefs are the result of processes that involve
the consumption of resources.
You need a linear logic for this, but you cannot impose it, you have
to get it from the material hypostases, so as to be able to
distinguish the quanta and the qualia. if not you are just doing
physics, and lost the relation with the mind body problem, as
formulated by the UDA.
No activity of a mind -thought- can be said to occur without the
occurrence of work. Thought is an activity and does not just
'exist'. Existence per se is property neutral, therefore to speak of
the content of thought we must not ignore the requirements of
discovering what it might be. The same solution to the homunculus
problem that Roger and I have discussed previously applies: if there
is a finite quantity of resources or, equivalently, physical work
involved in the implementation of a homunculus then there can only
be a finite number of them.
There is no problem in the infinite case that Bruno is
considering because it can be collapsed with Kleene's theorems, but
in so doing we make the possibility of distinguishing one mind from
another (within a plurality of minds) vanish.
The measure problem that Bruno complains about is the direct result
of assuming perfect knowledge aka omniscient mind. This is a soluble
problem but to find solutions we must give up the idea of accessible
I do not complain on the measure problem. It is what all everythingers
agree on. Then I show that comp makes the problem both:
1) equivalent with deriving physics from arithmetic/computer science,
2) completely formulable in arithmetical terms, by using the
traditional definition of knowledge.
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