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Basically, will knowing some fairly large set of words and phrases and their translations, could a computer perhaps predict what the word for perhaps a very basic word, like "eye?"
Alternatively, if given a large number of words and their translations, could a machine translate a word it had not encountered?
I assume you mean "correctly predict", not just make a prediction. If you impose certain requirements on the input data, it could. First, the corpus would have to not only have the language string, it would have to have a systematic English translation (e.g. "mouth" is translated as "mouth" and not as "beezer; piehole; cakehole; yap"). Second, it would have to contain enough paradigmatically related forms, for example "my N; I found an N; the N disappeared", so that the puzzle-solver can hold constant all of the the variables and solve for "my", "I", "found", etc. and then of course "eye". It would also have to contain at least one token of the word "eye" in a frame that is unambiguously parsable at least w.r.t. the slot containing the word "eye".
On the other hand, if you have an exhaustive corpus of every French word, phrase and sentence ever uttered / written but excluding any instance of "oeil", you have no way of predicting that the French word for eye is "oeil" (even if you know that the word for eyes is "yeux").
It is certainly possible to construct sets of words and translations that allow a prediction of the unknown word, e.g., by employing derivational morphology and all morphemes of the unknown word are in the given set of words and translations.
Chances are also quite good when the set of words is quite large and has a good coverage of the morphemes of the given language. But there are always words that cannot be translated using such methods, for examples see this question and its answers.
(EDIT) It is much more difficult to predict the word from a translation because the choice of the word is idiomatic. So there may be several ways of deriving the missing word, but by idiomatic choice of the native speakers it is something else! Many typical mistakes by L2 learners come from this problem.(/EDIT)
In general, context of surrounding words is more useful than pure morphology (newspaper readers can guess the meaning of new words routinely).
