Bart Jacobs - New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

lmcs:1600 - Logical Methods in Computer Science, October 1, 2015, Volume 11, Issue 3 - https://doi.org/10.2168/LMCS-11(3:24)2015
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Authors: Bart Jacobs

    Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using Lüders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems.


    Volume: Volume 11, Issue 3
    Published on: October 1, 2015
    Submitted on: December 10, 2014
    Keywords: Mathematics - Logic,Computer Science - Logic in Computer Science,Quantum Physics

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    Source : ScholeXplorer IsReferencedBy ARXIV 1810.02438
    Source : ScholeXplorer IsReferencedBy DOI 10.4204/eptcs.287.13
    Source : ScholeXplorer IsReferencedBy DOI 10.48550/arxiv.1810.02438
    Source : ScholeXplorer IsReferencedBy HANDLE 2066/200949
    • 10.4204/eptcs.287.13
    • 10.4204/eptcs.287.13
    • 10.4204/eptcs.287.13
    • 2066/200949
    • 1810.02438
    • 2066/200949
    • 10.48550/arxiv.1810.02438
    Lower and Upper Conditioning in Quantum Bayesian Theory
    Jacobs, B.P.F. ;

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