Computability in Europe 2008
Logic and Theory of Algorithms

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Regular Talk:
Anhomomorphic Logic: The Logic of Quantum Realism

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Speaker: Petros Wallden
Slot: Mon, 11:00-11:20, Drakopoulos (col. 1)


Anhomomorphic logic, is a novel interpretation of Quantum Theory (initiated by Sorkin) that comes as a development of the consistent histories approach and is an attempt to retain realism. When using logic to describe (classical) physics, we have a set of possible histories, a set of truth values ( e.g. {True, False}) and the possible maps (Phi_i) that are homomorphisms between the Boolean algebra of subsets of and of the truth values. These maps give rise to different realizations (here is where the measure and thus the dynamics enter the picture). It is well known that the above picture cannot hold in quantum theory. One can either restrict the subsets of allowed (standard consistent histories) or change the set of truth values to some Heyting algebra (Isham) or finally, weaken the requirement that the map is homomorphism. This latter approach is taken in "Anhomomorphic Logic". The weakening of the requirement to be a homomorphism is replaced by other conditions, that guarantee that most structure is preserved and the basic inference law (modus ponens) still aplies. Thus we have a deductive logic (which is not the case in what is usually referred to as "quantum logic"). In this talk, we will first introduce anhomomorphic logic in some detail. Then we will deal with some recent developments on the emergence of classicality and the closely related issue of recovery of probabilistic predictions. The former, essentially means that in some scale of coarse graining, we know that classical (boolean) logic applies. Thus we have to show that the anhomomorphic logic, upon some coarse grainings, results to homomorphic logic (i.e. classical). Finally, the issues of how probabilities arise, is present in several attempts to retain realism in quantum theory (such as many worlds), and in this approach it is resolved with the use of the concept of "approximate preclusion" and by taking a frequentist's view on probability rather than treating it as propensity.

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