David Ellerman

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.

A theory of property needs to give an account of the whole life-cycle of a property right: how it is initiated, transferred, and terminated. Economics has focused on the transfers in the market and has almost completely neglected the question of the initiation and termination of property in normal production and consumption. Yet the market also provides a laissez-faire mechanism: when the legal authorities do not intervene, then the initial right is, in effect, assigned to the first seller and the terminal liability to the last buyer. But does this mechanism satisfy the juridical principle of responsibility: assign de jure responsibility in accordance with de facto responsibility? The fundamental theorem states that if all transfers are covered by voluntary contracts and all contracts are fulfilled, then the laissez-faire mechanism satisfies the responsibility principle.Une théorie de la propriété se doit d’expliquer entièrement le cycle de vie du droit de propriété: les conditions de son émergence, de son transfert et de sa résiliation. La théorie économique s’est focalisée sur les mécanismes de transferts sur le marché et a presque totalement négligé la question de l’émergence et de la réalisation de la propriété dans les phénomènes de production et de consommation. Pourtant, le marché fournit également un mécanisme de laissez-faire: lorsque les autorités légales s’abstiennent d’intervenir, le droit initial est affecté au premier vendeur et la responsabilité terminale au dernier acheteur. Néanmoins, ce mécanisme satisfait-il le principe juridique de responsabilité? A savoir, assigner une responsabilité de jure en accord avec le principe de responsabilité de facto? En effet, le théorème fondamental exprime qui si tous les transferts s’effectuent par des contrats volontaires, et que tous les contrats sont remplis, alors le principe du laissez-faire satisfait le principe de responsabilité.