Generality of Proofs and Its Brauerian Representation

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

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Citations of this work

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.
Isomorphic formulae in classical propositional logic.Kosta Došen & Zoran Petrić - 2012 - Mathematical Logic Quarterly 58 (1):5-17.
Coherence for star-autonomous categories.Kosta Došen & Zoran Petrić - 2006 - Annals of Pure and Applied Logic 141 (1):225-242.
Syntax for split preorders.Kosta Došen & Zoran Petrić - 2013 - Annals of Pure and Applied Logic 164 (4):443-481.

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References found in this work

Ideas and Results in Proof Theory.Dag Prawitz & J. E. Fenstad - 1971 - Journal of Symbolic Logic 40 (2):232-234.
Functional completeness of cartesian categories.J. Lambek - 1974 - Annals of Mathematical Logic 6 (3):259.
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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