What can (partition) logic contribute to information theory?

Information as Distinctions: New Foundations for Information Theory

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of orde…

Counting Distinctions: On the Conceptual Foundations of Shannon's Information Theory

Categorical logic has shown that modern logic is essentially the logic of subsets (or "subobjects"). Partitions are dual to subsets so there is a dual logic of partitions where a "distinction" [an ordered pair of distinct elements (u,u') from the universe U ] is dual to an "element". An element being in a subset is analogous to a partition p on U making a distinction, i.e., if u and u' were in different blocks of p. Subset logic leads to finite probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the lo…

Introduction to Logical Entropy and its Relationship to Shannon Entropy

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an "amount of information," but, as he pointed out, "no concept of information itself was defined." Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set--just as the Boole-Laplace logical probability is the normalized quantitative measure of the e…

The Objective Indefiniteness Interpretation of Quantum Mechanics

The common-sense view of reality is expressed logically in Boolean subset logic (each element is either definitely in or not in a subset, i.e., either definitely has or does not have a property). But quantum mechanics does not agree with this "properties all the way down" picture of micro-reality. Are there other coherent alternative views of reality? A logic of partitions, dual to the Boolean logic of subsets (partitions are dual to subsets), was recently developed along with a logical version of information theory. In view of the subset-partition duality, partition logic is the alternative t…

低权重规范逻辑基来自配对划分码

Low-Weight Canonical Logical Bases from Pair-Partition Codes

本文为量子比特CSS码构造低权重规范逻辑基，通过四元系数与配对多项式实现，并在多个码上验证了权重优势。