to develop an easy rationalization for his murderous ways.  He is explaining to the audience why it is necessary for him to killBanquo.

Before, in Act I, he admitted that there was no good reason to kill Duncan.  He did so only to appease his "vaulting ambition."  Here, in Act III, he uses a bit of equivocation to create an "argument" for the necessity of killing Banquo.

His equivocating goes like this:

• Macbeth has murdered to become King of Scotland.

• The witches prophesied that Banquo's issue would be kings of Scotland.

• Therefore, Macbeth has murdered Duncan, only to have Banquo's son follow him as king of Scotland.

Which, of course, doesn't necessarily follow.  There could be many generations of kings between Macbeth and Banquo's future relations.  Macbeth has assumed that the witches meant that Fleance (Banquo's son) would succeed Macbeth as king.  This is fallcious (or faulty) reasoning.

Here are the lines of text that give this reasoning:

When first they put the name of king upon me,

. . .then prophet-like

They hail'd him father to a line of kings:

Upon my head they put a fruitless crown,

. . .No son of mine succeeding.  If't be so,

For Banquo's issue. . .

. . .the gracious Duncan have I murder'd

. . .To make them kings, the seed of Banquo kings!

There are certainly more possible instances of equivocation in this Act.  I hope that this answer and the links below will help you find them.

An equivocator is someone who lies and tells half-truths, or says something ambiguous to mislead another. In Shakespeare's play, Macbeth, the witches are generally considered the "great" equivocators in the play in the play. (In the second set of predictions they deliver in Act Four, they tell half-truths to trick Macbeth.) However, we can see instances, also where Macbeth is no better than the witches for telling lies.

The first instance where Macbeth outright lies is when he says he killed Duncan's guards because he was so upset over Duncan's murder at their hands. He actually kills them so they cannot raise doubts in anyone's mind that they were "framed."

MACBETH:
O, yet I do repent me of my fury,
That I did kill them. (II.iii.117-118)

He excuses his behavior, wondering how anyone could be levelheaded in the face of such horror: seeing his beloved King murdered.

Another instance is when Macbeth speaks to Banquo, ostensibly to ask him if he will be gone long when he goes riding—to remind him not to be late for dinner because Macbeth wants to speak to him. Macbeth is actually making plans to have his friend murdered because Banquo heard the witches' initial predictions that told Macbeth he would be king, and Banquo has told Macbeth that he will not be swayed from what he believes to be morally correct.

MACBETH:
We should have else desired your good advice,
Which still hath been both grave and prosperous
In this day's council; but we'll take tomorrow.
Is't far you ride? (III.i.23-26)

Later in the same scene, Macbeth meets with the men he has hired to murder Banquo. They are not professional killers, but simply common men whose lives have taken a turn for the worse. Macbeth has told the men that Banquo is to blame for their dire circumstances—when they had originally thought it was Macbeth. (The truth is that it was Macbeth who had beggared them.)

MACBETH:
...Know
That it was he, in the times past, which held you
So under fortune, which you thought had been
Our innocent self?
...“Thus did Banquo.” (81-84; 89)

Macbeth goes on to ask these men (much like his wife asked him with regard to Duncan) if they are kind-hearted enough to ignore what Banquo has "done," or if they will be man enough to do something about it.

Of course, the men believe Macbeth (as most of his peers do at the beginning), never expecting that this new King of Scotland is lying to them. He tells them that he could kill Banquo easily himself, but that he doesn't want to offend important men who are mutual friends to Macbeth and Banquo.

Macbeth does all he can not only to take the throne from Duncan, but to make sure it remains in his possession.

Well, an equivocation is a logical fallacy or something that constantly changes meaning depending on denotation.  An equivocator is some who who does the above but also fails to make a direct statement. When the porter does his large speech on the play of the word "equivocation" and "equivocate," he is talking about liars lying. The character of the Porter is intended to create a catharsis after the murder of the King, yet while he is a funny drunkard, he is also expressing ideas that someone is lying and will continue to lie and put forth false pretense, misleading everyone. He is referring to what Macbeth will do.

In Shakespeare's Macbeth, equivocation plays a central part in bringing about the death of Macbeth.

Equivocation is defined as...

...the misleading use of a term with more than one meaning...

It can also be described as "doublespeak" or "doubletalk," which is…

...language that deliberately disguises, distorts, or reverses the meaning of words.

Its use in Macbeth is intentional, and it refers primarily to the second set of predictions given by the witches. With the first set of predictions, the witches lure him with small truths: calling him by his present title, Glamis, his upcoming "promotion" with an added title of "thane of Cawdor," and finally with the clincher, the man who will be king. (This last prediction might be included as a form of equivocation—they fail to mention that in order to be king, Macbeth must first murder Duncan.)

When Hecate, the queen of the witches, learns that the witches have been trafficking with Macbeth, but only playing with him, she is angry. She wants to be a part of the "fun," and she believes that the evil they serve has not been "uplifted" in any way—she tells the weird sisters that they must make sure to lure Macbeth to his ultimate destruction.

It is here that we remember Banquo's warning to Macbeth when he told his friend that the powers of evil win a man with small truths, and then trick him with the big lie—the one that makes all the difference—when he already is a believer in the truth of their words…

BANQUO:
...But ’tis strange;
And oftentimes, to win us to our harm,
The instruments of darkness tell us truths,
Win us with honest trifles, to betray's
In deepest consequence— (I.iii.132-136)

As Banquo warned Macbeth in Act One, there is little truth to what the witches predict in Act Four; Macbeth will learn too late that he has been deceived.

Foolishly, Macbeth thinks he can control the witches. They serve a darker power, but they let he think he can command them. When Macbeth demands new predictions, they equivocate.

Each new prediction is delivered by an apparition or vision, and it says to watch out for Macduff—this is true enough:

FIRST APPARITION:
Macbeth! Macbeth! Macbeth! Beware Macduff;
Beware the Thane of Fife. (IV.i.79-80)

The second apparition "reels" a "hooked" Macbeth in; the vision tells Macbeth that only a man born of a woman can harm him. Literally, all men are born of a woman, but technically, a child can be "born" in more than one way. This is where they trick Macbeth:

SECOND APPARITION:
Be bloody, bold, and resolute; laugh to scorn
The power of man, for none of woman born
Shall harm Macbeth. (88-90)

Macbeth thinks he's safe, sure there is no one who does not fit that description. Later, Macduff will tell Macbeth that he was "born" by caesarean section.

The final apparition reinforces Macbeth's "false sense of security." This vision says that Macbeth will never be defeated until the woods can move to the hill. Macbeth figures woods cannot move. However, the technicality here is that Malcolm's army will camouflage itself with tree branches, so it looks as if the woods are moving.

THIRD APPARITION:
…Macbeth shall never vanquish’d be until
Great Birnam Wood to high Dunsinane Hill
Shall come against him. (103-105)

These equivocations lure Macbeth to his doom for he unwisely forgets the nature of the "women" who advise him. By the time he realizes this, it is too late.

Additional Source:

http://en.wikipedia.org/wiki/Equivocation

Doublespeak and duality apply to the prophecies that Macbeth received from the witches about danger.

Macbeth interprets the witches’ prophecies as demonstrating that he is not in danger.  His reactions to these seal his doom.  In his arrogance, he assumes he is safe.

A perfect example of this is the prophecy regarding Macduff.  Macbeth is aware that Macduff is coming after him, in revenge for Macbeth’s ordering the murders of his entire family.  Macbeth wants to know if Macbeth is a danger to him.

Be bloody, bold, and resolute; laugh to
scorn
The power of man, for none of woman born
Shall harm Macbeth. Descends.(90) (Act 4, Scene 1)

Macbeth’s reaction is “Then live, Macduff. What need I fear of thee?”  He is convinced that Macduff is no danger to him, and the witches reinforced that idea.  No man born of woman can be a danger to him, and since every man has to be born from a woman, he is safe.

Macbeth is still convinced of this after some of the other prophecies turn out to have double meanings.  He was supposed to be safe until the forest moved to him, and yet the soldiers pretended to be trees and so the forest basically did.

When Macbeth meets Macduff in battle, he taunts him that he is safe.  Macduff delivers a blow.

Despair thy charm,
And let the angel whom thou still hast served
Tell thee, Macduff was from his mother's womb
Untimely ripp'd.(20) (Act 5, Scene 8)

When Macbeth finds out that Macduff was born by C-section, he is scared.  He loses his courage, and suddenly realizes that he is doomed.  Psychologically, he is no longer able to fight.  The feeling of safety being pulled out from under him deflates his arrogance, and he essentially submits to dying in battle.

A paradox is a statement that seems to contradict itself but upon further examination contains a hidden truth. There are numerous examples of paradoxes in Shakespeare's classic play Macbeth. In the opening scene of the play, the Three Witches utilize a paradox by saying, "When the battle’s lost and won" (Shakespeare, 1.1.4). This is considered a paradox because the comment is initially confusing when one considers that a single battle cannot be both won and lost. However, in every battle, there is a winner and loser. Similarly, Macbeth will win numerous battles in life but experience many losses. A few lines later, the witches use another paradox by saying, "Fair is foul, and foul is fair" (Shakespeare, 1.1.12). This is considered a paradox because what humans consider fair is actually foul and the witches take pleasure in foul situations, which they believe are fair. This paradox is a common motif that runs throughout the play and underscores the theme that appearances can be deceiving.

The Three Witches once again utilize a paradox during Banquo's prophecy in act one, scene three. The First Witch tells Banquo that he will be "Lesser than Macbeth and greater" and the Second Witch tells him, "Not so happy, yet much happier" (Shakespeare, 1.3.66-67). Both of these statements seem to contradict themselves but contain hidden truths. While Banquo will not become a king, his descendants will reign for generations, which will make his name much greater than Macbeth's. He will also not experience the pain, heartache, and anxiety that Macbeth will endure after becoming king, which is why he will be much happier.

The witches' apparitions in act 4, scene 1, equivocate when they present their prophecies to Macbeth. The Second Apparition tells Macbeth,

Be bloody, bold, and resolute. Laugh to scorn The power of man, for none of woman born Shall harm Macbeth. (Shakespeare, 4.1.81-82)

This is considered an example of equivocation because the prophecy is purposely ambiguous and Macbeth does not discover its true meaning until he comes face-to-face with Macduff in the final battle. The ambiguity of this prophecy leads Macbeth to believe that no man can harm him since every man is born from a woman. However, he discovers that Macduff was not born and was instead "untimely ripped" from his mother's womb.