Logics of √’qMV algebras

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Logics of qMV algebras. Antonio Ledda Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks Universit di Cagliari. Siena, September 8 th 2008. Some motivation. - PowerPoint PPT Presentation

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<ul><li><p>Logics of qMV algebras </p><p>Antonio Ledda</p><p>Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks</p><p>Universit di Cagliari Siena, September 8th 2008</p></li><li><p>Some motivationqMV algebras were introduced in an attempt to provide a convenient abstraction of the algebra over the set of all density operators of the two-dimensional complex Hilbert space, endowed with a suitable stock of quantum gates.</p></li><li><p>The definition of qMV-algebraDefinitionukasiewiczs axiomSmoothness axioms</p></li><li><p>qMV-algebras</p></li><li><p>Adding the square root of the negationqMV algebras were introduced as term expansions of quasi-MV algebras by an operation of square root of the negation.</p></li><li><p>Adding the square root of the negation</p></li><li><p>quasi-Wajsberg algebras</p></li><li><p>Term equivalenceTheorem</p></li><li><p>The standard Wajsberg algebra St</p></li><li><p>The algebra F[0,1]</p></li><li><p>The standard qW algebras S and D</p></li><li><p>Equationally defined preorder</p></li><li><p>An example of equationally defined preorder</p></li><li><p>Logics from equationally preordered classes</p></li><li><p>Remark</p></li><li><p>A logic from an equationally preordered variety</p></li><li><p>The quasi-ukasiewicz logic q</p></li><li><p>A remark</p></li><li><p>Summary of the logic results</p></li><li><p>A logical version of qMV </p></li><li><p>Term equivalences</p></li><li><p>Logics of qMV algebras (1)</p></li><li><p>Logics of qMV algebras (2)</p></li><li><p>Most logics in the previous schema look noteworthy under some respect:Logics of qMV algebras (3)</p></li><li><p>1-cartesian algebras</p></li><li><p>Examples</p></li><li><p>Inclusion relationships</p></li><li><p>Placing our logics in the Leibniz hierarchy (1)Well-behaved logics is regularly algebraisable and is its equivalent quasivariety semantics;</p><p> is regularly algebraisable and is its equivalent quasivariety semantics;</p><p>(they are the 1-assertional logics of relatively 1-regular quasivarieties)</p></li><li><p>Placing our logics in the Leibniz hierarchy (2)Ill-behaved logicsNone of the other logics is protoalgebraic:</p><p> : the Leibniz operator is not monotone on the deductive filters of F120;</p><p> : it is a sublogic of such;</p><p> : the Leibniz operator is not monotone on the deductive filters of ;</p></li><li><p>Placing our logics in the Leibniz hierarchy (2)</p></li><li><p>Placing our logics in the Frege hierarchy</p><p>Selfextensional logicsNon-selfextensional logics</p></li><li><p>Some notationsWe use the following abbreviations:</p></li><li><p>The logics C and C1</p></li><li><p>The logics C and C1</p></li><li><p>A completeness result</p></li><li><p>The notion of (strong) implicative filter</p></li><li><p>Remark</p><p>In the definition of strong implicative filter conditions F2, F3, F4, F5 are redundant</p></li><li><p>A characterization of the deductive filters</p></li><li><p>Thank you for your attention!!</p></li></ul>

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