[FOM] A question about modal models

[Dana Scott]

I have thought a lot over the years about
"algebraic" (= lattice-theoretic) interpretations
of modal logic.  Perhaps I have stumbled on a
new model for the Lewis System S4?  If not,
then let me know about references, please.

Recall that a large class of models for S4 comes
from topological spaces.  Let X be a space, and use
the powerset of X, P(X), to interpret Boolean
logic in the usual way.  The S4-operator is then
the familiar topological interior operator.  This
goes back to McKinsey/Tarski (~1946) if not earlier.

Such models include the so-called Kripke models,
inasmuch as such a model IS a topological space
where the open sets are closed under arbitrary (not
just finite) intersections.  The minimal neighborhoods
of points (= worlds) then give the alternative relation.

There are also more abstract models, because we can start
with ANY complete Boolean algebra, L, and then single
out a sublattice H of L which is closed under finite
meets and arbitrary joins.  Then, as in the topological
case, we can define an S4-operator by joins:

	[]p = \/{ q in H | q =< p }.

All the analogues of the S4-laws then follow in this
generality.  The two degenerate cases are then

	H = {0} and H = L.

The example, which I hope has not been noticed before,
begins with the sigma-field, Meas, of all Lebesgue
measurable sets of the unit interval, [0,1], and passes to

	L = Meas/Null,

the quotient lattice (Boolean sigma-algebra) modulo the
ideal of sets of measure zero.  It is very well known that
L is actually a complete Boolean algebra with a strictly
positive probability measure coming from Lebesgue measure.

In thinking about this old friend one day, I said, "Wait!
What about the opens?"  And so I proved that the the sublattice

	H = Open/Null

is not only closed under finite meets and joins but is also
closed under arbitrary joins.  (Well, because they reduce
to countable joins.) Moreover, L and H are different.

We thus have a "pointless" (= atomless), nontrivial S4 model,
where every "proposition" has a probability.  Is this a
new observation or not?

	-- Dana Scott