is known to be sound w.r.t. Veltman semantics. That is,
every derivable principle holds on all Veltman frames. Close
inspection of the soundness proof shows the enormous amount of freedom
one has in defining the semantics for
.
Various
generalizations of Veltman semantics are known. (See e.g. [Sve91].) We will introduce here the notion of an
-frame, an idea of Dick de Jongh for catching Svejdar's models
in a general notion, as it was presented by Rineke Verbrugge in an
unpublished document [Ver]. In the classical Veltman
semantics there is the S relation. One can have xSx0y. The yhere is another world in the model. The main idea in the
semantics is to replace this y with a set of worlds. So, we could
have
for example. One can now define
and still
have
sound w.r.t. the new semantics. As we work with an
existential quantifier in the old definition we will exclude the empty
set as a possible S-successor: the axiom
demands
.
Again the axiom
demands that R can somehow be seen as embedded
in Sx. In our case this reads
.
The
axiom
imposes a sort of
reflexivity on our semantics; that is
.
The transitivity
clause leaves a lot of choice. The axiom states:
.
There is no first choice in how to adapt
transitivity in the new semantics. It is sufficient to set
.
One could
also replace the existential quantor by a universal quantor to obtain
the definition of [Ver]. This will be our choice as
well. Another possibility would be to demand
.
The axiom
did not
impose anything on the old semantics and the axiom maintains this
special status. Thus we have:
This new semantics yields strong enough a tool to allocate the principle P0 in the landscape of other interpretability principles. It turns out that P0 has the highest possible degree of independence with respect to the principles M0 and W. This result is stated in the next theorem.
In order to have a complete comparison we note that
and
.
P0 holds on every
respectively
frame. This fact combined with the modal completeness results
gives the derivability of P0 over both
and
.
Of
course the odds are for
to be an incomplete
logic, but we have not been able to prove this due to the lack of a
candidate for a valid but underivable principle.