Default reasoning has been widely studied in artificial intelligence. One of the the most influential piece of work within this area is that of Kraus et al. (1990). Kraus et al. investigated the properties of different sets of Gentzen-style proof rules for non-molmtonic consequence relations, and related these sets of rules to the model-theoretic properties of the associated logics. Their major result was that a particular set of proof rules generated the same set of consequences as a logic in which there is a preference order over models. This system of proof rules was termed System P by Kraus et al.. the P standing for "preferential".

Recently, conditional logics have been developed for application to problems in default reasoning. We present a uniform framework for the development and investigation of conditional logics to represent and reason with "normality", and demonstrate these logics to be equivalent to extensions of the modal system S4. We also show that two conditional logics, recently proposed to reason with default knowledge, are equivalent to fragments of two logics developed in this framework.

Clearly, the second approach is more cautious. Intuitively, it demands that there is a specific argument for τ that is contained in each rational stance a reasoner can take given Γ, DRules, and SRules. The first option doesn't bind the acceptability of τ to a specific argument: it is sufficient if according to each rational stance there is some argument for τ. In Default Logic, the main representational tool is that of a default rule, or simply a default.

The former will express domain constraints and will be denoted by rules B with no head (e.g., alive(p, t) A dead(p, t) ---)), while the latter will express contingent constraints and will be denoted as -43 (e.g., l[on(a, b, ti) Aon(b, c, ti)]). This distinction between background and evidence is implicit in probability theory and in Bayesian Networks (Pearl 1988b) and it used in several theories of default reasoning (Geffner 1992; Poole 1992). For simplicity, we will assume that background constraints B involve exactly two atoms.

In this paper we investigate nonmonotonic'modes of inference'. Our approach uses modal (conditional) logic to establish a uniform framework in which to study nonmonotonic consequence. We consider a particular mode of inference which employs a majority-based account of default reasoning--one which differs from the more familiar preferential accounts--and show how modal logic supplies a framework which facilitates analysis of, and comparison with more traditional formulations of nonmonotonic consequence.