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Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector spacewhich is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a categorytheoretic sense) to (...) 

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 misspecified 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 (...) 

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 reversethearrows categorytheoretic dualitywhich is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) 

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 principlesandparameters languageacquisition 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 (...) 

ince the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector spacewhich is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition is dual to the notion of a subset. Hence the Boolean logic of (...) 