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  1. Four Ways from Universal to Particular: How Chomsky's Language-Acquisition Faculty is Not Selectionist.David Ellerman - 2016 - Journal of Applied Non-Classical Logics 3 (26):193-207.
    Following the development of the selectionist theory of the immune system, there was an attempt to characterize many biological mechanisms as being "selectionist" as juxtaposed to "instructionist." But this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky's language-acquisition mechanism as all being "selectionist." Yet Chomsky's mechanism (and embryonic development) are significantly different from the selectionist mechanisms of biological evolution or the immune system. Surprisingly, there is a very abstract way using two dual mathematical logics to (...)
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  • Partitions and Objective Indefiniteness.David Ellerman - manuscript
    Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...)
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  • An Introduction to Partition Logic.David Ellerman - 2014 - Logic Journal of the IGPL 22 (1):94-125.
    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 (...)
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  • An introduction to logical entropy and its relation to Shannon entropy.David Ellerman - 2013 - International Journal of Semantic Computing 7 (2):121-145.
    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 (...)
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  • Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces.David Ellerman - 2022 - Foundations of Physics 52 (5):1-40.
    The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite (...)
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  • How Category Theory Works.David Ellerman - manuscript
    The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...)
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  • Logical Entropy: Introduction to Classical and Quantum Logical Information theory.David Ellerman - 2018 - Entropy 20 (9):679.
    Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper (...)
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  • The logic of partitions: Introduction to the dual of the logic of subsets: The logic of partitions.David Ellerman - 2010 - Review of Symbolic Logic 3 (2):287-350.
    Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...)
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  • On Classical and Quantum Logical Entropy.David Ellerman - manuscript
    The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...)
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  • On the duality between existence and information.David Ellerman - manuscript
    Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...)
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  • Partitions and Objective Indefiniteness in Quantum Mechanics.David Ellerman - manuscript
    Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are category-theoretically dual to one another (...)
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  • Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus.David Ellerman - 2017 - Synthese (12).
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  • On classical finite probability theory as a quantum probability calculus.David Ellerman - manuscript
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is not to (...)
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  • Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus.David Ellerman - 2017 - Synthese (12):4863-4896.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  • Logical information theory: new logical foundations for information theory.David Ellerman - 2017 - Logic Journal of the IGPL 25 (5):806-835.
    There is a new theory of information based on logic. The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform transformation from the corresponding formulas of logical information theory. Information is first defined in terms of sets of distinctions without using any probability measure. When a probability measure is introduced, the logical entropies are simply the values of the probability measure on the sets (...)
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  • Four ways from universal to particular: how Chomsky’s principles-and-parameters model is not selectionist.David P. Ellerman - 2016 - Journal of Applied Non-Classical Logics 26 (3):193-207.
    Following the development of the selectionist theory of the immune system, there was an attempt to characterise many biological mechanisms as being ‘selectionist’ as juxtaposed with ‘instructionist’. However, this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky’s principles-and-parameters language-acquisition mechanism together under the ‘selectionist’ umbrella, even though Chomsky’s mechanism and embryonic development are significantly different from the selectionist mechanisms of biological evolution and the immune system. Surprisingly, there is an abstract way using two dual mathematical (...)
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