Information as Distinctions: New Foundations for Information Theory David Ellerman University of California/Riverside January 25, 2013 Abstract 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 nite 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 ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. Contents 1 Introduction 2 2 Logical Entropy 3 2.1 Partition logic ........................................ 3 2.2 Logical Entropy ....................................... 4 2.3 A statistical treatment of logical entropy ......................... 8 2.4 A brief history of the logical entropy formula ...................... 8 3 Shannon Entropy 10 3.1 Shannon-Hartley entropy of a set ............................. 10 3.2 Shannon entropy of a probability distribution ...................... 11 3.3 A statistical treatment of Shannon entropy ....................... 11 3.4 Shannon entropy of a partition .............................. 12 3.5 Shannon entropy and statistical mechanics ........................ 13 3.6 The basic dit-bit connection ................................ 13 1