Entailment

Informal: Which formulas $B$ are true when $A$ is true?

Formal: For all Assignments $\phi$ which satisfy $A$, do these assignments also satisfy $B$ ?

$A$ entails/models $B$: $A\models B$

Entailment is the ideal outcome of reasoning.

Deduction is the process of an actual computer trying to reason about entailment.

$A\models B⟺(A\cup \{\neg B\})\models \perp .$

Examples

Modus Ponens