An overview of mathematical-logical approaches in formalizing natural languages

Question

_{Crossposted on Mathematics SE}

---

I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach), yet there aren't many resources that provide an overview of this very technical field. I wish to be able to provide myself a general map to navigate this subject more clearly, so I will attempt to explain my basic understanding of an overview of the fields that tackle with this subject, and I'd greatly appreciate any correction or addition to my limited insight.

From my understanding, (loosely speaking) there are two prominent mathematical-logical approaches in formalizing natural languages: Categorial Grammar and Montague Semantics. While Categorial Grammar uses methods borrowed from category theory to (mainly) study the syntax of natural languages, Montague Semantics, as the name might suggest, (mainly) focuses on the semantics of natural languages by implementing methods from Lambda Calculus. In terms of subareas of each of these two fields, I have not seen much discussed in terms of the subareas of Montague Semantics; however, Categorial Grammar seems ripe with subareas (although I have heard some of the fields mentioned below are only closely related to Categorial Grammar rather than being a strict subfield of it):

1. Combinatory Categorial Grammar (CCG)
2. Lambek Calculus
3. Type-Logical Grammar
4. Pre-Group Grammar
5. Proof-Theoretic Semantics

In addition to any corrections or additions, I would greatly appreciate any suggestions for resources, references or books that deal with these subjects or their prerequisites

Answer 1

Here is what I answered in Math.SE:

If you haven't read it already, I highly recommend reading L. T. F. Gamut's Logic, Language, and Meaning. If you already know some logic (say, up to first-order logic), the first volume may be a bit redundant (though the last two chapters, on pragmatics and the Chomsky hierarchy, may be new). The second volume, however, contains a very nice introduction to modal logic, type theory, and Montague semantics, as well as some other developments (such as Discourse Representation Theory). Obviously, it's just an introduction, but it is enough to give you an excellent overview of the subject.

Notice that many prominent semanticists in the Montague tradition (such as Barbara Partee) are also generativists, so you may need a background in Chomsky's theory to better understand their approach. There are a lot of books that work with this, but, for a general philosophical perspective, I quite liked John Collins's Chomsky: A Guide for the Perplexed, and, from a more technical one, Andrew Carnie's Syntax: A Generative Introduction (now in the 4th edition, together with the workbook). Incidentally, a classical textbook on this particular semantic approach is Heim & Kratzer's Semantics in Generative Grammar.

A quick observation about your summary: at least in my corner, Categorial Grammar does not refer to the application of Category Theory (the theory initially developed by Mac Lane and Eilenberg) to natural language. Rather, it refers to an approach that takes certain classes of expressions (categories, in the philosophical sense, since Aristotle) as basic and builds from them derived expressions. So a typical categorial grammar, for example, will take as basic expressions such as n (for names) and s (for sentences) and build derived categories of the form (n/s) (an expression that takes a name and builds a sentence, i.e. a predicate), (s, s/s) (an expression that takes two sentences and builds another sentence, i.e. a binary connective), etc. This type of grammar was developed in the 20s by Ajdukiewicz and later refined by Bar-Hillel in the 50s, and so was developed independently of Category Theory.

As for Montague, it is true that the introduction of the lambda calculus did revolutionize the field, but not by itself; the thing is that Montague showed how, by not trying to shoehorn everything into first-order logic (such as, say, Davidson was trying to do around the same time), and, instead, analyzing natural languages directly using more powerful and flexible means, one could produce an elegant compositional semantics for a remarkable fragment of natural language. (If you're interested in the history of the subject, several of Partee's papers are relevant, and they can be very accessible---cf. this one, for instance.)

Answer 2

This is a really a question about linguistics rather than mathematics, so you might be better off asking your question on Linguistics StackExchange rather than here. But anyway, I think what you're asking about is the field commonly known as formal semantics. I don't know a lot about this field, but you might start with the anthology Formal Semantics: The Essential Readings, edited by Paul Portner and Barbara Partee. This will at least give you a historical perspective on the subject, which might help you decide what sub-area you want to focus on.