Rational consequence relation

A rational consequence relation $\vdash$ is a logical consequence relation satisfying:

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 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.

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.

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.

References

• A mathematical paper quoting the GM rules