A. there are two ways of thinking about time and events in it; one involves the A series and the other the B series.
 

1. In the A series, events (or time) are located in time on the basis of their relation of being 'earlier than,' 'simultaneous with,' or 'later than' the present , the 'now.'  Since the future is what is later than the present, and the past what is earlier than the present, events are arranged in the order in which they become present, that is on the basis of their tenses.  A tense is a position in the time series defined by its location (and distance) with respect to the present, the 'now'.  For example, 'tomorrow' is a tense because it's a day which is one day later than the present day.  Notice that the 'now' moves.

NOTES
• We commonly use tenses.  For example, if I say “he will marry”, or “the war ended” I convey that idea that the first event is in the future and the second in the past.  It's important not to confuse grammatical tense with philosophical tense: "tomorrow I go to school" is philosophically tensed.
• The A series is dynamic because things/events go from being future to being present, and then past. In other words, suppose that 'tomorow I'll be 50' is true.  Then, it won't always be true; in fact, the day after tomorrow it will be false. Hence, the tenses of events and things change.  Similarly, the truth values (the property of being true or false) of tensed statements change.
2. In the B series, events are located in time on the basis of the relations of 'being earlier than', 'being simultaneous with', and 'being later than' with respect to each other.  In practice (although this is not necessary in principle), one selects an event and arranges other events on the basis of their temporal distance and relation to a fixed event such the birth of Christ, the foundation of SIUE, or whatever.  A date is a position in the time series defined by its location (and distance) with respect to a fixed point.  For example, '1945' is a date because it's a year 1945 years after the (traditional) birth of Christ.

NOTES:
• We commonly use dates, as in “the war ends in 1945”, "on the 5th I go home".
• The B series is static because if A is earlier than B, or occurs in 1945, that is always so. In other words, suppose that 'on March 11 2001 I'm 50' is true.  Then, it will always be true, even on March 12th.  That is, the dates of events and things don't change.  Similarly, the truth values (the property of being true or false) of tenseless statements don't change.

• Given the date of 'now', they are inter definable, that is, one can freely move from the A series to the B series and vice versa.
• They arrange things/events in the same order, that is, they are structurally identical.

• Tenses change but dates don't.
• Without date of 'now', the A series not reducible to the B series and vice versa.  One could know all the B-series history of the world without knowing which stage history has reached, i.e. what's happening now.  In this case, one would know dates but not tenses. Conversely, one could know the locations of all events with respect to now, without knowing the date of now.  One, then, would know tenses but not dates.

Notice that knowing dates but not tenses is not uncommon: I may know that the appointment is at 5 without knowing what time it is now because my watch stopped.  Note that this situation has a spatial analogue: have you ever looked at a map without knowing where the 'here', i.e., where you are, is? 
• “It is hot now” doesn't mean “its hot on the 2nd of August” even if ‘now’ is the 2nd of August.

C. McTaggart’s argument for the unreality of time involves 6 steps:
 
 1. If time exists, there is change.
 2. If the A series doesn't exist, there is no change.
 3. The A series doesn't exist.
 4. Hence, there is no time.
 5. The B series presupposes time.
 6. Hence, the B series is unreal as well.

The argument is obviously valid, and the only remaining issue is whether it's sound.
Step (5) is taken as obvious because “earlier” and “later” are time-relations, and consequently, if time is unreal, then that which presuppose time relations would be unreal as well.  Step (1) is taken for granted (notice that McTaggart needn't deny the possibility of a vacuum in time, since he thinks that the past recedes further and further, so that the tenses of events change all the time).
However, steps (2) and (3) are argued for at length.

McTaggart's argument for step 2 "If the A series doesn't exist, then there's no change":

1. The B series is static (the relations of “earlier” and “later” are permanent), and hence there is no change in it. Each event (e.g. death of Queen Anne) has a fixed position, and does not begin or cease to be because there is no “now’.
2. So, the only way an event can change is by being future, becoming present and finally becoming past and more past.
3. Hence, change involves the A series.

1. Russell's objection:
The A series exists only with relation to the knowing subject; it is a subjective way of thinking about time.  By contrast., the B series provides an objective view of time.  Hence, the A series can be eliminated, as one can see by noticing that:  

• A-statements can be rendered into B-statements; for example,

X is present = X is simultaneous with this assertion.
X is past = X is earlier than this assertion.
X is future = X is later than this assertion.
NOTE: Russell's point is not clear here.  If he meant that "X is past" means "X is earlier than this assertion", then he was wrong.  
• X changes if “X is F at time t1” is true and “X is F at time t2” is false (e.g., "the poker is hot at t1" and "the poker is not hot at t2"); that is, change is having different properties at different dates.

NOTE: this is the heart of Russell's point, namely the idea that the B series is sufficient to give a satisfactory account of change.

Since he allows only the B series, Russell cannot say that events change because they are frozen in the B- series. So he tries to find change in things, for example, in a poker, by saying “the poker is hot at t1” (A) and “the poker is not-hot at t2” (B).
But he can’t. (A) and (B) are eternally true: the property of being hot at date t1 and not hot at date t2 always belong to the poker.  Hence, there's no change.  B properties are analogous to properties in space: the fact that something has contrary properties at two different places doesn't constitute a change in the thing.  For example, 

1. the poker is hot at one end and not at the other; there's no change in the poker.
2. “At S1 the meridian of Greenwich is within the UK” is true and “At S2 the meridian of Greenwich is within the UK” is false. And yet there's no change.    

NOTE: McTaggart seems to overstate his case here.  Surely, one can say "the forest changes from tropical to temperate as the elevation increases." 

2. The Don Quixote's objection:
A non-existent time series, e.g. that in Don Quixote, is a B-series (events in the novel are 'temporally' ordered) but not an A series because it cannot be related to the present (the would be knight's misadventures are not in the past, and he did not live in Spain). But a B-series involves time. Hence, an A-series is not necessary for time.
 
McTaggart’s reply:
It is false that the events in Don Quixote constitute a B series.  For if something is in time, then it exists. But Don Quixote's world doesn't exist, ever. Hence, Don Quixote's world is not in time (it is not a time series)
It is true that we can imagine that world in time; but then we think of it in the A series, that is, in the past.
 

3. The multiple time-series objection:

There might be different time series (T1 and T2) which, as such, would be temporally unrelated.  Hence, the presents of the different time-series aren't connected by relations of past, present and future (that is, a time in T1 wouldn't be in any temporal relation to any time in T2). Hence, such relations are not needed for time to exist.

McTaggart’s reply:
Since each time series is supposed to exist, each present in each time series would have a position in terms of past and future in that time series, although there would not be such a thing as THE time.
 
 

McTaggart's arguments for step 3: "the A-series doesn't exist".

1. Negative argument:
This argument is not quite clear.  It seems that McTaggart is advancing a criticism of the notion of 'now', which is an essential component of any account of the A series.

1. The ordering relation among the members of the time-series (that is, the relations of "earlier than" and "later than") never changes.
2. But the relations of being past, being present and being future involve change.
3. Hence, these relations are between what's in the series and something outside it; moreover the link between things in time and this thing outside time constitutes the basic temporal characteristics of pastness, presentness and futurity.  That is, the basic features of time would depend on something outside it.
4. But it's hard to se what such a thing could be.

NOTE: presumably, the point is that the idea that the essential feature of time (the moving 'now') is not an item in time is preposterous. Hence, without a strong argument for the existence of the moving now, one should reject the whole idea. 

1. Past, present and future are incompatible determinations.
2. But each member M of the A-series has at least two of them, e.g. present and past.
3. Hence each member M of the A-series has incompatible features.
4. Hence, the A-series is incoherent.

1. M is past, present and future at different times, i.e. successively and not simultaneously.  For example suppose that M is present at t₂; then it was future at t₁, and will be past at t₃.
2. Hence, it isn't true that M has incompatible temporal determinations.

In order to say that M has the qualifications of past, present, and future successively, one must appeal to moments (t₁, t₂, t₃) or meta-tenses (was future, is present, will be past).  Let's look at their problems in succession.
1. Appealing to moments reproduces the original problem.  Each moment is in the A-series, and therefore itself past, present and future, i.e., with contradictory properties.  Hence, the same problem which afflicted M arises now for moments. This generates an infinite vicious regress.
2. The application of meta-tenses reproduces the original problem.  To see this, consider Mellor's rendition of Mctaggart:

Let Pe = e is past; Ne = e is present; Fe = e is future.  Then:
1. Each event e is Pe, Ne, Fe, which is impossible (this is the original contradiction).
2. The rejoinder is that Pe, Ne, and Fe are not simultaneous.  Rather, for example,  FPe (will be past), NNe (it’s now present), PFe (was future).  These temporal determinations, however, are compatible with each other.

Reply: but then PPe and FFe and NNe obtain as well, and thesehave the same problem Pe, Ne, and Fe had.  This generates an infinite vicious regress.

D. McTaggart’s own objection:
We are immediately (experientially) certain of the reality of objective (interpersonal) time. Hence, time is real.
McTaggart’s two replies:

1. We aren't immediately certain of the reality of time because the experienced present, the specious present, has a duration which is, or can be, different for different people.  Hence, if perceived time were objective time, the same event would be both present and not present, which is impossible.
2. Although the A and B series are unreal, it is reasonable to assume that there is some serial feature in nature which we perceive as temporal.  This is the C series, which is isomorphic to the B series, although its terms are not events and its ordering relation not temporal. Hence, the serial nature of the experience of what really exists is correctly mirrored in the (incoherent) temporal series.  So, the ordering is real.

NOTE: the C series is not temporal, and therefore it must not be confused with the B series.  Hence tenseless  theorists are wrong.

• The past can determine the future, e.g., if Smith died childless, then there  won't be a marriage of one of Smith's grandchildren.
• If a proposition about the past (e.g., x occurred) couldn't entail one about  the future (e.g., y will occur), then it couldn't entail any proposition about later past or present because the laws of entailment apply to every domain, without any temporal determinations.