# [FOM] Logical Correctness/ Pratt

Wed Dec 5 22:08:57 EST 2012

```Dear Vaughan:

Many thanks for your comment. I do not quite understand why you feel
that my questions are tricky. What is tricky about the sentence, p:
Robert is much richer than most of his friends? What is true is that p
has a fairly complex structure, but it is not tricky. Assessment of the
truth value of p in a given possible world is a nontrivial
problem.Please feel free to use your own wording in dealing with my
questions.

As I pointed out in my response to your earlier message, assessment of
the truth value of p in a possible world is, in general, a nontrivial
problem when p is a fuzzy proposition. But there are other problems
which are more fundamental. The principal problem is that my questions
do not fit the conceptual structure of modal logic. More concretely,
consider an expression of the form "h possible p," where h is a hedge
such as quite, more or less, very, etc. Expressions of this form are
common in everyday discourse. How would you deal with hedge-modified
possibilities?

There are two principal theories for dealing with possibility-qualified
propositions. The best known by far is modal logic. Far less known is
Dubois and Prade, /Possibility Theory/, Plenum Press, 1988). Possibility
theory parallels probability theory. In possibility theory, we have
possibility distributions and possibility measures, paralleling
probability distributions and probability measures. Hedged possibilities
fall within the province of possibility theory. My questions fit the
conceptual structure of possibility theory, but not that of modal logic.
This is the principal reason why they are hard to deal with within the
conceptual structure of modal logic.

To place my commentsin a proper perspective, I should like to suggest in
the following a very simple way of describing fuzzy Kripke semantics.
What I sketch is consistent with the fuzzy modal logic ofLluis Godo,Petr
Hajek and Francesc Esteva. One of many references is "A fuzzy modal
logic for belief functions," Fundamenta Informaticae, Vol. 57, N. 2-4,
pp. 127-146, 2003. Fuzzy modal logic is a very significant
generalization of modal logic, in part because it opens the doorto
reasoning and computation with possibility-qualified fuzzy propositions.

The point of departure in my suggestion is an annotated graph, G=(RW),
in which R is a relation and G is an annotated graph of R. Each node is
annotated with a collection of propositions. W is a collection of
annotations. G may be symmetric, transitive, reflexive, etc.The
structure of G may be represented as an incidence matrix. A designated
node in G is referred to as a reference node. A node, i, is reachable
from the reference node if there is a path from the reference node to i.

At this point, a crisp proposition, p, is brought into the picture. At
each node i, p has a truth value, t_i .With respect to G, p is possible
if there exists at least one node which is reachable from the reference
node, and at which p is true. With respect to G, p is necessary if p is
true at all nodes which are reachable from the reference node. What is
described, except for terminology, is a very simple abstract version of
the basics of Kripke semantics.

Next, assume that p is a fuzzy proposition. In this case, the truth
value, t_i , is not bivalent. For simplicity, we can assume that t_i
takes values in the unit interval. Then, possibility becomes a matter of
degree, and the truth value of possible p may be defined as the sup of
the t_i .

Next, assume that p is a fuzzy proposition, R is a fuzzy relationand G
is an annotated fuzzy (weighted) graph. In this case, each node, i,
becomes associated with a degree of reachability, r_i , from the
reference node. r_i may be defined as follows. Let w_i be the
conjunction of the weights of links on a path from the reference node to
i. Then, r_i may be represented as the sup of the w_i over all paths
from the reference node to i. Let u_i be the conjunction of t_i and r_i
. Then, the possibility of p is the sup of the u_i over all reachable
nodes from the reference node.

In sum, what is sketched above is asimple abstract version of the basics
of fuzzy Kripke semantics. Itserves to clarify the kind of problems
which can be addressed through the use of fuzzy modal logic.

Sincerely,

Lotfi

--
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)

729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720-1776
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433