"A proposition is _necessary_ with respect to a given constituent if it remains true when that constituent is altered in any way compatible with the proposition remaining significant."
We may now apply this definition to the definition of causality quoted above. It is obvious that the argument must be the time at which the earlier event occurs. Thus an instance of causality will be such as: "If the event [Math: e_{ 1}] occurs at the time [Math: t_{ 1}], it will be followed by the event [Math: e_{ 2}]." This proposition is intended to be necessary with respect to [Math: t_{ 1}], i.e. to remain true however [Math: t_{ 1}] may be varied. Causality, as a universal law, will then be the following: "Given any event [Math: t_{ 1}], there is an event [Math: e_{ 2}] such that, whenever [Math: t_{ 1}] occurs, [Math: e_{ 2}] occurs later." But before this can be considered precise, we must specify how much later [Math: e_{ 2}] is to occur. Thus the principle becomes:--
"Given any event [Math: e_{ 1}], there is an event [Math: e_{ 2}] and a time-interval [tau] such that, whenever [Math: e_{ 1}] occurs, [Math: e_{ 2}] follows after an interval [tau]."
I am not concerned as yet to consider whether this law is true or false. For the present, I am merely concerned to discover what the law of causality is supposed to be. I pass, therefore, to the other definitions quoted above.
The second definition need not detain us long, for two reasons. First, because it is psychological: not the "thought or perception" of a process, but the process itself, must be what concerns us in considering causality. Secondly, because it is circular: in speaking of a process as "taking place in consequence of" another process, it introduces the very notion of cause which was to be defined.
The third definition is by far the most precise; indeed as regards clearness it leaves nothing to be desired. But a great difficulty is caused by the temporal contiguity of cause and effect which the definition asserts. No two instants are contiguous, since the time-series is compact; hence either the cause or the effect or both must, if the definition is correct, endure for a finite time; indeed, by the wording of the definition it is plain that both are assumed to endure for a finite time. But then we are faced with a dilemma: if the cause is a process involving change within itself, we shall require (if causality is universal) causal relations between its earlier and later parts; moreover, it would seem that only the later parts can be relevant to the effect, since the earlier parts are not contiguous to the effect, and therefore (by the definition) cannot influence the effect. Thus we shall be led to diminish the duration of the cause without limit, and however much we may diminish it, there will still remain an earlier part which might be altered without altering the effect, so that the true cause, as defined, will not have been reached, for it will be observed that the definition excludes plurality of causes. If, on the other hand, the cause is purely static, involving no change within itself, then, in the first place, no such cause is to be found in nature, and in the second place, it seems strange--too strange to be accepted, in spite of bare logical possibility--that the cause, after existing placidly for some time, should suddenly explode into the effect, when it might just as well have done so at any earlier time, or have gone on unchanged without producing its effect. This dilemma, therefore, is fatal to the view that cause and effect can be contiguous in time; if there are causes and effects, they must be separated by a finite time-interval [tau], as was assumed in the above interpretation of the first definition.