What is the cut rule? I don't mean the rule itself but an explanation of what it means and why are proof theorists always trying to eliminate it? Why is a cut-free system more special than one with cut?



cut elimination
Suppose I have a proof of B starting from assumption A. And a proof of C starting from assumption B. Then the cut rule says I can deduce C from assumption A.


But I didn't need the cut rule. If I was able to deduce B from A I could simply "inline" the proof of B from A directly into the proof of C from B to get a proof of C from A.


So the cut rule is redundant. That's a good reason to eliminate it.


But eliminating it comes at a price. The proofs become more complex. Here's a [paper](http://www.ams.org/bull/1997-34-02/S0273-0979-97-00715-5/S0273-0979-97-00715-5.pdf) that quantifies just how much. So it's a hard rule to give up.


