What is the dramatic irony in Duncan's speech in act 1, scene 4, lines 11–14 of Macbeth?

Act 1, scene 4, lines 11–14 from Macbeth

William Shakespeare's Macbeth is a drama drawn in the sinister with shadows of evil as witches and dark hearts consumed with ambition construct a phantasmagoric world of deception, madness, and delusion.

In Act I the dark ambition of the traitorous Thane of Cawdor brings him defeat at the hands of "brave Macbeth." In Scene 4, King Duncan asks his son Malcolm if Cawdor has yet been executed for his rebellion; Malcolm replies that he has; furthermore, Malcolm continues, the villain met his death as "twere a careless trifle," of no importance. Hearing this, the king remarks that one can never tell by looking at someone what character that man possesses. Duncan adds that Cawdor was perceived by him as a man he could trust.

1. King Duncan's remarks about the traitorous Thane of Cawdor exemplify dramatic irony [a contradiction between what a character believes and what the audience knows to be true] because he will again be deceived by the next Thane of Cawdor, Macbeth, another Scottish nobleman whom he trusts. For, later, Macbeth, just as the traitorous Cawdor has done, will show a face that is deceptive.

2. In Scene 5, Lady Macbeth has read her husband's letter, telling her what has occurred with him after battle and his encounter with the "three weird sisters" who have hailed him as Thane of Cawdor and, then, as king. After a messenger informs Lady Macbeth that her husband is coming, she speaks in a subsequent soliloquy to the one in which she reads Macbeth's letter. Worried that Macbeth is too filled with "th' milk of human kindness," Lady Macbeth invokes the spirits to unsex her

and fill me, from the crown to the toe, top-full
Of direst cruelty! (1.5.42-43)

She entreats the spirits further, asking that she feel no compunctions about the murder of King Duncan that she plans. In an apostrophe to the night, she calls upon the darkness to hide her in her evil, "the dunnest smoke of hell," so that she will not see where she stabs the king (she will use a knife), nor will the spirits know afterwards that she has killed the king and cry, "Stop" if she remains under the cover of night.

Lady Macbeth beseeches the spirits in this manner because the Elizabethans believed in the Chain of Being, in which the king was right beneath the holy spirits and had divine rights. Thus, the semi-divine monarch is at the head of the hierarchy of life. If the king is killed, his death upsets world order, so Lady Macbeth wants Duncan's death concealed so that this order will not be disturbed.

Then, when Macbeth enters, after already having imagined her regicide, Lady Macbeth feels transported to the future and envisions her husband as the king, whom the witches have predicted he will be. This future of Macbeth's being king is what she implies when she says, "Greater than both" after saying Macbeth's present titles of "Great Glamis" and "Worthy Cawdor."

Why is Duncan's high praise of Macbeth in Act 1, Scene 4 considered dramatic irony?

Dramatic irony is where the audience knows something that one or more of the characters in the drama doesn't. In Act I Scene iv Duncan still believes, not unreasonably, that Macbeth is a brave and valiant warrior, a hero who's more responsible than anyone else for the Scots' crushing victory over the Norwegians.

Duncan is so grateful for Macbeth's heroism and bravery that he rewards him with the title of Thane of Cawdor. Yet unbeknownst to Duncan—and this is where dramatic irony comes into the picture—Macbeth has already had the seeds of ambition and treachery planted in his mind by the Weird Sisters' prophecy. The king is blissfully unaware of all this; as far as he's concerned, everything's normal. Macbeth is still a loyal and faithful servant. The very idea that Macbeth could be planning to betray him simply doesn't cross his mind, especially after he acquitted himself so well on the field of battle. But the audience knows better.