# Non-commutative Multiple-Valued Logic Algebras (Springer Monographs in Mathematics)

So, dynamical states do not coincide with property states. The determinate sublattice, which changes with the dynamics of the system, is a partial Boolean algebra, that is, the union of a family of Boolean algebras pasted together in such a way that the maximum and minimum elements of each one, and eventually other elements, are identified and, for every n -tuple of pair-wise compatible elements, there exists a Boolean algebra in the family containing the n elements.

Thus constructed, the structure avoids KS-type theorems.

Then, given a system S and a measuring apparatus M ,. Moreover, the quantum state can be interpreted as assigning probabilities to the different possible ways in which the set of determinate quantities can have values, where one particular set of values represents the actual but unknown values of these quantities. The problem with this interpretation is that, in the case of an isolated system, there is no single element in the formalism of QM that allows us to choose an observable R , rather than another.

This is why the move seems flagrantly ad hoc. Were we dealing with an apparatus, there would be a preferred observable, namely the pointer position, but the quantum wave function contains in itself mutually incompatible representations choices of apparatuses each of which provides non-trivial information about the state of affairs. The authors of this work have also contributed to the understanding of modality in the context of orthodox QL [, , , ].

From our investigation there are several conclusions which can be drawn. We started our analysis with a question regarding the contextual aspect of possibility. As it is well known, the KS theorem does not talk about probabilities, but rather about the constraints of the formalism to actual definite valued properties considered from multiple contexts.

What we found via the analysis of possible families of valuations is that a theorem which we called, for obvious reasons, the Modal KS MKS theorem can be derived which proves that quantum possibility, contrary to classical possibility, is also contextually constrained [].

### References

This means that, regardless of its use in the literature, quantum possibility is not classical possibility. In a paper written in [88], we concentrated on the analysis of actualization within the orthodox frame and interpreted, following the structure, the logical realm of possibility in terms of ontological potentiality.

The study of the structure of tensor products [57, , , , ] motivated a fruitful development of different algebraic structures that could represent quantum propositions, which in turn became a line of investigation by itself. Beginning with the proposal of test spaces by Foulis and Randall [, , , , , , ], which are related to orthoalgebras, the theory of structures as orthomodular lattices, partial Boolean algebras, orthomodular posets, effect algebras, quantum MV-algebras and the like became widely discussed.

The weakened structures allow consideration of unsharp propositions related, not to projections, but to the elements of the more general set of linear bounded operators—called effects —over which the probability measure given by the Born rule may be defined. An important line of research in the subject of quantum structures is the application of QL methods to languages of information processing and, more specifically, to quantum computational logic QCL [53, 80, , 82, , , , , , , ].

In this way several logical systems associated to quantum computation were developed. They provide a new form of quantum logic strongly connected with the fuzzy logic of continuous t -norms []. The groups in Firenze directed by Dalla Chiara, and Cagliari directed by Giuntini, have also developed different languages for quantum computation. A sentence in QL may be interpreted as a closed subspace of H. Instead, the meaning of an elementary sentence in QCL is a quantum information quantity encoded in a collection of qbits —unit vectors pertaining to the tensorial product of two dimensional complex Hilbert spaces—or qmixes —positive semi-definite Hermitian operators of trace one over Hilbert space.

Conjunction and disjunction are not associated to the join and meet lattice operations. On the one hand, NRL is, in a wide sense, a logic in which the relation of identity or equality is restricted, eliminated, replaced, at least in part, by a weaker relation, or employed together with a new non-reflexive implication or equivalence relation. There are other versions in higher-order logic, in which higher order variables appear. Some of the above principles are not in general valid in non-reflexive logics. They are total or partially eliminated, restricted, or not applied to the relation that is employed instead of identity.

Several of these principles are the motivations for the development of non-reflexive logics. In the Congress, Manin proposed as one of the new set of problems for the next century:. New quantum physics has shown us models of entities with quite different behaviour. Within this context, the weakening of the concept of identity—substituted by that of indiscernibility—allows the development of non-reflexive logics which, in a wide sense, are logics in which the relation of identity or equality is restricted, eliminated, replaced, at least in part, by a weaker relation, or employed together with a new non-reflexive implication or equivalence relation [68, 73, , 75].

There are also different approaches to the logic related to quantum set theories.

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Gaisi Takeuti proposed a quantum set theory developed in the lattice of projections-valued universe [, ] and Satoko Titani formulated a lattice valued logic corresponding to general complete lattices developed in the classical set theory based on the classical logic []. On the other hand, PL are the logics of inconsistent but non-trivial theories. The origins of PL go back to the first systematic studies dealing with the possibility of rejecting the PNC. PL was elaborated, independently, by Stanislaw Jaskowski in Poland, and by Newton da Costa in Brazil, around the middle of the last century on PL, see, for example: [72].

T is called trivial if any sentence of its language is also a theorem of T ; otherwise, T is said to be non-trivial. In classical logics and in most usual logics, a theory is inconsistent if, and only if, it is trivial. L is paraconsistent when it can be the underlying logic of inconsistent but non-trivial theories. Clearly, no classical logic is paraconsistent.

The notion of complementarity was developed by Bohr in order to consider the contradictory representations of wave representation and corpuscular representation found in the double-slit experiment see for example []. There is a great amount of work in progress in QL from new quantum structures, to the use of non-reflexive logics, paraconsistent logics, dynamical logics, etc. In the following section we shall review some of these advancements that have taken place in relation to QM.

IQSA gathers experts on quantum logic and quantum structures from all over the world under its umbrella. In fact, in the subject of quantum structures, MV-algebras, effect algebras, pseudo-effect algebras and related structures are being developed in relation to their use in QM. See [55, , , , , , ], just to cite a few examples.

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As mentioned above see Section 4. The standpoint of this approach is the observation that QL is essentially a dynamical logic, that it is about actions rather than propositions [30].

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Smets together with Alexandru Baltag have proposed two axiomatizations of the logic of quantum actions []. One of them takes the notion of action as fundamental and axiomatizes the underlying algebra, giving a quantale [22, 59]. The other takes the notion of state as fundamental and represents actions as relations between states. Contrary to orthomodular QL [78], these axiomatizations fulfill completeness with respect to infinite dimensional Hilbert spaces and have applications in computational science [28]. In fact, the application to computational science and more broadly to information processing needs to manage composite systems, one of the profound difficulties that faces orthodox QL.

Also, the relation between category theory and QL is being explored from different perspectives. They make claims about the necessity of reviewing the basic suppositions that are taken from granted, for example, the nature of space-time, the use of real numbers as values of physical quantities and the meaning of probability.

## Non-commutative Multiple-Valued Logic Algebras - eBook

From a logical point of view, contrary to the intractable QL, any topos in which the physical theory is represented comes with an intrinsic intuitionistic logic that is obviously more tractable. Moreover, compound systems also find their place in the topos approach []. Classical theories are included in this new formalization and for all of them the corresponding topos is that of sets endowed with classical logic as a trivial intuitionistic one.

Also Elias Zafiris and Vassilios Karakostas are making new research in categorial semantics []. On the other side, the line of investigation initiated at Oxford by Samson Abramsky and continued by Coecke among others proposed an axiomatization which may be useful for managing the formal language of physical processes involved in new quantum technologies as quantum computation and teleportation. Quantum computers exploit the existence of superpositions to drastically decrease the time and recourses required to deal with certain problems such as triangle-finding, integer factorization or the searching of an entry in an unordered list [, ].

Teleportation uses non-separability to safely transmit information from one place to another by means of an entangled state and a classical communication channel [45]. The categorical approach of the Oxford group uses monoidal categories [1, 2, 3, 4, 67, ] and simple diagrams to view quantum processes and composite systems in a consistent manner. They apply these tools to research in the subject of computing semantics [63, 64, 65], in particular in the subject of linear logic [] which is essential for computing science.

Also Cristina and Amilcar Sernadas in Lisbon are working on the connection of category theory and linear logic [, , 49]. Research on computational semantics is being developed in connection with epistemic logics by members of the Italian group. The relation between quantum structures and epistemic logics is also being studied by a group in Amsterdam.

They are applying a modal dynamic-epistemic QL for reasoning about quantum algorithms and, in general, for considering quantum systems as codifying actions of information production and processing [29, 30, 31]. Dynamics of concepts as studied by cognitive science are also being considered with the aid of quantum structures. In fact, D. Aerts and co-workers have applied the formalism of QM for modeling the combination of concepts, showing the indeterministic and holistic characters of this process [13, 16, 17, 20, 21]. This approach has technological applications in connection with quantum computation and robotics [18, 19].

As remarked by Coecke the meaning of the superposition principle might be the key to understand QM:. Birkhoff and von Neumann crafted quantum logic in order to emphasize the notion of quantum superposition. In terms of states of a physical system and properties of that system, superposition means that the strongest property which is true for two distinct states is also true for states other than the two given ones. Birkhoff and von Neumann as well as many others believed that understanding the deep structure of superposition is the key to obtaining a better understanding of quantum theory as a whole.

In line with this intuition, in [74], one of the authors of this paper together with N. It was claimed that, even though most interpretations of QM attempt to escape contradictions, there are many hints—coming mainly from present technical and experimental developments in QM—that indicate it could be worthwhile to engage in a research of this kind. Arenhart and Krause [23, 24, 25] have raised several arguments against the paraconsistent approach to quantum superpositions which have been further analyzed in [86].

Recently, some new proposals to consider quantum superpositions from a logical perspective have been put forward [76, ].

Possible relations between two of the mentioned types of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach may be represented by taking into account monadic Boolean algebras. The square of opposition has been considered, in relation to QL, as a useful tool to identify paraconsistent negations [38, 40]. This representation is called the modal square of opposition.

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An extension of the square to a case in which the underlying structure is replaced by the algebra of QL has been provided in [] and it may be useful to identify paraconsistent negations in the structure of QM see also for discussion [89].