The experimental logic of Moore and Mealy type automata is investigated. key words: automaton logic; partition logic; comparison to quantum logic; intrinsic measurements 1

We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain theoretic structure. In the same

Abstract. It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and completeness of the calculuses for the models. We also prove that all the operations in an orthomodular lattice are five-fold defined. In the end we discuss possible repercussions of our results to quantum computations and quantum computers.

this paper is to give an answer to the above question. More precisely, we only aim at clarifying one aspect of von Neumann's dissatisfaction with the Hilbert space formalism: the one which is related to the contrast between the standard Hilbert space formalism and the `case II 1 ' structure { and the analysis is not complete even in this direction (see the end of the nal Section for a few more advantages von Neumann saw in the II 1 structure). Other issues related to von Neumann's criticism of the standard formalism of quantum mechanics { for instance the question of why von Neumann considered it necessary to replace the axioms of the Hilbert space formalism by the axioms of Jordan algebras [24] { are entirely beyond the scope of the present investigation.

be formulated in terms of projective geometry. In three-dimensional Hilbert space, elementary logical propositions are associated with one-dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis {⃗u, ⃗v, ⃗w}, successive application of the binary logical operation (x, y) ↦ → (x ∨ y) ⊥ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis {⃗u, ⃗v, ⃗w} is orthogonal to the other ones. PACS numbers: 02.10.Ab,02.40.-k

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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

Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 5 times slower. We also provide an algorithm and programs for checking of Greechie diagram passage by equations defining varieties of orthomodular lattices and give examples from Hilbert lattices. At the end we discuss some additional characteristics of Greechie diagrams. PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd

We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simpler. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplied proof of Shultz's Theorem on characterization of spaces of nitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for...