When seeming and being are confused in Shakespeare's Hamlet, how does Hamlet's failure to recognize the difference between the two help lead to his downfall?
The theme of being and seeming, or real and seeming, is a detailed and important one in Hamlet. In this format, I can offer a couple of illustrative examples.
The first is the trouble Hamlet has knowing whether the Ghost is a real phantom or an unreal hallucination: is it a troubled spirit from the nether realms or is it a vapor from his own troubled mind? This confusion helps lead to his downfall because he cannot easily decide whether it is spiritually and morally right to obey the Ghost's commands for revenge. Thus he both loses opportunities for revenge and provides opportunities for others to cause him more trouble, like Laertes challenge (IV.vii).
... I, the son of a dear father murder'd,
Prompted to my revenge by heaven and hell,
Must, like a whore, unpack my heart with words,
Another is the symbolic underscoring of the theme that is represented in Hamlet's confrontation with Gertrude. During this confrontation, he sets out the particulars of his description of the real King of Denmark, his dead father king, and the seeming King of Denmark, the usurper, King Claudius. This might be styled a "confusion" on Hamlet's part because the real King is dead and not the king while the seeming King is alive and really the king. This helps leads to his downfall because his ideas and words make him appear to be mad so he has no allies, only enemies.
Look here, upon this picture, and on this,
The counterfeit presentment of two brothers.
See, what a grace was seated on this brow;
... the front of Jove himself;
... Look you now, what follows:
Here is your husband; like a mildew'd ear,
Blasting his wholesome brother. (III.iv)