A system is physically closed when there is no material exchange with an environment. A system is logically closed when no process within it refers to anything outside its definitions. That would include any background against which to measure change. Such a system, however, is a deductive system, an idealization. It appears time-reversible because time has no meaning in a deductive system: every theorem is eternally latent within the axioms. Equations are reversible (by substituting –t for t), but real systems are not. No real system (except, by definition, the universe as a whole) can be considered logically closed or absolutely isolated; there is always an “outside” that constitutes a background against which processes can appear irreversible. Processes within systems considered isolated may appear reversible. In the real world, however, there is no such thing as time-reversibility.

No doubt the concept of a time-reversible system was inspired by special cases, such as celestial motions and simple cyclical mechanisms, where reversing the direction of motion does not affect the overall behavior of the system. Yet, even such reversibility leads to problems, such as the arrow of time or the prevalence in the real world of irreversible processes. (Not to mention the fact that actually reversing the direction of motion of a planet, for example, would be catastrophic!) The fundamental laws of physics are generally “time-reversal invariant,” even though many physical processes at the macroscopic scale seem irreversible, as expressed in the 2nd Law of Thermodynamics. However, strictly speaking, this reversibility is not a property of nature but a mathematical property of equations. One can calculate the behavior of the model backward or forward in time at will. But the model is an invention, not a real system. The problem is not to account for irreversible processes in a theoretically reversible world. Irreversible processes predominate because no part of the real world is actually a deductive system, as represented in such a model.