are usually not happy Drug abusers are usually young We may consider it reasonable to conclude: Young drug abusers are usually not happy This would not be a valid conclusion under a monotonic deduction system (omitting of course the word 'usually'), since the third sentence would contradict the first two. In contrast the conclusion follows immediately using the Gabbay-Makinson rules: applying the rule CMO to the last two sentences yields the result. Consequences The following consequences follow from the above rules: MP Modus ponens MP is proved via the rules AND and RWE. CON Conditionalisation CC Cautious Cut The notion of Cautious Cut simply encapsulates the operation of conditionalisation, followed by MP. It may seem redundant in this sense, but it is often used in proofs so it is useful to have a name for it to act as a shortcut. SCL Supraclassity SCL is proved trivially via REF and RWE. Rational consequence relations via atom preferences Let be a finite language. An atom is a formula of the form (where and ). Notice that there is a unique valuation which makes any given atom true (and conversely each valuation satisfies precisely one atom). Thus an atom can be used to represent a preference about what we believe ought to be true. Let be the set of all atoms in L. For SL, define . Let be a sequence of subsets of . For , in SL, let the relation be such that if one of the following holds: for each for some and for the least such i, . Then the relation is a rational consequence relation. This may easily be verified by checking directly that it satisfies the GM-conditions. The idea behind the sequence of atom sets is that the earlier sets account for the most likely situations such as "young people are usually law abiding" whereas the later sets account for the less likely situations such as "young joyriders are usually not law abiding". Notes By the definition of the relation , the relation is unchanged if we replace with , with ... and with . In this way we make each disjoint. Conversely it makes no difference to the rcr if we add to subsequent atoms from any of the preceding . The representation theorem It can be proven that any rational consequence relation on a finite language is representable via a sequence of atom preferences