Combining Possibilities and Negations

Combining non-classical or`sub-classical' logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility in the presence of conjunction and disjunction, and then we combine the resulting systems with intuitionistic logic. We will nd that Kracht's results on the undecidability o f classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic. Many people are interested in logics of modal operators likènecessarily' and`possibly,' and their cousins taken from temporal, epistemic, doxastic and many other concerns. Quite a few people are also interested in negative modal operators, like classical boolean negation, but with some kind of`modal' force. The idea with these sorts of operators is that to evaluatènot p' a t a p o i n t w orld, information state, moment or whatever you check the status of p at some other class of points. Intuitionistic negation is one such negation: to check for`not p' a t a p o i n t, you examine whether p fails at all of that point's successors. The de Morgan negation of relevant logics is also this sort of negation. For`not p' is true at a point, you need p to fail to be true at another particular point related to the rst point. In this paper we will examine what happens when you put these things together. Speciically, we will see how combining logics of possibility and negation or combining two possibilities, or two negations can result in undecidability. We will also see what happens when these logics are combined with intuitionistic propositional logic. The jumping-oo points of this work are numerous. The results we will discuss are not only in the general scene of combining logics, but they are also an instance of combining logical techniques. From the side of classical modal logic, we will be using Marcus Kracht's elegant results giving examples of simple undecidable modal logics 7. He shows that given a nitely presented semigroup, you can construct a nitely axiomatised modal logic with a modality for every variable used in the presentation which encodes the semigroup in a natural way. Deciding theoremhood in the logic is suucient