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CATEGORICAL LOGIC

## New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Resource type

Author/contributor

- Jacobs, Bart (Author)

Title

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Abstract

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 L\"uders 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.

Publication

Logical Methods in Computer Science

Volume

11

Issue

3

Pages

24

Date

2015-10-01

Journal Abbr

Log.Meth.Comput.Sci.

DOI

10/ggdf99

ISSN

18605974

Accessed

2019-11-23T16:40:56Z

Library Catalog

Extra

ZSCC: NoCitationData[s0] arXiv: 1205.3940

Citation

Jacobs, B. (2015). New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic.

*Logical Methods in Computer Science*,*11*(3), 24. https://doi.org/10/ggdf99
CATEGORICAL LOGIC

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