1205.3940 (Bart Jacobs)

New Directions in Categorical Logic, for Classical, Probabilistic and
Quantum Logic    [PDF]

Bart Jacobs

Traditionally in categorical logic predicates on an object/type X are represented as subobjects of X. Here we break with that tradition and use maps of the form p : X --> X + X with [id, id] o p = id as predicates. This new view gives a more dynamic, measurement-oriented view on predicates, that works well especially in a quantitative setting. In classical logic (in the category of sets) these new predicates coincide with the traditional ones (subsets, or characteristic maps X --> {0,1}); in probabilistic logic (in the category of sets and Markov chains), the new predicates correspond to fuzzy predicates X --> [0,1]; and in quantum logic (in Hilbert spaces) they correspond to effects (positive endomaps below the identity), which may be understood as fuzzy predicates on a changed basis. It is shown that, under certain conditions about coproducts +, predicates p : X --> X + X form effect algebras and carry a scalar multiplication (with probabilities). Suitable substitution functors give rise to indexed/fibred categories. In the quantum case the famous Born rule - describing the probability of observation outcomes - follows directly from the form of these substitution functors: probability calculation becomes substitution in predicate logic. Moreover, the characteristic maps associated with predicates provide tests in a dynamic logic, and turn out to capture measurement in a form that uniformly covers the classical, probabilistic and quantum case. The probabilities incorporated in predicates (as eigenvalues) serves as weights for the possible measurement outcomes.

View original: http://arxiv.org/abs/1205.3940