Weaker Axioms, More Ranges

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.