[Please remember that analysis of a poem is subjective, and this answer is based upon what I perceive. Also note, however, that any interpretation of a piece of literature—to my mind—is valid as long as you can support your viewpoint with logical reasoning from the text. With this in mind, be confident in your ability to analyze poetry.]
Generally, a paradox is a literary device that states a truth that seems contradictory and untrue at first glance. However, the explanation of paradox also describes more subtle forms of the device, very different than the one noted above. The definition goes on to state...
Sometimes [paradox] is applied to a self-contradictory false proposition.
Studying this kind of paradox, we see that truth is not always a part of all of its forms. Based upon the author's troubled life, this darker form of paradox might seem a more appropriate form of the device to use. But as subtle as the forms of paradox might be, so are Plath's artful perceptions and her implementation of them into her work.
Look to Sylvia Plath's poem "Mirror." Line four is paradoxical:
I am not cruel, only truthful...
I find this a paradoxical statement. How can one tell the truth and not be cruel in some circumstances? By nature, many truths are things that people would rather not hear. From the start, we should look for a "self-contradictory false proposition."
The mirror that is speaking in the poem has been personified—given human characteristics—and so it cannot be accepted now as truthful and not cruel. The presence of personification robs the mirror of its impartiality and credibility. The identification of personification can be found in the following:
I have looked at it so long
I think it is a part of my heart. But it flickers.
Faces and darkness separate us over and over. (7-9)
How can a mirror have a heart? This in itself seems paradoxical: how can an inanimate object have feelings? When the mirror is personified, then its statement regarding its complete objectivity is called into question.
Knowing as we do that mirrors must reflect what they see, and couldn't possibly hold judgment as inanimate objects becomes irrelevant when our narrator is a mirror.
By the poem's end, the mirror reports what it sees, but it passes judgment. When truth is presented, there is only the statement of fact—no judgment.
Each morning it is her face that replaces the darkness.
In me she has drowned a young girl, and in me an old woman
Rises toward her day after day, like a terrible fish. (16-18)
The mirror states that the woman has wasted her youth studying her image in the mirror, looking for signs that she is aging. This is a subjective observation. The mirror's narrative is now biased. It speaks of the woman's face now—but that the mirror at one time reflected the image of a young girl...that the woman has drowned (a harsh and emotionally-charged word choice). The face that the mirror reflects back (it says) is old. The mirror goes on to compare the visage of the woman rising into the mirror's sight each day to that of a "terrible fish." This is a clear example of being cruel, which the mirror claimed it could not be at the poem's beginning.
The clues we find to identify this kind of paradox are glaringly obvious in the diction (word choice) the author employs. In life, hearing the truth may sometimes feel like cruelty. However, what the mirror observes is subjective, judgmental and unkind. By definition, the form of paradox used in this poem "presents a self-contradictory false proposition."