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 $\theta \vdash \theta$

and the so-called Gabbay-Makinson rules:

LLE

Left Logical Equivalence ${\frac {\theta \vdash \psi \quad \theta \equiv \phi }{\phi \vdash \psi }}$

RWE

Right-hand weakening ${\frac {\theta \vdash \phi \quad \phi \models \psi }{\theta \vdash \psi }}$

CMO

Cautious monotonicity ${\frac {\theta \vdash \phi \quad \theta \vdash \psi }{\theta \wedge \psi \vdash \phi }}$

DIS

Logical or (ie disjunction) on left hand side ${\frac {\theta \vdash \psi \quad \phi \vdash \psi }{\theta \vee \phi \vdash \psi }}$

AND

Logical and on right hand side ${\frac {\theta \vdash \phi \quad \theta \vdash \psi }{\theta \vdash \phi \wedge \psi }}$

RMO

Rational monotonicity ${\frac {\phi \not \vdash \neg \theta \quad \phi \vdash \psi }{\phi \wedge \theta \vdash \psi }}$

Uses

The rational consequence relation is non-monotonic, and the relation $\theta \vdash \phi$ 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 ${\frac {\theta \vdash \phi \quad \theta \vdash \left(\phi \rightarrow \psi \right)}{\theta \vdash \psi }}$

CON

Conditionalisation ${\frac {\theta \wedge \phi \vdash \psi }{\theta \vdash \left(\phi \rightarrow \psi \right)}}$

CC

Cautious Cut ${\frac {\theta \vdash \phi \quad \theta \wedge \phi \vdash \psi }{\theta \vdash \psi }}$

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 ${\frac {\theta \models \phi }{\theta \vdash \phi }}$

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 $\bigwedge _{i=1}^{n}p_{i}^{\epsilon }$ (where $p^{1}=p$ and $p^{-1}=\neg p$). 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 $\theta \in$ SL, define $S_{\theta }=\{\alpha \in At^{L}|\alpha \models ^{SC}\theta \}$.

Let ${\vec {s}}=s_{1},\ldots ,s_{m}$ be a sequence of subsets of $At^{L}$. For $\theta$, $\phi$ in SL, let the relation $\vdash _{\vec {s}}$ be such that $\theta \vdash _{\vec {s}}\phi$ if one of the following holds:

1. $S_{\theta }\cap s_{i}=\emptyset$ for each $1<=i<=m$
2. $S_{\theta }\cap s_{i}\neq \emptyset$ for some $1<=i<=m$ and for the least such i, $S_{\theta }\cap s_{i}\subseteq S_{\phi }$.

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 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: Failed to parse (unknown function "\eq"): {\displaystyle \vdash_\vec{s} \eq \vdash}

References

• A mathematical paper in which the GM rules are defined