My aim is to start with the conceptual principle that properties are not intrinsic to a system , but are relational, only taking values upon appropriate interaction with another system. This is given the mathematical form of a quantum algebra (essentially, a union of Boolean algebras) to replace the Boolean algebra of properties of classical physics. By adding some experimental results, we aim to construct the formalism of quantum theory. We prove that the move from Boolean to quantum algebras transforms the classical concepts of observables, states, and symmetries into Hermitian, density, and unitary operators, and allows us to derive the Schrodinger equation and the von Neumann-Luders Projection Rule. The mysterious EPR correlations have a transparent explanation in the context of quantum algebras. I expect to study other paradoxes of quantum mechanics in this reconstruction in the future. In the usual interpretation, the Measurement Problem is an inconsistency arising between the unitary evolution of the state of the isolated system, consisting of the measured system and the apparatus, and the measured (reduced) state. This problem is resolved in the proposed reconstruction. We stand the problem on its head; instead of starting with the unitary evolution of an isolated system, and asking when reduction takes place, we start with the reduced properties of a quantum algebra. We determine the conditions under which unitary evolution of a system takes place, and show that even for isolated systems consisting of interacting subsystems a form of spontaneous symmetry-breaking occurs which breaks the unitary evolution, thus resolving the inconsistency in favor of the reduced states.