I've read somewhere that the cut rule in sequent calculus
$$\frac{A \vdash \mathbf{C}, B \qquad A',\mathbf{C} \vdash B'}{A,A' \vdash B,B'} (\text{cut})$$
states that the $\mathbf{C}$ on the right is stronger than $\mathbf{C}$ on the left.


I would like to know what is this notion of strongness and how is $\mathbf{C}$ on the right stronger than $\mathbf{C}$ on the left.



Notion of strongness in cut rule
First, by "stronger than", Girard means "at least as strong as". So the identity rule can be read as saying "If you have a C on the left side, you can have C on the right side as well (because you can use the identity rule to derive them)", and in that sense "a C on the left is as good as having a C on the right".


And by that reading, the cut-rule could be interpreted as saying "If you have a C on the right, you can have a C on left (because you can cut them away using cut)".


