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An Introduction to Partition Logic

Ellerman, David (2013) An Introduction to Partition Logic. Logic Journal of the IGPL.

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Abstract

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets.
Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Ellerman, Daviddavid@ellerman.org
Keywords: logic of partitions, Boolean logic of subses, propositional logic, subset-quotient set duality, quantum mechanics
Subjects: Specific Sciences > Mathematics
Specific Sciences > Physics > Quantum Mechanics
Depositing User: David Ellerman
Date Deposited: 11 Jan 2014 15:45
Last Modified: 11 Jan 2014 15:45
Item ID: 10211
Journal or Publication Title: Logic Journal of the IGPL
Publisher: Oxford University Press
Official URL: http://www.oxfordjournals.org/our_journals/igpl/ab...
DOI or Unique Handle: https://doi.org/10.1093/jigpal/jzt036
Subjects: Specific Sciences > Mathematics
Specific Sciences > Physics > Quantum Mechanics
Date: September 2013
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/10211

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