Nonmonotonic logics are logics capable of formalizing defeasible inferences, i.e. inferences leading to conclusions that are withdrawn in case additional information, or additional premises, contradicts the defeasible conclusion. In other terms, the set of defeasible conclusions does not increase by incrementing the set of beliefs, it might rather decrease.
In1990, H. J. Levesque introduced a modal logic of belief capable of formalizing defeasible reasoning. There are two keys in his system that make the formalism possible. First, the introduction of a complementary belief operator that, combined with the usual belief operator, make it possible to express the exact content of an agent’s beliefs. Second, an axiom schema stating that a proposition is a logical possibility, provided that the proposition is consistent in the framework.
The condition that the proposition is consistent in the framework causes the logic to not being closed under uniform substitution. Moreover, the mentioned axiom means that the reasoning is carried out not entirely at the object level. Motivated by, among others, these points of criticism, Arild Waaler, dep. of Informatics, University of Oslo, introduced a logic of belief where the axiom of Levesque is replaced by a particular formula, the logical space, from which the logical possibilities, and necessities are derived entirely at the object level. This also has the effect that the notion of necessity is a notion of personal necessity. I.e. necessity surpasses the level of analytic relations between concepts.
This thesis aims at generalizing the notions of Waaler to the multi-agent case. The advantages of operating with a logical space in place of a multi-modal Levesque axiom are many. First, the advantages given for the single-modality case hold for the multi-agent case also. Second, and more importantly, the condition that the axiom of Levesque applies to consistent formulae is highly problematic in the multi-modal context. In the single agent case, this formula is of the language of propositional language, and the question of consistency is a propositional logical question. This is not the case in the multi-modal context, because the axiom says that a formula not mentioning the beliefs of a given agent is a logical possibility of this agent. But the formula is in general a modal formula, and the question of consistency of this formula must be solved within the multi-modal axiom system. Intrinsic to this is a danger of a vicious circularity, but by replacing the axiom with a logical space, we are able to go around this problem.
However, the construction of a logical space for the multi-modal case is highly non-trivial. We need to be able to express every single possibility of what a state of affairs might be from a given agent point of view. In the multi-modal case, a state of affairs must capture the belief set of every agent, where these belief sets in turn involves expressions in the modal language.
The construction of the logical space is core of the thesis. In general, the thesis provides a study of modal logics, defeasible logics, multi-modal logics and multi-modal defeasible logics. Additionally, a modal reduction theorem is presented, a result that proves the ability to reduce any belief representation to a set of representations, each explicitly expressing the belief set, and each compatible with the initial representation. Comparative studies, relating the system of this thesis to other formalisms are also provided. Finally, we suggest an extension of the system allowing the logical space to be deducible instead of explicitly given.