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 . (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)