right side, and the "future" operators with greater values of appear on the left side. This definition can be extended to inhomogeneous histories as well. Central to the consistent histories is the notion of consistency. A set of histories is consistent (or strongly consistent) if for all . Here represents the initial density matrix, and the operators are expressed in the Heisenberg picture. The set of histories is weakly consistent if for all . Probabilities If a set of histories is consistent then probabilities can be assigned to them in a consistent way. We postulate that the probability of history is simply which obeys the axioms of probability if the histories come from the same (strongly) consistent set. As an example, this means the probability of " OR " equals the probability of "" plus the probability of "" minus the probability of " AND ", and so forth. Interpretation The interpretation based on consistent histories is used in combination with the insights about quantum decoherence. Quantum decoherence implies that irreversible macroscopic phenomena (hence, all classical measurements) render histories automatically consistent, which allows one to recover classical reasoning and "common sense" when applied to the outcomes of these measurements. More precise analysis of decoherence allows (in principle) a quantitative calculation of the boundary between the classical domain and the quantum domain. According to Roland Omnès, In order to obtain a complete theory, the formal rules above must be supplemented with a particular Hilbert space and rules that govern dynamics, for example a Hamiltonian. In the opinion of others this still does not make a complete theory as no predictions are possible about which set of consistent histories will actually occur. In other words, the rules of consistent histories, the Hilbert space, and the Hamiltonian must be supplemented by a set selection rule. However, Robert B. Griffiths holds the opinion that asking the question of which set of histories will "actually occur" is a misinterpretation of the theory; histories are a tool for description of reality, not separate alternate realities. Proponents of this consistent histories interpretation—such as Murray Gell-Mann, James Hartle, Roland Omnès and Robert B. Griffiths—argue that their interpretation clarifies the fundamental disadvantages of the old Copenhagen interpretation, and can be used as a complete interpretational framework for quantum mechanics. In Quantum Philosophy, Roland Omnès provides a less mathematical way of understanding this same formalism. The consistent histories approach can