Why is causal order not related to directionality of time?

Question

Hans Reichenbach argues for the causality and causal chain to define a topological coordinative definition of time order. Here is an excerpt from his textbook, The Philosophy of Space and Time, Dover(1957), pp.138

Another example: We throw a stone from A to B. If we mark the stone with a piece of chalk at A, it will carry the same mark when it strives at B (event E2). If we mark the stone only on its arrival at B, then the stone leaving A (event E1) has no mark.

This distinction appears trivial, but it is extremely significant. A theory of causality which ignores this elementary difference has neglected the most essential aspect. The procedure which we have described is used constantly in everyday life to establish a time order, and we have no other method in many scientific investigations where time intervals are too short to be directly observable. We must therefore include the mark principle in the foundations of the theory of time.

We have in the above principle a criterion for causal order that does not employ the direction of time, and we can therefore use it in our definition of time order. There exists a topological coordinative definition for time order. We can base it in general on the concept of the causal chain, in which the order of events corresponds to the order of time. Occasionally one speaks also of signals or signal chains. It should be noted that the word "signal" means the transmission of signs and hence concerns the very principle of causal order which we have discussed.

My issue is, isn't he implicitly assuming prior and later time, i.e, the time order in defining the causal events? Isn't the mark principle takes into account the time ordering? How is this decoupled from the time ordering?

Answer 1

The excerpt you show isn't sufficient to tell what Reichenbach is talking about. The bit you need is in the previous two pages, where he defines the mark principle and what he means by cause and effect.

If $E_1$ is the cause of $E_2$, then a small variation (a mark) in $E_1$ is associated with a small variation in $E_2$, whereas small variations in $E_2$ are not associated with variations in $E_1$.

If we wish to express even more clearly that this formulation does not contain the concept of temporal order, we can express it in the following form, where the events that show a slight variation are designated by $E^*$.

We observe only the combinations $E_1E_2$, $E_1^*E_2^*$, $E_1E_2^*$ and never the combination $E_1^*E_2$.

In this arrangement the two events are asymmetrical and therefore it defines an order. That event which appears in the unobserved combination without an asterisk, namely $E_2$, is called the effect and furthermore the temporally later event.

It's an very bad definition of cause/effect, but it doesn't assume temporal order. You can perform an experiment lots of times, collect statistics on correlations between events, and when you find a pair of events where one combination doesn't ever occur, you can declare one to be the effect of the other, and later in time.

It (sort of) works for the examples he picks. However, it's not hard to find examples where this definition gives completely the wrong answer, as we ordinarily understand causality. The 'mark principle' is tantamount to saying that marks are made but never erased. Either no mark is made, or a mark has already been made before either event, or there is initially no mark and later there is, but you never see a mark disappear. So you just need a pair of events where marks get erased as a counterexample.

I think the underlying idea of the 'mark principle' is to take advantage of the second law of thermodynamics. By 'marks' or 'small variations' he really means entropy, and he is relying on entropy always increasing. Things get scrambled but never unscrambled - small variations ('marks') arise but never spontaneously disappear. (You could also call them 'memories' or 'records' instead, which might be a bit clearer about the intention, although the language is obviously less neutral.) However, I think the second law and the definition of entropy have a quite different meaning to his simple 'mark' definition.