- Rational consequence relation
A rational consequence relation vdash is a
logical consequence relation satisfying the properties listed below.Properties
A rational consequence relation satisfies::; REF : Reflexivity heta vdash heta
and the so-called Gabbay-Makinson rules:
:; LLE : Left Logical Equivalence frac{ heta vdash psi quad heta equiv phi}{phi vdash psi}:; RWE : Right-hand weakening frac{ heta vdash phi quad phi models psi}{ heta vdash psi}:; CMO : Cautious monotonicity frac{ heta vdash phi quad heta vdash psi}{ heta wedge psi vdash phi}:; DIS :
Logical or (ie disjunction) on left hand side frac{ heta vdash psi quad phi vdash psi}{ heta vee phi vdash psi}:; AND :Logical and on right hand side frac{ heta vdash phi quad heta vdash psi}{ heta vdash phi wedge psi}:; RMO : Rational monotonicity frac{phi otvdash eg heta quad phi vdash psi}{phi wedge heta vdash psi}Uses
The rational consequence relation is non-monotonic, and the relation heta vdash phi 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:
*"Young people are usually happy"
*"Drug abusers 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 frac{ heta vdash phi quad heta vdash left( phi ightarrow psi ight)}{ heta vdash psi}::MP is proved via the rules AND and RWE.:;CON : Conditionalisation frac{ heta wedge phi vdash psi}{ heta vdash left(phi ightarrow psi ight)}
:;CC : Cautious Cut frac{ heta vdash phi quad heta wedge phi vdash psi}{ heta vdash psi}::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 frac{ heta models phi}{ heta vdash phi}::SCL is proved trivially via REF and RWE.
Rational Consequence Relations via Atom Preferences
Let L = {p_1, ldots , p_n} be a finite language. An atom is a formula of the form igwedge_{i=1}^n p^epsilon_i (where p^1 = p and p^{-1} = eg p). 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 At^L be the set of all atoms in L. For heta in SL, define S_ heta = {alpha in At^L | alpha models^{SC} heta }.
Let vec{s} = s_1, ldots , s_m be a sequence of subsets of At^L. For heta, phi in SL, let the relation vdash_vec{s} be such that heta vdash_{vec{s phi if one of the following holds:
#S_ heta cap s_i = emptyset for each 1 leq i leq m
#S_ heta cap s_i eq emptyset for some 1 leq i leq m and for the least such i, S_ heta cap s_i subseteq S_phi.Then the relation vdash_vec{s} 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 vdash_vec{s}, the relation is unchanged if we replace s_2 with s_2 setminus s_1, s_3 with s_3 setminus s_2 setminus s_1 ... and s_m with s_m setminus igcup_{i=1}^{m-1} s_i. In this way we make each s_i disjoint. Conversely it makes no difference to the rcr vdash_vec{s} if we add to subsequent s_i atoms from any of the preceding s_i.
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 vdash there is a sequence vec{s} = s_1, ldots , s_m of subsets of At^L such that the associated rcr vdash_vec{s} is the same relation: vdash_vec{s} = vdash
Notes
#By the above property of vdash_vec{s}, the representation of an rcr vdash need not be unique - if the s_i are not disjoint then they can be made so without changing the rcr and conversely if they are disjoint then each subsequent set can contain any of the atoms of the previous sets without changing the rcr.
References
* [http://www.maths.manchester.ac.uk/~jeff/theses/lee-hill.ps A mathematical paper in which the GM rules are defined]
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