# Nijmegen Quantum Logic Group

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## Measurable Spaces and Their Effect Logic

25 Jun 2013 [ proceedings (LICS) · preprint ]

Continuous probabilistic computation is shown to fit in the effect-theoretic framework.

So-called effect algebras and modules are basic mathematical structures that were first identified in mathematical physics, for the study of quantum logic and quantum probability. They incorporate a double negation law $p^{\perp\perp} = p$. Since then it has been realised that these effect structures form a useful abstraction that covers not only quantum logic, but also Boolean logic and probabilistic logic. Moreover, the duality between effect and convex structures lies at the heart of the duality between predicates and states. These insights are leading to a uniform framework for the semantics of computation and logic. This framework has been elaborated elsewhere for set-theoretic, discrete probabilistic, and quantum computation. Here the missing case of continuous probability is shown to fit in the same uniform framework. On a technical level, this involves an investigation of the logical aspects of the Giry monad on measurable spaces and of Lebesgue integration.