Abstract
Partition logics often allow a dual probabilistic interpretation: a classical one for which probabilities lie on the convex hull of the dispersion-free weights and another one, suggested independently from the quantum Born rule, in which probabilities are formed by the (absolute) square of the inner product of state vectors with the faithful orthogonal representations of the respective graph. Two immediate consequences are the demonstration that the logico-empirical structure of observables does not determine the type of probabilities alone and that complementarity does not imply contextuality.
Highlights
Partition logics as nonboolean structures pasted from Boolean subalgebrasA partition logic (Svozil 1993; Schaller and Svozil 1994, 1996; Dvurecenskij et al 1995; Svozil 2005) is the logic obtained (i) from collections of such partitions, each partition being identified with an m-atomic Boolean subalgebra of 2n, and (ii) by “stitching” or pasting these subalgebras through identifying identical intertwining elements
T Partitions provide ways to distinguish between elements of a given finite set Sn = {1, 2, . . . , n}
Every partition can be identified with some Boolean subalgebra 2m—in graph theoretical terms a clique—of 2n whose atoms are the elements of that partition
Summary
A partition logic (Svozil 1993; Schaller and Svozil 1994, 1996; Dvurecenskij et al 1995; Svozil 2005) is the logic obtained (i) from collections of such partitions, each partition being identified with an m-atomic Boolean subalgebra of 2n, and (ii) by “stitching” or pasting these subalgebras through identifying identical intertwining elements In quantum logic, this is referred to as pasting construction, and the partitions are identified with, or are synonymously denoted by, blocks, subalgebras or cliques, which are representable by orthonormal bases or maximal operators. Constructing 5 contexts/cliques from the occurrences of the dispersion-free value 1 on the respective 10 atoms results in the partition logic based on the set of indices of the dispersion-free states S11 = {1, . It is an interesting problem to find other potential probability measures based on different approaches which are linear in mutually exclusive events
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