In the previous post I explained what an essentially ordered series is and gave an argument to the effect that every such series must have a first member. While I still think the argument works, discussions about the post with others have led me to realize that we can give a considerably simpler argument for the same conclusion.
Start with a case of one thing causally influencing another, say Alice picking up her mug from the table. The thing doing the causing (Alice) is the agent and the thing being influenced (the mug) is the patient. Both the agent and the patient contribute something that enables the causation to occur, namely the actuality in the agent and the potential in the patient, and the causation itself consists in these contributions coming together in such a way that the potential can be actualized by the actuality.
A causal series is a series of successive causations. When this occurs, the members of the series fall into one of three categories: the first member (if there is one) is an agent without being a patient; the last member is a patient without being an agent; and each intermediate member is in some sense both an agent and a patient. For any intermediate member, its agency and its patiency will coincide either accidentally or essentially. They coincide accidentally when the actuality whereby the member is an agent is to some extent distinct from the potential whereby it is a patient. For instance, when Bob lifts Charlie so that he can grab an item off the top shelf, Charlie has the potential to be lifted and the actuality of grabbing, where the latter is distinct from the former. An intermediate member’s agency and patiency coincide essentially when the actuality whereby it is an agent just is the actualization of the potential whereby it is a patient. For instance, when Alice moves a stone in a shape on the ground using a stick, the actuality in the stick whereby it moves the stone just is the actualization of its potential to be moved in that way.
This essential coincidence of agency and patiency captures a unique middle ground between the “pure” agent and patient that we started with, and for this reason we introduce a third term to describe this sort of member: instrument. Similar to an agent, an instrument has an actuality in virtue of which it actualizes the potential of something else. Unlike an agent, this actuality is something the instrument receives from another as part of the functioning of the causal series rather than something that it contributes itself. Similar to a patient, then, the only thing the instrument contributes to the series is the potential. But unlike a patient, the potential contributed by the instrument is the potential for actualizing a potential in the next member of the series.
By contrast, the accidental coincidence of agency and patiency does not require us to introduce another sort of member, since in this case the intermediate member is separately a patient of the member preceding it and an agent of the member succeeding it.
We are now in a position to introduce the distinction between two types of causal series. An accidentally ordered series is a causal series in which the agency and patiency of every intermediate member coincide accidentally, whereas an essentially ordered series is one in which they coincide essentially. In other words, none of the intermediate members in an accidentally ordered series are instruments, while in an essentially ordered series all of them are instruments.
Given this distinction, we can see quite easily that every essentially ordered series must have a first member. After all, if any such series didn’t have a first member, then the only members in the series would be the instruments and the patient. But instruments and patients only contribute potentials to the series, and therefore this series would not have any actuality in it. But without any actuality, no potentials can be actualized to bring about any effect. And therefore, this series wouldn’t in fact cause anything at all.
 More technically, a contribution of a member M to a causal series S is some intrinsic feature of M which makes it possible for S to obtain and which it does not receive as part of S obtaining.
 There can still be some overlap or dependency and still be considered distinct, so long as the actualization of the potential is not identical to the actuality.
 We are restricting ourselves to causal series where there is at least one intermediate member so that we can give succinct versions of these definitions.