proof theory

[beroal]

I have started reading a textbook (Negri and von Plato 25) on proof theory. I would say, it is an easy read. My impression is that the theory is very meticulous. Check an inference rule, then check another inference rule, then check another inference rule… IMHO, it would be more productive to find common properties of inference rules and reason from those properties. In the textbook, it is not justified why we study specific inference systems. How do we know that those inference systems are correct and useful? I guess that the authors should prove that inference systems are deductively equivalent to a gold standard like natural deduction in sequent calculus style.

Works Cited

Negri, Sara, and Jan von Plato. Structural Proof Theory. Cambridge University Press, 2001.