2,160 research outputs found

    Propositional superposition logic

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    We extend classical Propositional Logic (PL) by adding a new primitive binary connective φ∣ψ\varphi|\psi, intended to represent the "superposition" of sentences φ\varphi and ψ\psi, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form ⟨M,f⟩\langle M,f\rangle, where MM is an ordinary two-valued assignment for the sentences of PL and ff is a choice function for all pairs of classical sentences. In the new semantics φ∣ψ\varphi|\psi is strictly interpolated between φ∧ψ\varphi\wedge\psi and φ∨ψ\varphi\vee\psi. By imposing several constraints on the choice functions we obtain corresponding notions of logical consequence relations and corresponding systems of tautologies, with respect to which ∣| satisfies some natural algebraic properties such as associativity, closedness under logical equivalence and distributivity over its dual connective. Thus various systems of Propositional Superposition Logic (PLS) arise as extensions of PL. Axiomatizations for these systems of tautologies are presented and soundness is shown for all of them. Completeness is proved for the weakest of these systems. For the other systems completeness holds if and only if every consistent set of sentences is extendible to a consistent and complete one, a condition whose truth is closely related to the validity of the deduction theorem.Comment: 55 page

    Characterization of quantum states in predicative logic

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    We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations. We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.Comment: 14 pages, Boston, IQSA10, to appea
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