This rule of proof (so far as it is used to establish direct causation) depends, first, upon observation of an invariable connection between the given phenomenon and one other circumstance; and, secondly, upon I. (a) and II. (b) among the propositions obtained from the unconditionality of causation at the close of the last chapter.

To prove that A is causally related to p, suppose two instances of the occurrence of A, an antecedent, and p, a consequent, with concomitant facts or events—and let us represent them thus:

Antecedents:ABCADE
Consequents:pqrpst;

and suppose further that, in this case, the immediate succession of events can be observed. Then A is probably the cause, or an indispensable condition, of p. For, as far as our instances go, A is the invariable antecedent of p; and p is the invariable consequent of A. But the two instances of A or p agree in no other circumstance. Therefore A is (or completes) the unconditional antecedent of p. For B and C are not indispensable conditions of p, being absent in the second instance (Rule II. (b)); nor are D and E, being absent in the first instance. Moreover, q and r are not effects of A, being absent in the second instance (Rule II. (d)); nor are s and t, being absent in the first instance.

It should be observed that the cogency of the proof depends entirely upon its tending to show the unconditionality of the sequence A-p, or the indispensability of A as a condition of p. That p follows A, even immediately, is nothing by itself: if a man sits down to study and, on the instant, a hand-organ begins under his window, he must not infer malice in the musician: thousands of things follow one another every moment without traceable connection; and this we call 'accidental.' Even invariable sequence is not enough to prove direct causation; for, in our experience does not night invariable follow day? The proof requires that the instances be such as to show not merely what events are in invariable sequence, but also what are not. From among the occasional antecedents of p (or consequents of A) we have to eliminate the accidental ones. And this is done by finding or making 'negative instances' in respect of each of them. Thus the instance

ADE
pst

is a negative instance of B and C considered as supposable causes of p (and of q and r as supposable effects of A); for it shows that they are absent when p (or A) is present.

To insist upon the cogency of 'negative instances' was Bacon's great contribution to Inductive Logic. If we neglect them, and merely collect examples of the sequence A-p, this is 'simple enumeration'; and although simple enumeration, when the instances of agreement are numerous enough, may give rise to a strong belief in the connection of phenomena, yet it can never be a methodical or logical proof of causation, since it does not indicate the unconditionalness of the sequence. For simple enumeration of the sequence A-p leaves open the possibility that, besides A, there is always some other antecedent of p, say X; and then X may be the cause of p. To disprove it, we must find, or make, a negative instance of X—where p occurs, but X is absent.

So far as we recognise the possibility of a plurality of causes, this method of Agreement cannot be quite satisfactory. For then, in such instances as the above, although D is absent in the first, and B in the second, it does not follow that they are not the causes of p; for they may be alternative causes: B may have produced p in the first instance, and D in the second; A being in both cases an accidental circumstance in relation to p. To remedy this shortcoming by the method of Agreement itself, the only course is to find more instances of p. We may never find a negative instance of A; and, if not, the probability that A is the cause of p increases with the number of instances. But if there be no antecedent that we cannot sometimes exclude, yet the collection of instances will probably give at last all the causes of p; and by finding the proportion of instances in which A, B, or X precedes p, we may estimate the probability of any one of them being the cause of p in any given case of its occurrence.

But this is not enough. Since there cannot really be vicarious causes, we must define the effect (p) more strictly, and examine the cases to find whether there may not be varieties of p, with each of which one of the apparent causes is correlated: A with p1 B with p11, X with p111. Or, again, it may be that none of the recognised antecedents is effective: as we here depend solely on observation, the true conditions may be so recondite and disguised by other phenomena as to have escaped our scrutiny. This may happen even when we suppose that the chief condition has been isolated: the drinking of foul water was long believed to cause dysentery, because it was a frequent antecedent; whilst observation had overlooked the bacillus, which was the indispensable condition.