It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is so possible taking into account the logical openness relationship between observer and system. We are going to show how considering the truth-values of quantum propositions within the context of the fuzzy sets is here more useful for systemics . In conclusion we propose an example of formal quantum coherence.
It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is so possible taking into account the logical openness relationship between observer and system. We are going to show how considering the truth-values of quantum propositions within the context of the fuzzy sets is here more useful for systemics . In conclusion we propose an example of formal quantum coherence.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
The logical structures studied in this paper are generalizations of the propositional calculus. The classical propositional calculus is essentially Boolean algebra or, alternatively, the theory of functions on an arbitrary set S with values in a two-element set. The generalization consists in allowing partial functions on the set S , i.e., functions defined on certain subsets of S , and defining an equivalence relation among these functions such that any two constant functions with the same constant value belong to the same equivalence class. The generalization is equally natural for functions with values in the field of real numbers and we shall consider this case first. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The multiple valued logic of inexactness and approximation we proposed in the past as framework for scientific thought in general is in this paper considered as logic for quantum theory in particular. In this logic truth-values are viewed as degrees of error and a deduction is considered valid if the error in the conclusion can be minimized whenever the error in the premise is brought under control. Applications of this logic to quantum mechanics encompass notions from earlier (non-classical) formal-logical approaches (modal, intuitionistic and multi-valued) and enable rigorous formulations of ideas relating to uncertainty, to states of quantum mechanical systems and to the familiar Hilbert space presentation of quantum physics.
This paper applies to quantum systems a modelling for the logic of knowledge, originally developed for reasoning about distributed systems, but since then applied to game theory, computer security and artificial intelligence. A formal model of quantum message passing systems is developed and the question of how one might define the semantics of a modal operator for knowledge in this model is considered. It is argued that there are at least two plausible semantics, depending on whether the agents are permitted to make use of their quantum state in determining what they know, and on whether one is dealing with single instances of quantum systems, or ensembles. The framework is illustrated using a number of examples from the quantum computing literature, including protocols for quantum key distribution and teleportation. 1
We propose a linear algebraic method, named Eigenlogic, for two-, many-valued and fuzzy logic using observables in Hilbert space. All logical connectives are represented by observables where the truth values correspond to eigenvalues and the atomic input propositional cases, i.e. the “models” of a propositional system, to the respective eigenvectors. In this way propositional logic can be formalized by using combinations of tensored elementary quantum observables. The outcome of a “measurement” of a logical observable will give the truth value of the associated logical proposition, and becomes “interpretable” when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate.Recently the concept of “quantum predicate” has been proposed in leading to similar concepts.We develop logical observables for binary logic and extend them to many-valued logic.For binary logic and truth values {0, 1} logical observables are commuting projector operators.For truth values {+1,−1} the logical observables are isometries formally equivalent to the ones of a composite quantum spin 12 system, these observables are reversible quantum logic gates.The analogy of many-valued logic with quantum angular momentum is then established using a general algebraic method, based on classical interpolation framework suggested by the finite-elementmethod. Logical observables for three-valued logic are formulated using the orbital angular momentum observable Lz with ` = 1. The representative 3-valued 2-argument logical observables for theternary threshold logical connectives Min and Max are then explicitly obtained.Also in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem.The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projector observable and turns out to be a probability measure. Fuzziness arises because ofthe quantum superposition of atomic propositional cases, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true.This method could be employed for developing algorithms in high-dimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist modelsof cognition. For practical implementation, due to the exponential growth of the vector space dimension, adapted logical reduction methods must then be used. LSA (Latent Semantic Analysis)algorithms are often used in quantum-like approaches, this was done using the HAL (Hyperspace Analogue to Language) algorithm.Our approach is also of interest for quantum computation because several of the observables in Eigenlogic are well-known quantum gates (for example CONTROL-Z) and other ones can be derived by unitary transformations. Ternary-logic quantum gates using qu-trits lead to less complex circuits, our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.