Rational consequence relation

In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listed below.
Properties.
A rational consequence relation formula_1 satisfies:
and the so-called Gabbay-Makinson rules:
Uses.
The rational consequence relation is non-monotonic, and the relation formula_9 is intended to carry the meaning "theta usually implies phi" or "phi usually follows from theta". In this sense it is more useful for modeling some everyday situations than a monotone consequence relation because the latter relation models facts in a more strict boolean fashion - something either follows under all circumstances or it does not.
Example.
The statement "If a cake contains sugar then it tastes good" implies under a monotone consequence relation the statement "If a cake contains sugar and soap then it tastes good." Clearly this doesn't match our own understanding of cakes. By asserting "If a cake contains sugar then it usually tastes good" a rational consequence relation allows for a more realistic model of the real world, and certainly it does not automatically follow that "If a cake contains sugar and soap then it usually tastes good."
Note that if we also have the information "If a cake contains sugar then it usually contains butter" then we may legally conclude (under CMO) that "If a cake contains sugar and butter then it usually tastes good.". Equally in the absence of a statement such as "If a cake contains sugar then usually it contains no soap" then we may legally conclude from RMO that "If the cake contains sugar and soap then it usually tastes good."
If this latter conclusion seems ridiculous to you then it is likely that you are subconsciously asserting your own preconceived knowledge about cakes when evaluating the validity of the statement. That is, from your experience you know that cakes which contain soap are likely to taste bad so you add to the system your own knowledge such as "Cakes which contain sugar do not usually contain soap.", even though this knowledge is absent from it. If the conclusion seems silly to you then you might consider replacing the word "soap" with the word "eggs" to see if it changes your feelings.
Example.
Consider the sentences:
We may consider it reasonable to conclude:
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:
Rational consequence relations via atom preferences.
Let formula_14 be a finite language. An atom is a formula of the form formula_15 (where formula_16 and formula_17). 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 formula_18 be the set of all atoms in L. For formula_19 SL, define formula_20.
Let formula_21 be a sequence of subsets of formula_18. For formula_23, formula_24 in SL, let the relation formula_25 be such that formula_26 if one of the following holds:
Then the relation formula_25 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".
The representation theorem.
It can be proven that any rational consequence relation on a finite language is representable via a sequence of atom preferences above. That is, for any such rational consequence relation formula_1 there is a sequence formula_21 of subsets of formula_18 such that the associated rcr formula_25 is the same relation: formula_48