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

1. In the A series, time is thought of in terms of past, present (now), and future, and events are located in time on the basis of their relation to the 'now' (the present).   That is, they 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 distance from the moving 'now'.)

NOTES
• We commonly use tenses.  For example, if I say “he will marry”, or “the war ended” I convey the 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 dynamical because thing/event go from being future to being present, and then past. Hence, the tenses of events and things 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'.  In practice (although this is not necessary in principle), one selects an event and arranges other events on the basis of their dates (a date is a position in the time series defined by its distance from a fixed point, e.g., the birth of Christ, the foundation of SIUE, or whatever).

NOTES:
• We commonly use dates, as in “the war ends (tenseless) in 1945”, "on the 5th I go home".
• The B series is static because if A earlier than B, or occurs in 1945, that is always So. That is, dates dont 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 a more common case, I may know that the appointment is at 5 without knowing what time it is now because my watch stopped (I know the date but not the tense).  Conversely,  one could know the locations of all events with respect to now, without knowing the date of now; more commonly, I'm at the meeting place now, but I don't know if I'm on time because the meeting was supposed to be at 5, and my watch stopped again.  I know the tense but not the date of my being at the meeting place.

Note that these situations have spatial analogues: 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 the 2nd of August” even if ‘now’ is the 2nd of August.

The argument is obviously valid, and the only remaining issue is whether it's sound (i.e., has true premises).
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:  

 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. So, 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 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.

Ultimately, McTaggart believes that the world is serially ordered according to what he calls the C-series, which is non-temporal but isomorphic with (with the same structure as) the B-series. To understand this, consider the following  view on the structure of time.  Time is a set of moments ordered by the relation T, “being before than.”  Then the standard topology is given by the following axioms, where x, y, z are moments:

1. (x) -Txx  (irriflexivity, i.e., no moment is before itself)
2. (x)(y)(Txy->-Tyx) (asymmetry, i.e., if x is before y, then y is not before x)
3. (x)(y)(z)((Txy & Tyz)->Txz) (transitivity, i.e., if x is before y and y before z, then x is before z)
4. (x)(y)(Txy v Tyx) (connectedness, i.e., every moment is before or after any other moment)

(x)(y)(Ez)(Txy->(Txz & Tzy) (density, i.e., between any two moments there is a third one)

Few philosophers have been persuaded by McTaggart that time doesn't exist.  However, his arguments have produced atwo camps among those who believe that time is real:

1. The world is dated but not tensed: there are dated facts but no tensed facts (Russell; Quine; Mellor; Oaklander, etc.).  Philosophers who accept this view are called "detensers"
2. The world is both dated and tensed: there are both dated and tensed facts (Swinburne; Smith, etc.). Philosophers who accept thos view are called "tensers"

NOTE: Pre-philosophically, the tensed view seems plausible: "WWII ended in 1945" seems to be made true by a dated fact and by a tensed fact.

• In the tensed view of time, the 'now' flows along the series S of events.  An event E, for example, acquires and loses the property of “being present”.  This, in the tensed view, is the essence of change.  However, the event E acquiring presentness is itself an event.  Yet, it cannot be an event in the series S. Hence, it must be an event in some other series.  This involves introducing a meta-time, or giving up explaining change (Broad).

 
• How fast does the 'now' flow?  The natural answer is something like "at one second per second".  Detensers find this answer unacceptable on two grounds:
1. A movement involves two series whose items are correlated (e.g., the positions of a car and those of moments of time).  So, the positions of the 'now' must correlate to moments of time in some meta-time.  Hence, one ends up with two time series.
2. The answer 'at one second per second' is tautologous, amounting to saying that there is one second in every second.

Replies:
• No need to appeal to a meta-time.  One could just say that time flows at the rate of one second for every n meters covered by the car.
• Since the flow of time is that to which other things are compared to, asking how fast time flows is similar to asking how long the standard meter is Paris is long, a meaningless question (standards don't admit of self-predication)
 
• McTaggart’s argument against the reality of the A series (i.e., of tense)
• The tensed view of time is incompatible with a well established theory, namely  Special Relativity.  We shall not go into this; however, it's fair to point out that whether the tensed view is incompatible with Special Relativity is a matter of controversy (for example, Smith has argued that it is not).

1. Can the tenseless view account for the diachronic identity of things?

NOTE: it must be able to do that, for to have change the same thing must have different properties at different times. 
Merely having different properties at different dates doesn't constitute change: the property of “being hot at t1”  and “cold at t2” belong to the poker at every date.  The situation analogous to the poker  being hot at one end and cold at the other, which constitutes no change in the poker.  

1. Our different attitudes towards past, present and future. When the dentist session is over, and I say “Thank goodness that's over!” (T),  I am not thanking goodness for the tenseless fact F that the conclusion of the root canal is contemporaneous with T’s utterance, for F is a fact also before and during the root canal. Rather, I'm thanking goodness for the fact that the drilling stopped, i.e. was present and now is past (a tensed fact). (Prior).

Reply:  An event ceases to exist if and only if there is a date after it, or it is earlier than my statement about it.  At date t₁ I dread the root canal.  At t₂ I have the perception of the root canal, which is simultaneous with the event dreaded at t₁.  At t₃ I have memory of the dreaded experience, and I feel relief because my memory is later than the dreaded experience.  Hence, there's no need of tensed facts to explain my attitude.
 
2. Many have the definite impression of perceiving the flow of time.

Reply: We don't perceive the flow ot time.  It is the succession of different attitudes and perceptions (anticipation and dread, perception and pain, memory and relief) that gives rise to the impression to the flow of time.