Journal of Symbolic Logic

Generality of proofs and its Brauerian representation

Kosta Došen and Zoran Petrić

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference.

This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category whose arrows are equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of logic, with the associated map into the category of equivalence relations of generality.

This category is isomorphically represented in the category whose arrows are binary relations between finite ordinals, where composition is the usual simple composition of relations. This representation is related to a classical representation result of Richard Brauer.

Article information

J. Symbolic Logic, Volume 68, Issue 3 (2003), 740- 750.

First available in Project Euclid: 17 July 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F07: Structure of proofs 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10] 18A15: Foundations, relations to logic and deductive systems [See also 03- XX] 16G99: None of the above, but in this section

identity criteria for proofs generality of proof categories of proofs Brauer algebras representation


Došen, Kosta; Petrić, Zoran. Generality of proofs and its Brauerian representation. J. Symbolic Logic 68 (2003), no. 3, 740-- 750. doi:10.2178/jsl/1058448435.

Export citation


  • R. Brauer On algebras which are connected with the semisimple continuous groups, Annals of Mathematics, vol. 38 (1937), pp. 857--872.
  • K. Došen Identity of proofs based on normalization and generality,2002, (available at: \ttfamily
  • K. Došen, Ž. Kovijanić, and Z. Petrić, A new proof of the faithfulness of Brauer's representation of Temperley-Lieb algebras, 2002, (available at: \ttfamily
  • K. Došen and Z. Petrić The maximality of cartesian categories, Mathematical Logic Quarterly, vol. 47 (2001), pp. 137--144, (available at: \ttfamily
  • S. Eilenberg and G. M. Kelly A generalization of the functorial calculus, Journal of Algebra, vol. 3 (1966), pp. 366--375.
  • V. F. R. Jones A quotient of the affine Hecke algebra in the Brauer algebra, Enseignement des Mathématiques, vol. 40 (1994), no. 2, pp. 313--344.
  • C. Kassel Quantum groups, Springer, Berlin,1995.
  • G. M. Kelly and S. Mac Lane Coherence in closed categories, Journal of Pure and Applied Algebra, vol. 1 (1971), pp. 97--140 and 219.
  • J. Lambek Deductive systems and categories I: Syntactic calculus and residuated categories, Mathematical Systems Theory, vol. 2 (1968), pp. 287--318.
  • J. Lambek and P. J. Scott Introduction to higher-order categorical logic, Cambridge University Press, Cambridge,1986.
  • Z. Petrić Coherence in substructural categories, Studia Logica, vol. 70 (2002), pp. 271--296, (available at: \ttfamily
  • D. Prawitz Ideas and results in proof theory, Proceedings of the second Scandinavian logic symposium (J. E. Fenstad, editor), North-Holland, Amsterdam,1971, pp. 235--307.
  • M. E. Szabo A counter-example to coherence in cartesian closed categories, Canadian Mathematical Bulletin, vol. 18 (1975), pp. 111--114.
  • H. Wenzl On the structure of Brauer's centralizer algebras, Annals of Mathematics, vol. 128 (1988), pp. 173--193.