In this paper, I address the problems of complement polyvalence, of complement polyadicity and of complement permutation. These problems are common across languages, in my experience, ranging from English and French, to Latin and Sanskrit, to Chinese. I explain what these problems are and propose a single, rather simple solution to them. (The solution involves a modified Lambek Calculus.)
The problem of complement polyvalence is where the very same word admits complements of different syntactic categories, though the word's meaning is invariant under such changes. A well-known example of such a word in English is the copula to be, whose complements include adjective phrases, noun phrases, prepositional phrases and adverbial phrases. The problem of complement polyadicity is the problem presented by words whose complements are optional. Correlated with their omission is a systematic shift in sense. Here are some examples from English: to eat vs. to devour, to arrive vs. to reach, to dress vs. to clothe and to meet vs. to encounter. The problem of complement permutation is where a single word permits its complements to permute though preserving its sense. These words include, but are not limited to, verbs with indirect objects: to give (Alice gave a ball to Bill; Alice gave Bill a ball).