Since the AGM system cannot include conditional beliefs, it cannot elucidate the question of what logical relationships hold between conditional defaults.
In particular, we show that the normative conditional for default reasoning (developed in a companion paper) and our subjunctive conditional are identical.
The truth-conditions of the default conditional and the definition of nonmonotonic consequence can be fine-tuned to match many of our intuitions, but in the end of the day, the theory of Commonsense Entailment offers no simple answer to the question of what its conditional or its consequence relation are supposed (ideally) to represent.
When this insight is applied to the nonmonotonic logic of extreme probabilities, we can use causal information to identify which defaults function independently of others: that is, we can decide when the fact that one default conditional has an exception is irrelevant to the question of whether a second conditional is also violated (see Koons 2000, 320 323).
Logics of extreme probability (beginning with the work of Ernest Adams) did not permit the nesting of default conditionals for this reason: the conditionals were supposed to represent something like subjective conditional probabilities of the agent, to which the agent was supposed to have perfect introspective access.
Default conditionals are not allowed to be nested within other default conditionals, or within the scope of the usual Boolean operators of propositional logic (negation, conjunction, disjunction, material conditional).
Several conditional theories of default reasoning have recently been proposed for the representation of statements about normal states of affairs or prototypical properties.
In particular, they require the default conditional operator, ⇒, to have wide scope in every formula in which it appears.
We can then distinguish between a default conditional's following with logical necessity from a default theory and its following defeasibly from that same theory.
