Source: Notre Dame J. Formal Logic
Volume 49, Number 2
We prove, by using the concept of schematic interpretation, that the natural embedding from
the category ISL, of intuitionistic sentential pretheories and i-congruence classes of
morphisms, to the category CSL, of classical sentential pretheories and c-congruence
classes of morphisms, has a left adjoint, which is related to the double negation
interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of
Excluded Middle. Moreover, we prove that from the left to the right adjoint there is a
pointwise epimorphic natural transformation and that since the two endofunctors at CSL,
obtained by adequately composing the aforementioned functors, are naturally isomorphic to the
identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On
the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from
pretheories can be extended to equivalences of categories from the category CSL,
respectively, ISL, to the category Bool, of Boolean algebras, respectively,
Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from
Heyt to Bool has, in addition to its well-known right adjoint (that is, the
canonical embedding of Bool into Heyt) a left adjoint, that from the left to the
right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a
string of adjunctions constitutes an adjoint cylinder.
 Balbes, R., and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, 1974.
 Brown, D. J., R. Suszko, and S. L. Bloom, "Abstract logics", Dissertationes Mathematicae, vol. 102 (1973), pp. 5--41.
 Cleave, J. P., A Study of Logics, vol. 18 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
 Cohn, P. M., Universal Algebra, 2d edition, vol. 6 of Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1981.
Mathematical Reviews (MathSciNet): MR620952
 Gentzen, G., "Die Widerspruchsfreiheit der reinen Zahlentheorie", Mathematische Annalen, vol. 112 (1936), pp. 493--565.
 Gödel, K., "Zur Intuitionistischen Arithmetic und Zahlentheorie", Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1931/32), pp. 34--38.
 Lawvere, F. W., "Cohesive toposes and Cantor's `lauter Einsen'. Categories in the foundations of mathematics and language", Philosophia Mathematica. Series III, vol. 2 (1994), pp. 5--15.
 Luckhardt, H., Extensional Gödel Functional Interpretation. A Consistency Proof of Classical Analysis, vol. 306 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1973.
 Mac Lane, S., Categories for the Working Mathematician, 2d edition, vol. 5 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.
 Monk, J. D., Mathematical Logic, vol. 37 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1976.
 Prawitz, D., and P.-E. Malmnäs, "A survey of some connections between classical, intuitionistic and minimal logic", pp. 215--29 in Contributions to Mathematical Logic (Colloquium, Hannover, 1966), edited by H. Schmidt et al., North-Holland, Amsterdam, 1968.
 Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, vol. 41 of Monografie Matematyczne, Państwowe Wydawnictwo Naukowe, Warsaw, 1963.
 Smith, J. D. H., Mal'cev Varieties, vol. 554 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1976.
 Tarski, A., "Grundzüge des Systemenkalküls. Erster Teil", Fundamenta Mathematicae, vol. 25 (1935), pp. 503--26.
 Wójcicki, R., Theory of Logical Calculi. Basic Theory of Consequence Operations, vol. 199 of Synthese Library, Kluwer Academic Publishers Group, Dordrecht, 1988.