What is the difference between $P\vdash Q$ and $\frac{P}{Q}$?

Question

I think both are used to write inference rules.What is the difference?

Answer 1

Usually, the second one is the way to express inference rules, while the first one says that (conclusion) $Q$ is derivable from (premise) $P$. This does not imply that there is a rule that allow us to directly infer $Q$ from $P$.

Here is an example with Natural Deduction: we have the rule $\dfrac {P∧Q}{P}$ and we have $P∧Q ⊢ Q∧P$, but we have no inference rule $\dfrac {P∧Q}{Q∧P}$.