Formalizing Natural Languages

Question

I've been interested in the subject of metalanguages [in mathematical logic] and how (if) we can formalize them. Most metalanguages I've encountered use some variation of a natural language (such as English, German or French). Therefore, a relevant question is "how can we mathematically formalize natural languages" or at least, certain aspects of it. However, there are very few sources (As far as I've seen) that concern themselves with the subject. I've only seen two papers that have dealt with a similar subject, thus far; One from Richard Montague, and the other from Thomas Graf.

If you could kindly introduce me to references such as papers, books or any resources that deal with this subject itself and/or its prerequisites (such as syntax, semantics, formal languages), I'd be most thankful.

I'm an undergraduate mathematics student in my fourth semester; I'm not as familiar as I'd like to be in both logic and linguistics; So, even the most trivial resource would be most useful to me.

Thank you, very much.

Answer 1

This is a ridiculously complicated area, and the need to eventually narrow down your interest is a high priority. I would start with classical generative syntax as practiced by Noam Chomsky, and I would limit the initial investigation to the pre-Aspects version of the theory. Chomsky's works started out very mathematical and formal and became progressively less mathematical: here is a list, you might stop at 1964. There are some more-formal works in syntax in Head-Driven Phrase Structure Grammar, a model promulgated by Pollard and Sag e.g. in Information-based syntax and semantics.

In computer science, there is a lot written on formal grammar, but often that work is remote from facts of natural language. Formal semantics is the area of linguistics which maintains the tightest connection between natural language facts and formulaic expressions. This is not exactly to denigrate the technical progress that has been made in statistically-based machine translation, thus you can argue that the entire area of computational linguistics is a kind of mathematical approach to language. Those links may provide enough information that you could see if it makes sense for you to further pursue computational linguistics, or formal semantics, or formal syntax.

Answer 2

My favorite is Coppock & Champollion's Semantics Boot Camp. It is formally precise yet written in a very accessibly way. The first chapters provide an introduction to elementary logic and set theory, the mid section covers the standard approach of formalizing simple natural language expressions with typed lambda calculus buid up on a phrase structure grammar, and the remaining chapters discuss some advanced phenomena.

Y. Winter (2016) "Elements of Formal Semantics" is a somewhat more concise introduction that you might like if you have a mathematical background.

B. Partee, A. ter Meulen & R. Wall (1990) "Mathematical Methods in Linguistics" is a classic that also covers some algebra and formal language theory which the other two don't have.

From there on just follow the references for more specific topics, or read some handbooks (collections of overview articles on certain phenomena/resarch fields).