Every essentially ordered series has a first member

A causal series or chain is an ordered collection of members, where earlier members cause or explain later members. Every member other than the first is preceded by one other member, and every member other than the last is succeeded by one other member. We can distinguish between accidentally ordered series and essentially order series. These don’t exhaust all the casual series there could be, but they do pick out important groups.

The difference between these two types of causal series can be illustrated with paradigm examples. A paradigm case of accidentally ordered series is that of parents begetting children: x begets y, y begets z, and so on. On the other hand, a paradigm case of essentially ordered series is that of someone moving a stone in a shape on the ground using a stick.

In both of these cases the later members depend on earlier members, but they do so in different ways. In the stick-and-stone case the stick moves the stone only insofar as it is moved by the person, whereas in the begetting case it is not true that y begets z only insofar as x begets y. Put another way, in the begetting case there is a measure of independence between members that is absent in the stick-and-stone case. Or to put it yet another way, in the stick-and-stone case we have an example of caused causing — the person causes the stick’s causing the stone to move — whereas this is not so in the begetting case.

Contrary to the impression we might get from these examples, the difference is not simply about whether the causes are simultaneous or not. Consider the example of Bob holding Charlie up while Charlie reaches for and picks up something off a high shelf. In this case, Charlie’s reaching is simultaneous with Bob’s lifting, but the latter does not cause the former and so this is an example of an accidentally ordered series.

In my opinion, the best description of the difference between the two types is the third difference we mentioned above: essentially ordered series involve caused causations while accidentally ordered series do not. To put it a bit more generally, we could say that in an accidentally ordered series each member causes the next without causing the members after it, whereas in an essentially ordered series each member causes the next and all subsequent members at once, by causing the later ones through the earlier ones.

Among other things, this description is best because it explains the other two descriptions we gave. The independence that obtains in accidentally ordered series has to do with the fact that members are not caused by members not directly preceding them. And what we mean by “only insofar as” is captured by the fact that intermediate members cause later members because they are caused to do so, like the movement of the stone is caused through the movement of the stick.

Given this difference between these two types of series, we can argue that every essentially ordered series must have a first member as follows:

  1. The entirety of any essentially ordered series forms an act.
  2. This act must arise from one of the members within the series.
  3. The member from which this act arises must be the first member of the series.
  4. Therefore, every essentially ordered series has a first member.

Regarding premise (1), every causal relation belongs to some act but it is possible for multiple causal relations to belong to a single act. When x causes y’s causing z, there are three causal relations (x causing y, y causing z, x causing y’s causing z), but only one act: x causes y and z at once, by causing z through y. We might think about y causing z in isolation from anything x does, but properly speaking it is part of the act arising from x. On the other hand, it is of course possible for multiple causes to work together while belonging to separate acts. Every member in every accidentally ordered series acts separately from the other, since we have said that it causes the next member of the series without causing any of the subsequent members.

With this in mind, premise (1) says that for any essentially ordered series, there is some act to which the entire series belongs. If this weren’t the case, then there would need to be two or more acts such that any part of the series is contained in at least one of them, but the entire series is not contained in any of them. To see why this cannot be the case, assume to the contrary that x and y are two members of the series, such that x is part of some act X but not part of some act Y and y is part of Y but not X. We can assume, without loss of generality, that y is subsequent to x in the series. Now, because we’re in an essentially ordered series, x causes the next member as well as all subsequent members at once, by causing the later ones through the earlier ones. Among these, therefore, x will cause y and y’s causing whatever it does. Therefore, y will be part of the act to which x belongs, namely X, which is a contradiction. Since every member must be part of some act (otherwise it would not be doing anything), it follows that all members must belong to the same act. Therefore, the entire series forms an act.

Regarding premise (2), we note that whatever a causal series does is reducible to what its members do, and therefore we have three options for whence the act of the series arises: (a) from nothing, (b) from one member, or (c) from a multitude of members. It can’t arise from nothing, since nothing comes from nothing. It also can’t arise from a multitude of members, since any multitude will contain at least one member from which it cannot arise, even in part. After all, any multitude will have at least two members with one being subsequent to the other, and therefore one caused to do what it does by the other. In this case, the act will in no way arise from this later member, not even partially. Thus, the act does not arise from that particular multitude, but some proper part of it. Since the same could be said of any multitude, it follows that the act arises from no multitude. This leaves us with option (b), that the act arises from one member within the essentially ordered series.

Regarding premise (3), if the act arises from a single member then that member is not caused to act by members prior to it. Since every member other than the first is caused to act by members prior to it, this member from which the act arises cannot have any members prior to it. Thus, it is the first member in the series.

The conclusion in (4) is sufficient for many of the uses of essentially ordered series, such as some arguments for God’s existence or Aristotle’s argument that there must be some chief good (see the last section here). But we could also use it to say something about the finitude of essentially ordered series. Since every member has one predecessor and one successor, and since there must always be a first member, it follows that the only way for the series to be infinite is if it has no last member. Conversely, then, any essentially ordered series with a last member must be finite.

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