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AGGREGATION AND IDEMPOTENCE

Published online by Cambridge University Press:  07 August 2013

LLOYD HUMBERSTONE
LLOYD HUMBERSTONE*
Affiliation:
Monash University
*
*DEPARTMENT OF PHILOSOPHY SCHOOL OF PHILOSOPHICAL HISTORICAL AND INTERNATIONAL STUDIES MONASH UNIVERSITY VICTORIA 3800, AUSTRALIA
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Abstract

A 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 4 , December 2013 , pp. 680 - 708
Copyright
Copyright © Association for Symbolic Logic 2013 

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