An Introduction to Effectus Theory

TL;DR

Effectus theory is a categorical framework that generalizes quantum, probabilistic, and classical logic using effect algebras, providing a unified foundation for understanding effects, states, and measurement in various logical systems.

Contribution

This paper introduces the foundational concepts of effectus theory, including duality, total and partial forms, and the role of comprehension and quotients in measurement.

Findings

Effectus theory unifies quantum, probabilistic, and classical logic.

Effectuses can be equivalently described in total and partial forms.

Boolean effectuses correspond to extensive categories.

Abstract

Effectus theory is a new branch of categorical logic that aims to capture the essentials of quantum logic, with probabilistic and Boolean logic as special cases. Predicates in effectus theory are not subobjects having a Heyting algebra structure, like in topos theory, but `characteristic' functions, forming effect algebras. Such effect algebras are algebraic models of quantitative logic, in which double negation holds. Effects in quantum theory and fuzzy predicates in probability theory form examples of effect algebras. This text is an account of the basics of effectus theory. It includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. A basic result says that effectuses can be described equivalently in both `total' and `partial' form. So-called `commutative' and `Boolean' effectuses are distinguished, for probabilistic and classical models. It is shown how these Boolean effectuses are essentially extensive categories. A large part of the theory is devoted to the logical notions of comprehension and quotient, which are described abstractly as right adjoint to truth, and as left adjoint to falisity, respectively. It is illustrated how comprehension and quotients are closely related to measurement. The paper closes with a section on `non-commutative' effectus theory, where the appropriate formalisation is not entirely clear yet.