Generality of proofs and its Brauerian representation

Journal of Symbolic Logic 68 (3):740-750 (2003)

Authors
Kosta Došen
University of Belgrade
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.
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DOI http://projecteuclid.org/euclid.jsl/1058448435
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References found in this work BETA

Introduction to Higher Order Categorical Logic.J. Lambek & P. J. Scott - 1989 - Journal of Symbolic Logic 54 (3):1113-1114.
Identity of Proofs Based on Normalization and Generality.Kosta Došen - 2003 - Bulletin of Symbolic Logic 9 (4):477-503.
A Brauerian Representation of Split Preorders.Z. Petric & K. Dosen - 2003 - Mathematical Logic Quarterly 49 (6):579.
The Maximality of Cartesian Categories.Z. Petric & K. Dosen - 2001 - Mathematical Logic Quarterly 47 (1):137-144.
Bicartesian Coherence.Kosta Došen & Zoran Petrić - 2002 - Studia Logica 71 (3):331-353.

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Citations of this work BETA

Identity of Proofs Based on Normalization and Generality.Kosta Došen - 2003 - Bulletin of Symbolic Logic 9 (4):477-503.
Equality of Proofs for Linear Equality.Kosta Došen & Zoran Petrić - 2008 - Archive for Mathematical Logic 47 (6):549-565.
Isomorphic Formulae in Classical Propositional Logic.Kosta Došen & Zoran Petrić - 2012 - Mathematical Logic Quarterly 58 (1):5-17.

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Generality of Proofs and Its Brauerian Representation.Kosta Došen & Zoran Petrić - 2003 - Journal of Symbolic Logic 68 (3):740 - 750.
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