Categorical abstract algebraic logic: Gentzen pi -institutions and the deduction-detachment property

Given a π-institution I, a hierarchy of π-institutions I(n) is constructed, for n ≥ 1. We call I(n) the n-th order counterpart of I. The second-order counterpart of a deductive π-institution is a Gentzen π-institution, i. e. a π-institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I(2) of I is also called the “Gentzenization” of I. In the main result of the paper, it is shown that I is strongly Gentzen, i. e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction-detachment property.