Affiliations: Department of Applied Mathematics, University of
Crete, Knossos Avenue, 714 09 Heraklion, Crete, Greece | Department of Business Administration, University of
Patras, 26500 Patras, Greece
Abstract: In the family of many-valued modal languages proposed by M. Fitting
in 1992, every modal language is based on an underlying Heyting algebra which
provides the space of truth values. The lattice of truth values is explicitly
represented in the language by a set of special constants and this allows for
forming weak, generalized, many-valued analogs of all classical modal axioms.
Weak axioms of this kind have been recently investigated from the canonicity,
completeness and correspondence perspective. In this paper, we provide some
results on the effect of adopting weak versions of the axioms D,
T, 4, 5 and w5 in the family of
many-valued modal non-monotonic logics, à la McDermott and Doyle,
introduced in [4] and further investigated in [7]. For
many-valued modal languages built on finite chains, we extend the results of
[7] by proving two quite general range theorems. We then hint on
the relation between the modal non-monotonic logics obtained: we prove that
there exist ranges which selectively pick out some of the expansions produced
by the many-valued autoepistemic logics introduced in [4,9],
actually the ones with a confidence-bounded set of beliefs. However, an exact
characterization of the relation between the various ranges created by the weak
many-valued modal axioms still remains to be explored.
