Logic and Qualitative Probability

While theories of probability have long been adequately axiomatised, these
generally take a quantitative approach. Yet humans are also able to reason
about probability in a qualitative sense, such as when comparing the relative
likelihoods of sets of events and making inferences based on those comparisons.
This suggests that probabilistic reasoning ultimately has a strong
qualitative basis, which can be supplemented by various other components
in order to obtain complete theories of probability.
We analyse a recent approach to capturing qualitative probabilistic reasoning
in the form of the logic IP and the corresponding multi-measure semantics.
By restricting our attention to a fragment of the language of comparative
probability that exclusively contains probabilistic comparisons between
propositional formulas in disjunctive normal form, we reduce the logic IP to
its qualitative probabilistic essence. We show that the resulting restricted
logic IP-DNF is nevertheless sound and complete with respect to the multimeasure
semantics, and also show that the computational complexity of the
satisfiability problem for the logic IP-DNF is in polynomial time, whereas
that of the logic IP is NP-complete.
In the course of this analysis we encounter strong similarities between
qualitative probabilistic reasoning and counting and arithmetic. This leads
us to hypothesize that the multi-measure semantics, because it lacks the
facilities for differentiating between cardinalities, cannot be finitely axiomatised,
but instead exclusively possesses infinitary axiom-schemes such as the
logics IP and IP-DNF, which are both based on the infinitary Scott-axioms.
We proceed to prove this hypothesis, and ultimately conclude that a purely
qualitative approach to probabilistic reasoning seems insufficient.