series: orders events in terms of past, present, future
B- series: orders events in terms of earlier, later
C- series: orders events in terms of between
(C-series + direction → B-series; B-series + now → A-series.)
Part I: time → A-series
Reasons to think time requires the A-series:
1. An argument McTaggart rejects. We perceive events as ordered not only in terms of earlier and later (the B-series) but also as past, present, or future (the A-series), and in fact we are unable to perceive events without perceiving them as in the A-series. As McTaggart points out, though, this does not establish that the A-series is essential to time. For instance, we might be able to think about or understand time in a way that we cannot perceive it. (Compare Descartes on the superiority of understanding over perception. Perception often involves a kind of distortion or illusion: we perceive the stick in water as bent, we perceive the sun as rising, even though we understand that these descriptions are not accurate. In fact, perceiving the sun as rising may be a particularly good analogy here: that perception is based on our particular perspective on the world, which it in a sense projects onto the world. B-series advocates hold a similar view, namely that the apparent passage of time is a feature of our perceptual situation rather than of the objective world itself.)
2. The argument McTaggart thinks is successful.
1. time → change
2. change → A-series
3. time → A-series
(a) premise 1: time → change
McTaggart simply takes premise 1 for granted, saying that everyone acknowledges it. Certainly it is a central feature of Aristotle's view (in particular, Aristotle defends the contrapositive, ¬change → ¬time). Question: if we accept Shoemaker's argument that time without change is possible, does this undermine McTaggart's argument? Or do McTaggart and Shoemaker mean something different by "change"?
(b) premise 2: change → A-series
McTaggart defends the idea that change requires the A-series by arguing against analyses that try to do without it. He suggests that without the A-series we can't make sense either of an event ceasing to exist and another event beginning to exist, or of an event changing from an event of one kind to an event of another.
I find the argument here puzzling, and in any case to me it seems beside the point. It seems to me we shouldn't be surprised by the result that events do not change. Events are changes! What does the changing is not events but things.
But McTaggart considers precisely this response, in connection with Russell's view of time. McTaggart replies that if everything is expressed in terms of the B-series, then objects as well as events have all their properties eternally. The verb 'to be' is normally tensed, but tense presupposes the A-series, so to do a serious job of restricting ourselves to the B-series, we have to abandon tense. (I don't think M actually says this, but I think it's a useful amplification of his argument. Compare van Inwagen on 'IS' vs. 'is'.) If we are allowed only tenseless verbs, then we need to either (i) add an argument-place to every predicate for time, or (ii) talk about time-slices of objects instead of enduring objects ("continuants" in the current lingo), or (iii) talk about temporally-indexed properties (like red-at-5:10-on-2/10/2005) instead of properties simpliciter. But any claim about objects that is expressed in any of these three ways will be eternally true, not something that changes in truth value from one time to another. So again, on M's view, genuine change has been left out of the picture.
3. The relation between the A-series and the B-series
McTaggart also argues that in fact the B-series is sufficient for time, B-series → time. But then since time → change and change → A-series, we also have (by transitivity) that B-series → A-series. So, to summarize the main premises so far:
time → change
change → A-series
B-series → time
And from these premises we can straightforwardly infer:
time → A-series
B-series → A-series
4. The C-series
Is any ordering of events possible without either the A-series or the B-series? McTaggart's answer is yes. He introduces the C-series, which simply orders times in terms of which times are between which other times. He suggests that this gives us an ordering but not a direction. (This is because "between" is a symmetric relation, in the following sense: if Between(x, y, z), then Between(x, z, y) -- where "Between(x,y,z)" means "x is between y and z.")
McTaggart considers and responds to two objections to the idea that time → A-series. First, there is the suggestion that in fiction we imagine times with no connection to (actual) past, present, and future. McTaggart's response is that if we imagine a story as real, then we need to fit it into our A-series framework. (And in this connection it may be worth noticing how many fictions begin with "once upon a time," "a long time ago in a galaxy far, far away," and so on.) Second, McTaggart mentions an argument of Bradley's that there might be multiple independent time-series. McTaggart responds that if there is more than one present, then there is more than one time.
Part II: A-series → ⊥
(I'm using ⊥ to mean "contradiction," so the Part II heading above means that an A-series is impossible.)
The other half of McTaggart's argument is that the A-series leads immediately to a contradiction. So it is impossible to consistently apply the A-series to anything. Since time requires the A-series, we reach McTaggart's conclusion that time is unreal.
The basic argument is that past, present, and future are incompatible properties, but every event must have them all.
Of course, a given event does not have all these properties at the same time. Maybe instead of saying
E is past & E is present & E is future
we should say instead
E is present & E has been future & E will be past
however, if we treat things like being currently present or having been in the future as properties of an event, then events must have incompatible sets of these second-level properties as well. True, E now has the property PF (in the past it was in the future), but soon it will also have the properties PPres and PP. (OK, depending on how these are interpreted maybe they aren't contradictory: if P means "there is a past time at which" then all three of these properties can be simultaneously true of the same thing after it has occurred. But it's also true that there was a time at which ~PP, for instance.) Moving to second-level properties leaves us with the problem that contradictory statements are true of events. And the same thing will be true at the third and higher levels. So we seem to have an infinite regress problem.
Of course, these problems only arise if we insist that being present is a property rather than a relation -- that presentness is absolute rather than relational. (And even if it is a property, the problems arise only if we insist that the property expressed by 'present' is the same property in all contexts, as opposed to the view that 'present' is an indexical term.) Does the view that 'now' or 'present' is relational or indexical give us an easy way around McTaggart's argument for the unreality of time? Not really, as far as I can see, since to offer a relational or indexical account of presentness would be to concede that the A-series does not express anything that cannot be expressed without it. If time requires the A-series, but expressing all the expressible facts about the world does not require the A-series, then the world is not temporal. So the indexical or relational analysis concedes McTaggart's point that time is unreal. (That is, if we concede the first half of his essay, the argument that time requires the A-series.)