The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

Abstract

In this paper we argue that the Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a 'quantum implicative connective', has a fundamental dynamic nature and encodes the so-called 'causal duality' (Coecke, Moore and Stubbe 2001) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. In particular: The action of the Sasaki hook (a S → −) for fixed antecedent a assigns to some property "the weakest cause before the measurement of actuality of that property after the measurement", i.e. (a S → b) is the weakest property that guarantees actuality of b after performing the measurement represented by the projector that has the 'subspace a' as eigenstates for eigenvalue 1 , say, the measurement that 'tests' a . From this we conclude that the logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within 'dynamic (operational) quantum logic' (DOQL), what leads us to the claim made in the title of this paper. More explicitly, although (as many argued in the past) the Sasaki hook should not be seen as an implicative hook, the formal motivation that persuaded others to do so, i.e. the Sasaki adjunction, does have a physical significance (in terms of causal duality). It is within the context of DOQL that we can then derive that the labeled dynamic hooks (forwardly and backwardly) that encode quantum measurements act on properties as (a 1, taking values in the 'disjunctive extension' DI(L) of the property lattice L (Coecke 2001a) , where a ∈ L is the tested property and (− → L −) is the Heyting implication that lives on DI(L) . Since these hooks (− ϕa → −) and (− ϕa ← −) extend to DI(L) × DI(L) they constitute internal operations . In an even more radical perspective one could say that the transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective 'operational resolution' (Coecke 2001a) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement). QUANTUM LOGICALITYWe claim that logical considerations on quantum behavior and as such, further development of the research field, have been 'corrupted' by two features. Once these two features are neutralized, the way towards an essentially dynamic quantum logic (i.e. a unified logic of 'changes' both for classical and quantum systems), or otherwise put, a true quantum process semantics, is opened. Moreover, the solution to the second 'corrupt feature' indicates that the logicality encoded in pure quantum theory is of a fundamental dynamic nature. Structures somewhat similar to those emerging in the context of categorical grammar (Lambek 1958), linear logic (Girard 1987...

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Modeling correlated information change: from conditional beliefs to quantum conditionals

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