A presentation of quantum logic based on an ”and then” connective

When a physicist performs a quantic measurement, new in-formation about the system at hand is gathered. This pre-sentation studies the logical properties of how this new in-formation is combined with previous information. It presents Quantum Logic as a propositional logic under two connec-tives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to im-ply all the properties considered. No independence or com-pleteness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativ-ity, or the monotonicity of the and then connective. Entail-ment is defined in this logic and can be proved to be a par-tial order. In orthomodular lattices, the operation proposed by Finch in (Finch 1969) satisfies all the properties studied in this paper. All properties satisfied by Finch’s operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch’s operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean al-gebras are defined, ideals are characterized and a homomor-phism theorem is given 1