Generality of proofs and its Brauerian representation

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. (2000): 03F07, 03G30, 18A15, 16G99 Mathematics Subject Classification

A Brauerian representation of split preorders

“…This representation is related to Brauer's representation of Brauer algebras [1], which it generalizes in a certain sense (see [6], Section 6).…”

“…In the presence of ⊤ and ⊥, we would obtain a particular brand of bicartesian category, according to the coherence result of [4]. If we replace the split preorders G(f ) considered here for conjunctive-disjunctive logic by their transitive and symmetric closures, as we did above for conjunctive logic, we will not obtain any more as the category C the free category with nonempty finite products and coproducts (see [6]). …”

“…This is the reason why we call our representation of split preorders Brauerian. It is a generalization of Brauer's representation, and of the Brauerian representation of Gen of [6].…”

“…In this paper we will generalize the results of [6]. We will represent the category whose arrows are split preorders in general in the category Rel whose arrows are binary relations between certain sets.…”

“…We will represent the category whose arrows are split preorders in general in the category Rel whose arrows are binary relations between certain sets. If our split preorders are relations on finite sets W , we may, as in [6], take that in Rel we have as arrows binary relations between finite ordinals. Our representation in Rel, which generalizes [6], generalizes further Brauer's representation, and hence its name.…”