Deduction

Which statements $B$ can be derived from $A$ using a set $\mathcal{C}$ of Inference Rules?

A\vdash_{\mathcal{C}}B

Example

We have a set of Inference Rules $\mathcal{C}$ which contains the only rule:

\begin{prooftree} \AxiomC{} \AXC{$A\quad A\to B$} \UIC{$B$} \end{prooftree}

We can now say that we could use $P$ and $P\to Q$ to derive $Q$ :

P, P \Rightarrow Q \vdash_{\mathcal{C}} Q

or in other words, $Q$ is derivable from $P$ and $P\to Q$ with the set of Inference Rules $\mathcal{C}$ .