We will treat this puzzle by assuming that there are
married couples in the country. Then
the language L = (Pr, Sp, T) adequate for this puzzle will be:
where 
denotes ith husband, 
means that
's wife is unfaithful and 
denotes
ith day.
Let 
denote that k-fold cartesian
product of the vector space 
with addition 
. We define
by 
,
where 
and 
denotes
, resp.). We put 
and
.
We also use 
to denote arbitrary element in 
.
Now, let 
denote what the King publicized on the first day, and
denote a knowledge base
for 
under the situation 
.
Let us put:
and
where 
. We also put 
.
Then, as a formalization of the puzzle, we postulate the following
identities:
Since the meta-notions such as knowledge base and provability 
cannot be expressed directly in our language, we were forced
to interpret the King's decree into in a somewhat indirect
fashion.
Now, if we read Eq(*) as the definition of , then
we find that the definition is circular, since in order
that may be definable by Eq(*) it is necessary that
are already defined, whereas
are defined in terms of in 
.
So, we will treat these equations as a system 
, 
,
of equations with the unknowns
and .
We will solve 
under the following conditions:
(*) For any 
, 
is consistent.
(**) For any and , is
a knowledge base for .
We think these conditions are natural in view of the intended
meanings of and .
Let us define a norm on
by 
,
where 
.
For any 
and 
, we put
and for any 
, we put
We also put 
.
We define a KT5-model 
as follows:
Then we have the following theorem.
Theorem 8. Under the conditions (
) and (
),
has the unique solution
,
where the solution is characterized by the condition:
Thus we have seen that 
may be regarded as the formal
counterpart of the King's decree in our
formal system. The puzzle is then reduced to the
problem of showing that:
We note that we can moreover prove the following:
It is clear that 
and 
follow at once from the condition stated
in theorem 8.