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 Università di Cagliari. Siena, September 8 th 2008. Some motivation. - PowerPoint PPT PresentationTRANSCRIPT
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Logics of √’qMV algebras
Antonio Ledda
Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks
Università di Cagliari
Siena, September 8th 2008
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Some motivation
qMV 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.
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The definition of qMV-algebra
Definition
Łukasiewicz’s axiom Smoothness axioms
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qMV-algebras
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Adding the square root of the negation
√’qMV algebras were introduced as term
expansions of quasi-MV algebras by an
operation of square root of the
negation.
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Adding the square root of the negation
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quasi-Wajsberg algebras
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Term equivalence
Theorem
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The standard Wajsberg algebra St
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The algebra F[0,1]
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The standard qW algebras S and D
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Equationally defined preorder
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An example of equationally defined preorder
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Logics from equationally preordered classes
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Remark
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A logic from an equationally preordered variety
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The quasi-Łukasiewicz logic qŁ
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A remark
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Summary of the logic results
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A “logical” version of qMV
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Term equivalences
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Logics of qMV algebras (1)
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Logics of qMV algebras (2)
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Most logics in the previous schema look noteworthy under some respect:
Logics of qMV algebras (3)
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1-cartesian algebras
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Examples
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Inclusion relationships
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Placing our logics in the Leibniz hierarchy (1)
Well-behaved logics is regularly algebraisable and is its
equivalent quasivariety semantics;
is regularly algebraisable and is its equivalent quasivariety semantics;
(they are the 1-assertional logics of relatively
1-regular quasivarieties)
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Placing our logics in the Leibniz hierarchy (2)
Ill-behaved logicsNone of the other logics is protoalgebraic:
: the Leibniz operator is not monotone on the deductive filters of F120;
: it is a sublogic of such;
: the Leibniz operator is not monotone on the deductive filters of ;
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Placing our logics in the Leibniz hierarchy (2)
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Placing our logics in the Frege hierarchy
Selfextensional logics
Non-selfextensional logics
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Some notations
We use the following abbreviations:
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The logics C and C1
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The logics C and C1
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A completeness result
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The notion of (strong) implicative filter
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Remark
In the definition of strong implicative filter conditions F2, F3, F4, F5 are redundant
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A characterization of the deductive filters
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Thank you for your attention!!
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