Productivity and Structure: Negative Prefixes in English
One reason that it is likely that productivity can tell us something about language structure is that the more productive a morphological derivational process is, the more likely it is to have a compositional output, one whose meaning is transparently predictable from the meaning of its input. The converse is also true: the less productive a derivational process, the more likely it is to result in a non-compositional, semantically idiosyncratic, non-transparent output.
To illustrate this point, consider negative prefixes in English. Zimmer (1964) looked at three of them, non-, un-, and in-, and discovered that the most productive of the set, non-, also has the most semantically transparent derivatives. This is shown by the contrast between the two columns in (4):

What’s the difference? Non-Christian means ‘not Christian’. While unchristian can mean that, too, more often than not it means something like ‘not behaving in a Christian manner’ or even ‘uncivilized and barbaric’. Likewise, non-human simply means ‘not human’, while inhuman refers to the absence of human qualities like pity or kindness. A person can be both human and inhuman, but not human and non-human. Thus, while the non-words in general simply negate their bases, the in- and un- words have the meaning ‘completely opposite to X’, where X is the meaning of their bases, in the way that east and west or long and short are opposed. To put it in a more technical way, non- is a logical or contrary negator. Using logical notation, we could represent non-Christian as in (5), where ¬ means ‘not’:

Un- and in- are contradictory negators, whose addition to a word X results in a new word meaning ‘opposite of X’.
Zimmer’s observation extends to other derivational affixes. The suffix -ness is more productive in English than -ity (Aronoff 1976). Consider the pair collectivity∼collectiveness. While both may mean ‘the quality or condition of being collective’, only collectivity has the additional meaning ‘the people considered as a body or whole’. Overall, when we compare many such pairs, the -ness derivative has more transparent semantics. Sometimes the -ity derivative sounds or looks odd, while the -ness derivative is pretty much always acceptable. Compare conduciveness with conducivity. Most English speakers would say that the former is more acceptable than the latter. We can even go beyond morphology to make the observation that syntax, which is always productive, is by definition compositional.