Syntactic Categories in the Language of Mathematics

We have been studying the structure of simple mathematical statements in English. It has turned out that both phrase structure grammar and type theory are needed. Then we have given an interpretation and generalization of phrase structure grammar in type theory—in other words, an extension of type theory with a system of categories of phrase structure grammar. We have ended up with a formalism that comprises both informal English (that is, strings of words), a system of syntactic categories and syntax trees, and a formal mathematical language. We have shown how syntax trees are sugared into English and interpreted in the mathematical formalism.