A movement paradox is a grammatical phenomenon which, particularly according to proponents of lexical functional grammar, presents some problems for a transformational approach to syntax. Take the following example sentences:

We talked about the fact that he was sick for days.

The fact that he was sick, we talked about for days.

These two sentences are related by movement in a transformational analysis: in the second sentence, the phrase "the fact that he was sick" has been moved to the front of the sentence for emphasis. However, there are pairs of sentences like this in which the sentence without movement is ungrammatical while the sentence with movement is not:

*We talked about that he was sick for days.

That he was sick, we talked about for days.

This may be difficult to explain in an analysis based on movement, since it is not obvious how the second sentence can be grammatical if it is derived from a movement operation applying to an ungrammatical structure, i.e. a structure like that of the first sentence. A more striking example of a movement paradox can be found in English morphology:

*I aren't allowed to do that

Aren't I allowed to do that?

In a transformational analysis, the question is formed by the movement of aren't to the front of the sentence, which does not explain why aren't is acceptable as a contraction of am not only after movement. The problem may be alleviated if one assumes that morphological processes apply after movement, but this raises a further problem:

*Am not I allowed to do that?

If morphological processes applied after movement then the sentence above would presumably be the one from which "Aren't I allowed to do that" would derive, but it is clearly ungrammatical.

Lexical functional grammar avoids these difficulties to some extent because it does not have a movement operation.

Reference

Bresnan, Joan. 2001. Lexical-Functional Syntax. Oxford: Blackwell.