Original Paper

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

  1. George Voutsadakis

Article first published online: 29 SEP 2005

DOI: 10.1002/malq.200310132

Issue

Mathematical Logic Quarterly

Mathematical Logic Quarterly

Volume 51, Issue 6, pages 570–578, November 2005

Additional Information

How to Cite

Voutsadakis, G. (2005), Categorical abstract algebraic logic: Gentzen π -institutions and the deduction-detachment property. Mathematical Logic Quarterly, 51: 570–578. doi: 10.1002/malq.200310132

Author Information

  1. School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA

Publication History

  1. Issue published online: 29 SEP 2005
  2. Article first published online: 29 SEP 2005
  3. Manuscript Accepted: 15 JUL 2005
  4. Manuscript Revised: 9 JUL 2005
  5. Manuscript Received: 8 DEC 2003

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Keywords:

  • Algebraic logic;
  • equivalent deductive systems;
  • equivalent institutions;
  • algebraizable logics;
  • algebraizable institutions;
  • lattice of theories;
  • category of theories;
  • Leibniz operator;
  • metalogical properties;
  • deduction-detachment theorem;
  • Gentzen systems;
  • Gentzen institutions

Abstract

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)

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