Journal of Symbolic Logic

Generality of proofs and its Brauerian representation

Kosta Došen and Zoran Petrić

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Abstract

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

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

Dates
First available in Project Euclid: 17 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1058448435

Digital Object Identifier
doi:10.2178/jsl/1058448435

Mathematical Reviews number (MathSciNet)
MR2000074

Zentralblatt MATH identifier
1058.03062

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.jsl/1058448435


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