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

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 et al., 2001) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. The action of the Sasaki hook ( $axrightarrow{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., ( $axrightarrow{S}b$ ) 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. 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,’ what leads us to the claim made in the title of this paper. The Sasaki adjunction has a physical significance in terms of causal duality. The labeled dynamic hooks (forwardly and backwardly) that encode quantum measurements, act on properties as $(a_1 xrightarrow{{varphi _a }}a_2 ): = (a_1 to _L (axrightarrow{S}a_2 ))$ and $(a_1 xleftarrow{{varphi _a }}a_2 ): = ((axrightarrow{S}a_2 ) to _L a_1 )$ , taking values in the ‘disjunctive extension’ $DI(L)$ of the property lattice L, where $a in L$ is the tested property and $( - to _L - )$ is the Heyting implication that lives on DI(L). Since these hooks $( - xrightarrow{{varphi _a }} - )$ and $( - xleftarrow{{varphi _a }} - )$ extend to DI(L)×DI(L) they constitute internal operations. The transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective ‘operational resolution’ (Coecke, 2002a) 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)