Database: Transitivity of entailment

Proposition 61: Transitivity of entailment (inference rule)

Given that \phi entails \psi and \psi entails \xi , we may infer \phi entails \xi : \cfrac{\phi \vdash \psi \quad \psi \vdash \xi}{\phi \vdash \xi}.

Proof: Assume \phi . By the cut rule, infer \psi holds. By the cut rule again, infer \xi holds. But then we have showed \phi entails \xi . \square

This database entry builds on the following:

  1. Terminology: Inference, entailment
  2. Terminology: Inference rule
  3. Theorem: The cut rule