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This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole's theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as commuting projection operators and the truth values, which take the binary values { 0 , 1 } , are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to apply the quantum "eigenlogic" formulation to behavioural analysis. Agents, represented by Braitenberg vehicles, are investigated in the context of the quantum robot paradigm. The agents are processed through quantum logical gates with fuzzy and multivalued inputs; this permits to enlarge the behavioural possibilities and the associated decisions for these simple vehicles. Design/methodology/approach: In eigenlogic, the eigenvalues of the observables are the truth values and the associated eigenvectors are the logical interpretations of the propositional system. Logical observables belong to families of commuting observables for binary logic and many-valued logic. By extension, a fuzzy logic interpretation is proposed by using vectors outside the eigensystem of the logical connective observables. The fuzzy membership function is calculated by the quantum mean value (Born rule) of the logical projection operators and is associated to a quantum probability. The methodology of this paper is based on quantum measurement theory. Findings: Fuzziness arises naturally when considering systems described by state vectors not in the considered logical eigensystem. These states correspond to incompatible and complementary systems outside the realm of classical logic. Considering these states allows the detection of new Braitenberg vehicle behaviours related to identified emotions; these are linked to quantum-like effects. Research limitations/implications: The method does not deal at this stage with first-order logic and is limited to different families of commuting logical observables. An extension to families of logical non-commuting operators associated to predicate quantifiers could profit of the "quantum advantage" due to effects such as superposition, parallelism, non-commutativity and entanglement. This direction of research has a variety of applications, including robotics. Practical implications: The goal of this research is to show the multiplicity of behaviours obtained by using fuzzy logic along with quantum logical gates in the control of simple Braitenberg vehicle agents. By changing and combining different quantum control gates, one can tune small changes in the vehicle's behaviour and hence get specific features around the main basic robot's emotions. Originality/value: New mathematical formulation for propositional logic based on linear algebra. This methodology demonstrates the potentiality of this formalism for behavioural agent models (quantum robots). [ABSTRACT FROM AUTHOR]

Purpose The paper aims to provide a semiotic interpretation of the role played by entanglement in quantum-based models aimed to information retrieval and suggests possible improvements. Actual models are capable of retrieving documents relevant to a query composed of a keyword and its acceptation expressed by a given context. The paper also considers some analogies between this technique and quantum-based approaches in other disciplines to discuss the consequence of this quantum turn, as epistemology and philosophy of language are concerned.Design/methodology/approach We use quantum geometry to design a formal model for textual semiotics. In particular, the authors refer to Greimas’s work on semantics and information theory, to Eco’s writings on semantic memory and to Lotman’s work on a cybernetic notion of culture.Findings Quantum approaches imply a particular point of view on meaning. Meaning is not a real, positive quality of a given word. It is a net of relations constructed in the text, whose value is progressively determined during the reading process. Furthermore, reading is not a neutral operation: to read is to determine meaning. If it is said that, from a general semiotic point of view, meaning is stored in quantum semantic memories and is read/written by semantic machines, then the operation of “reading/writing” is analogous to the operation of measuring in quantum theory: in other terms, meaning is a value, and this implies an instance (not necessarily human) according to which values are valuable.Research limitations/implications The authors are not proposing a complete quantum semantics. At the present, quantum information retrieval can detect the presence of semantic relations. The authors suggest a way to characterize them, leaving open the problem on how to formalize the document as a vector in four-state semantic space.Practical implications A quantum turn shows deep semiotic implications on the approach to language, which shows an immanent semantic organization not reducible to syntax and morphology. This organization is probabilistic and indeterministic and explains to what extent text fixes the meaning of its lexical units.Social implications In the authors’ perspective, signification is not the exclusivity of a human subject. Criticizing Turing test, the great semiotic and cybernetic scholar Jurij Lotman wrote that if we identify “intelligent” and “human”, we raise the failings of an actual form of intelligence to the rank of an essential characteristic. On this line, meaning is considered as a feature of social, artificial and biological systems.Originality/value The adoption of quantum formalism seems in line with cybernetic framework, involving a probabilistic, non-cartesian point of view on meaning aimed to critically discuss the human–machine relation. Furthermore, Quantum theory (QT) implies a phenomenological point of view on the conditions of possibility of meaning. [ABSTRACT FROM AUTHOR]