Rational consquence relation

From Wikipedia, the free encyclopedia

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

1 Uses
2 Example
3 Consequences
4 References

[edit] 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.

[edit] 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.

[edit] Consequences

The following consequences follow from the above rules:

CON: Conditionalisation $\frac{\theta \wedge \phi \vdash \psi}{\theta \vdash \left(\phi \rightarrow \psi \right)}$
CC: Cautious Cut $\frac{\theta \vdash \phi \quad \theta \wedge \phi \vdash \psi}{\theta \vdash \psi}$