Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-22T14:46:47.702Z Has data issue: false hasContentIssue false
  • English
  • Français

Identity of Proofs Based on Normalization and Generality

Published online by Cambridge University Press:  15 January 2014

Kosta Došen
Kosta Došen*
Affiliation:
Mathematical Institute, Serbian Academy of Sciences and Arts Knez Mihailova 35, P.F. 367, 11001 Belgrade, SerbiaE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic.

The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well.

The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 4 , December 2003 , pp. 477 - 503
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Brauer, R., On algebras which are connected with the semisimple continuous groups, Annals of Mathematics, vol. 38 (1937), pp. 857–872.CrossRefGoogle Scholar
[2] Buss, S.R., The undecidability of k-provability, Annals of Pure and Applied Logic, vol. 53 (1991), pp. 75–102.CrossRefGoogle Scholar
[3] Carbone, A., Interpolants, cut elimination and flow graphs for the propositional calculus, Annals of Pure and Applied Logic, vol. 83 (1997), pp. 249–299.CrossRefGoogle Scholar
[4] Cockett, J.R.B. and Seely, R.A.G., Weakly distributive categories, Journal of Pure and Applied Algebra, vol. 114 (1997), pp. 133–173 (updated version available at: http://www.math.mcgill.ca/rags).CrossRefGoogle Scholar
[5] Došen, K., Logical constants as punctuation marks, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 362–381; slightly amended version in What is a logical system? (D. M. Gabbay, editor), Oxford University Press, Oxford, 1994, pp. 273-296.CrossRefGoogle Scholar
[6] Došen, K., Cut elimination in categories, Kluwer, Dordrecht, 1999.CrossRefGoogle Scholar
[7] Došen, K., Abstraction and application in adjunction, Proceedings of the Tenth Congress of Yugoslav Mathematicians (Kadelburg, Z., editor), Faculty of Mathematics, University of Belgrade, Belgrade, 2001, pp. 33–46 (available at: http://arXiv.org/math.CT/0111061).Google Scholar
[8] Došen, K., Models of deduction, to appear in Proceedings of the conference “Proof-Theoretic Semantics, Tübingen 1999” (Schroeder-Heister, P., editor), Oxford University Press, Oxford.Google Scholar
[9] Došen, K., Simplicial endomorphisms, preprint (available at: http://arXiv.org/math.GT/0301302), 2003.Google Scholar
[10] Došen, K. and Petrić, Z., Isomorphic objects in symmetric monoidal closed categories, Mathematical Structures in Computer Science, vol. 7 (1997), pp. 639–662.CrossRefGoogle Scholar
[11] Došen, K., The maximality of the typed lambda calculus and of cartesian closed categories, Publications de l'Institut Mathématique (N.S.), vol. 68(82) (2000), pp. 1–19 (available at: http://arXiv.org/math.CT/9911073).Google Scholar
[12] Došen, K., The maximality of cartesian categories, Mathematical Logic Quarterly, vol. 47 (2001), pp. 137–144 (available at: http://arXiv.org/math.CT/9911059).3.0.CO;2-F>CrossRefGoogle Scholar
[13] Došen, K., Coherent bicartesian and sesquicartesian categories, Proof theory in computer science (Kahle, R. et al., editors), Lecture Notes in Computer Science, vol. 2183, Springer, Berlin, 2001, pp. 78–92 (available at: http://arXiv.org/math.CT/0006091).CrossRefGoogle Scholar
[14] Došen, K., Bicartesian coherence, Studia Logica, vol. 71 (2002), pp. 331–353 (version with some corrections in the proof of maximality available at: http://arXiv.org/math.CT/0006052).CrossRefGoogle Scholar
[15] Došen, K., Generality of proofs and its Brauerian representation, to appear in The Journal of Symbolic Logic (available at: http://arXiv.org/math.LO/0211090), 2002.Google Scholar
[16] Došen, K., A Brauerian representation of split preorders, to appear in Mathematical Logic Quarterly (available at: http://arXiv.org/math.LO/0211277), 2002.Google Scholar
[17] Došen, K., Self-adjunctions and matrices, Journal of Pure and Applied Algebra, vol. 184 (2003), pp. 7–39 (unabridged version available at: http://arXiv.org/math.GT/0111058).CrossRefGoogle Scholar
[18] Eilenberg, S. and Kelly, G.M., A generalization of the functorial calculus, Journal of Algebra, vol. 3 (1966), pp. 366–375.CrossRefGoogle Scholar
[19] Feferman, S., Review of [44], The Journal of Symbolic Logic, vol. 40 (1975), pp. 232–234.CrossRefGoogle Scholar
[20] Freyd, P., Aspects of topoi, Bulletin of the Australian Mathematical Society, vol. 7 (1972), pp. 1–76.CrossRefGoogle Scholar
[21] Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1935), pp. 176-210, 405–431; English translation: Investigations into logical deduction, The collected papers of Gerhard Gentzen (M. E. Szabo, editor), North-Holland, Amsterdam, 1969, pp. 68-131.CrossRefGoogle Scholar
[22] Girard, J.-Y., Taylor, P. and Lafont, Y., Proofs and types, Cambridge University Press, Cambridge, 1989.Google Scholar
[23] Jones, V.F.R., A quotient of the affine Hecke algebra in the Brauer algebra, L'Enseignement Mathématique (2), vol. 40 (1994), pp. 313–344.Google Scholar
[24] Kassel, C., Quantum groups, Springer, Berlin, 1995.CrossRefGoogle Scholar
[25] Kauffman, L.H. and Lins, S.L., Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematical Studies, vol. 134, Princeton University Press, Princeton, 1994.CrossRefGoogle Scholar
[26] Kelly, G.M. and Lane, S. Mac, Coherence in closed categories, Journal of Pure and Applied Algebra, vol. 1 (1971), pp. 97-140, 219.CrossRefGoogle Scholar
[27] Kreisel, G., A survey of proof theory II, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J.E., editor), North-Holland, Amsterdam, 1971, pp. 109–170.CrossRefGoogle Scholar
[28] Lambek, J., Deductive systems and categories I: Syntactic calculus and residuated categories, Mathematical Systems Theory, vol. 2 (1968), pp. 287–318.CrossRefGoogle Scholar
[29] Lambek, J., Deductive systems and categories II: Standard constructions and closed categories, Category theory, homology theory and their applications I, Lecture Notes in Mathematics, vol. 86, Springer, Berlin, 1969, pp. 76–122.CrossRefGoogle Scholar
[30] Lambek, J., Deductive systems and categories III: Cartesian closed categories, intuitionist propositional calculus, and combinatory logic, Toposes, algebraic geometry and logic (Lawvere, F.W., editor), Lecture Notes in Mathematics, vol. 274, Springer, Berlin, 1972, pp. 57–82.CrossRefGoogle Scholar
[31] Lambek, J., Functional completeness of cartesian categories, Annals of Mathematical Logic, vol. 6 (1974), pp. 259–292.CrossRefGoogle Scholar
[32] Lambek, J. and Scott, P.J., Introduction to higher-order categorical logic, Cambridge University Press, Cambridge, 1986.Google Scholar
[33] Lawvere, F.W., Adjointness in foundations, Dialectica, vol. 23 (1969), pp. 281–296.CrossRefGoogle Scholar
[34] Leinster, T., Topology and higher-dimensional category theory: The rough idea (available at: http://arXiv.org/math.CT/0111058), 2001.Google Scholar
[35] Leinster, T., A survey of definitions of n-category, Theory and Application of Categories, vol. 10 (2002), pp. 1–70 (available at: http://arXiv.org/math.CT/0107188).Google Scholar
[36] Lickorish, W.B.R., An introduction to knot theory, Springer, Berlin, 1997.CrossRefGoogle Scholar
[37] Lane, S.Mac, Natural associativity and commutativity, Rice University Studies, Papers in Mathematics, vol. 49 (1963), pp. 28–46.Google Scholar
[38] Lane, S.Mac, editor, Coherence in categories, Lecture Notes in Mathematics, vol. 281, Springer, Berlin, 1972.Google Scholar
[39] Martin-Löf, P., An intuitionistic theory of types: Predicative part, Logic Colloquium '73 (Rose, H.E. and Shepherdson, J.C., editors), North-Holland, Amsterdam, 1975, pp. 73–118.Google Scholar
[40] Martin-Löf, P., About models for intuitionistic type theories and the notion of definitional equality, Proceedings of the Third Scandinavian Logic Symposium (Kanger, S., editor), North-Holland, Amsterdam, 1975, pp. 81–109.CrossRefGoogle Scholar
[41] Moschovakis, Y.N., What is an algorithm?, Mathematics unlimited—2001 and beyond (Engquist, B. and Schmid, W., editors), Springer, Berlin, 2001, pp. 919–936.CrossRefGoogle Scholar
[42] Petrić, Z., Coherence in substructural categories, Studia Logica, vol. 70 (2002), pp. 271–296 (available at: http://arXiv.org/math.CT/0006061).CrossRefGoogle Scholar
[43] Prasolov, V.V. and Sosinskii, A.B., Knots, links, braids and 3-manifolds (in Russian), MCNMO, Moscow, 1997.Google Scholar
[44] Prawitz, D., Ideas and results in proof theory, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J.E., editor), North-Holland, Amsterdam, 1971, pp. 235–307.CrossRefGoogle Scholar
[45] Prawitz, D., Philosophical aspects of proof theory, Contemporary philosophy: A new survey, vol. 1 (Fløistad, G., editor), Nijhoff, The Hague, 1981, pp. 235–277.Google Scholar
[46] Pudlák, P., The lengths of proofs, Handbook of proof theory (Buss, S., editor), North-Holland, Amsterdam, 1998, pp. 547–637.CrossRefGoogle Scholar
[47] Riguet, J., Relations binaires, fermetures, correspondances de Galois, Bulletin de la Société mathématique de France, vol. 76 (1948), pp. 114–155.CrossRefGoogle Scholar
[48] Simpson, A.K., Categorical completeness results for the simply-typed lambda-calculus, Typed lambda calculi and applications (Dezani-Ciancaglini, M. and Plotkin, G., editors), Lecture Notes in Computer Science, vol. 902, Springer, Berlin, 1995, pp. 414–427.CrossRefGoogle Scholar
[49] Soloviev, S.V., The category of finite sets and cartesian closed categories (in Russian), Zapiski Nauchnykh Seminarov (LOMI), vol. 105 (1981), pp. 174–194 (English translation in Journal of SovietMathematics, vol. 22 (1983), pp. 1387-1400).Google Scholar
[50] Statman, R., λ-definable functionals and βη-conversion, Archiv für mathematische Logik und Grundlagenforschung, vol. 23 (1983), pp. 21–26.CrossRefGoogle Scholar
[51] Szabo, M.E., Algebra of proofs, North-Holland, Amsterdam, 1978.Google Scholar
[52] Thiele, R., Hilbert's twenty-fourth problem, American Mathematical Monthly, vol. 110 (2003), pp. 1–24.CrossRefGoogle Scholar
[53] Troelstra, A.S., Non-extensional equality, Fundamenta Mathematicae, vol. 82 (1975), pp. 307–322.CrossRefGoogle Scholar
[54] Turaev, V.G., Operator invariants of tangles and R-matrices (in Russian), Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, vol. 53 (1989), pp. 1073–1107 (English translation in Mathematics of the USSR-Izvestiya, vol. 35, 1990, pp. 411-444).Google Scholar
[55] Wenzl, H., On the structure of Brauer's centralizer algebras, Annals of Mathematics, vol. 128 (1988), pp. 173–193.CrossRefGoogle Scholar
[56] Widebäck, F., Identity of proofs, doctoral dissertation, University of Stockholm, Almqvist & Wiksell, Stockholm, 2001.Google Scholar
[57] Yetter, D.N., Markov algebras, Braids (Birman, J.S. and Libgober, A., editors), Contemporary Mathematics, vol. 78, American Mathematical Society, Providence, 1988, pp. 705–730.CrossRefGoogle Scholar
[58] Zach, R., Completeness before Post: Bernays, Hilbert, and the development of propositional logic, The Bulletin of Symbolic Logic, vol. 5 (1999), pp. 331–366.CrossRefGoogle Scholar