Complete exogenous quantum propositional logic
نویسنده
چکیده
After an overview of EQPL (exogenous quantum propositional logic), the proof of its weak completeness is outlined, using a non trivial extension of the Fagin-Halpern-Megiddo technique. A new logic (EQPL – exogenous quantum propositional logic), embodying all that is stated in the postulates of quantum physics, was proposed in [2, 3, 4] for modeling and reasoning about quantum systems. The logic was designed from the semantics upwards, starting with the key idea of adopting superpositions of classical models as the models of the proposed quantum logic. This novel approach to quantum reasoning is quite different from the traditional approach to the problem that, as initially proposed by Birkhoff and von Neumann, focuses on the lattice of closed subspaces of a Hilbert space. Our exogenous semantics approach has the advantage of closely guiding the design of the language around the underlying concepts of quantum physics while keeping the classical connectives and was inspired by the possible worlds approach originally proposed by Kripke for modal logic. It is also akin to the society semantics introduced in for many-valued logic and to the possible translations semantics proposed in for paraconsistent logic. The possible worlds approach was also used in for probabilistic logic. Our semantics of quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics which is still closely related to the lattice-oriented operations. Contrarily to traditional quantum logics that replace the classical connectives by new connectives inspired by the lattice-oriented operations, by adopting superpositions of classical models as the models of the quantum logic, we are led to a natural extension of the classical language containing the classical connectives (like modal languages are extensions of the classical language). Furthermore, the new logic allows quantitative reasoning about amplitudes and probabilities, being in this respect much closer to the possible worlds logics for probability reasoning than to the traditional quantum logics. For other developments in this direction, also motivated by applications in quantum computation and information, see [5].
منابع مشابه
Weakly complete axiomatization of exogenous quantum propositional logic
A finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The axiomatization is shown to be weakly complete relative to an oracle for analytical reasoning. The proof is carried out using a non trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.
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