of the same observable at increasing times (for memory) and at decreasing times (for anticipation). It should be an interesting project to find the conditions under which (and the accuracy to which) the Markov property for a completely observed time series of a sufficiently isolated system, lost in the quantum domain, is recovered in a thermodynamic limit of a large number of particles.

We now show in detail that is inconsistent with quantum mechanics. According to the standard interpretation of probabilities as expectations of projection operators we have $P_{ab}=\<a|\Pi_b|a\>$ and $P_{bc}=\<b|\Pi_c|b\>$, where $\Pi_b=|b\>\<b|$ is the projector to state $b$ (and similarly for $c$). In order that $P_{abc}$ makes sense in the standard interpretation we need $P_{abc}=\<a|\Pi_{bc}|a\>$ with a projector $\Pi_{bc}$ expressing the proposition $\Pi_b \& \Pi_c$ that $B$ is in state $b$ and $C$ is in state $c$.

In orthodox quantum logics ([Birkhoff & von Neumann]{} [@BirN], [Svozil]{} [@Svo]), one considers a projector as representing the subspace of eigenstates to the eigenvalue 1 of the projector and takes the logical operation $\&$ as subspace intersection. One finds that, whenever $\Pi_b \ne \Pi_c$, we have $\Pi_b \& \Pi_c = 0$ and thus $P_{abc}=0$. This is different from , proving that has 