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Quantum mechanics (QM) is recognized as one of the most successful physical theories however, it is a common place to say that we do not know what the theory talks about. And, although the features that once were considered as problems ?like the Schrodinger cat, related to the fact that the Hilbert space comes with an addition that allows superpositions, or entanglement, first shown by EPR? are nowadays resources of technical applications, nevertheless we continue lacking a semantics and a conceptual language for QM.In classical physics, every system may be described by specifying its actual properties. Mathematically, this is done by means of representing the state of the system by a point (p, q) in the corresponding phase space Γ and its properties by subsets of Γ, which combine following the usual set-theoretic operations. Consequently, the propositional structure associated to properties about a classical system follows the rules of classical logic. In the orthodox formulation of QM, the representation of a pure state of a system is given by a ray in Hilbert space H and physical properties by closed sub-spaces of H, which with adequate definitions of meet and join operations build an orthomodular lattice (Maeda & Maeda 1970). This lattice, denoted by L(H), is called the Hilbert lattice associated to H and motivates the standard quantum logic (QL) introduced in the thirties by Birkhoff and von Neumann (Birkhoff & von Neumann, 1936).The traditional version of QL needs careful consideration for several reasons. From an algebraic point of view, QL is founded on the orthomodular lattice structure. But it is well known that the variety of orthomodular lattices is strictly larger than the variety generated by the Hilbert lattices. Thus, standard QL does not fully capture the concept of the Hilbert lattice. From a physical point of view several features appear: if P represents a proposition about the system, in general there are superposition states in which it is wrong to say that either P or its negation ¬P hold, in accord with the association of the join operation not with set theoretical union but with the smallest closed subspace including the projection represented by P and its orthocomplement. However, the orthomodular structure satisfies the equation P ∨ ¬P = 1 which is a kind of law of the excluded middle. Thus, as discussed in (Dalla Chiara, Giuntini & Greechhie, 2004), it seems necessary to distinguish the logical law of excluded middle from the semantic principle in which the truth of the disjunction implies the truth of at least one of the members. Moreover, in spite of the fact that the meet of its elements is well defined in the lattice, there are conjunctions of (actual) properties that make no sense because the corresponding operators do not commute. Thus, the orthomodular structure shows a kind of conflict with the underlying physical content of the theory. There is also a well known difficulty with traditional forms of QL in relation to composite systems, namely the lack of a canonical formalism for dealing with the properties of a the whole system when given the description of its components. In fact, if H1 and H2 are the representatives of two systems, the postulates of QM say that the tensor product H = H1 ⊗ H2 stands for the representative of the composite. But the naive construction of the lattice of propositions of the whole as the tensor product of the lattices of the individuals fails (Aerts, 1984a; 1984b; Randall & Foulis 1981) due to the lack of a product of lattices, or even posets (Aerts & Daubechies, 1979). Mathematically, this is the expression of the fact that the category of Hilbert lattices as objects and lattice morphisms as arrows has not a categorial product because of the failure of orthocomplementation (Aerts, 1984a; Gudder 1978). Attempts to vary the conditions that define the product of lattices have been made (Pulmannová, 1985), but in all cases it results that the Hilbert lattice factorizes only in the case in which one of the factors is a Boolean lattice or when systems have never interacted, rendering the construction physically useless. For a complete review, see (Dvure ̆censkij, 1995).In the last years several approaches using category theory have been used to search for an adequate and rigorous language for quantum systems. First, both from a neo-realist point of view (Caspers, Heunen, Landsman & Spitters, 2009; A. D ̈oring and C.J. Isham, 2010; C. Heunen, N. Landsman and B. Spitters, 2009; Isham, 2010) or not (Zafiris & Karakostas, 2013), there are attempts that relate algebraic QM to topos theory, which comes with an associated intrinsic (intuitionistic) logic. Moreover, some of these attempts equip the structure with an external intuitionistic logic. In these approaches, the quantum analogue of classical phase space is captured by the notion of frame. There are also other attempts related to category theory for describing several aspect of the quantum systems. For example, contextuality and non-locality may be modeled using the framework of sheaf theory too (Abramsky & Brandenburger, 2011) and monoidal categories are used for representing processes (Abramsky & Coecke, 2004; 2008). This approach also enables a consistent description of compound systems (B. Coecke, C. Heunen and A. Kissinger, 2000), a deep difficulty for standard QL.

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