In an earlier post I wondered whether the newer pregroup grammar framework should supersede type-logical grammar for syntax. I have now decided such niceties are not important. The fact is, we have more than enough machinery to handle the pure syntax of language "algebraically" or "logically," however you view it. My own work has contributed to the learning algorithms for this. I think it is time to move on.
What's next? Morphosyntax. Many, perhaps most, languages involve a very significant amount of morphology which contributes essentially syntactic information. Some, such as Navajo, have a very simple syntax and a cripplingly complex morphology does most of the actual work. Yet mathematical linguists rarely treat morphosyntax. I don't really know how to handle it, myself. My basic suggestion is that we need a new class of mathematical models that would be capable of handling a language such as Navajo. These would be able to relate semantics directly to the morphology and syntax at the same time. Preferably without making reference to morphemes. Most morphology is not very well described morphemically, but it all submits rather well to a relational view in which the morphology is modeled as a system of form-meaning relations among words.
This is a challenge I hope to take up one day in the not-too-distant future. I of course invite anyone to tell me what is already out there in the vein of mathematical morphosyntax. My acid test is, does it work for Navajo?