Overview

In the last few weeks, we’ve introduced syntactic and semantic rules for English that aim to characterize two things at once: (i) the distributional properties of English expressions (e.g., that the $np$ Ziggy can precede the $vp$ meows, in order to make an $s$); and (ii) the semantic contributions these expressions make to the meanings of the larger expressions that contain them. Further, our rules are stated to be sensitive to the syntactic categories of the expressions whose behavior they characterize. For example, the rule

\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}}⟩ ⊢ vp$} \BinaryInfC{$⟨x^{⌢}y, ⟦y⟧_{\mathcal{M}}(⟦x⟧_{\mathcal{M}})⟩ ⊢ s$} \end{prooftree}\]

says:

that you can form an $s$ by sticking an $np$ to the left of a $vp$;
that you can get the meaning of the $s$ by applying the meaning of the $vp$ as a function to the meaning of the $np$ as an argument.

So, syntactic categories have an important role: they say where things can go, and they also say what things can do (semantically) when they go there.

These notes further develop this idea, in two directions. First, we introduce a language of semantic types. Semantic types tell us what kind of meaning an expression can have—for example, whether it is an entity, a truth value, or some type of function (and if so, what type). Second, we introduce a language of syntactic categories (or syntactic types, if you like). Syntactic categories tell us what an expression’s distributional properties are—that is, what kinds of expressions can appear to its left and to its right. As we’ll see, there is a close relation between syntactic categories and semantic types. This relation holds because of the fact that where an expression can go inside some larger expression determines what type of meaning it can have.

We’ve already encoded a version of this idea in our grammar with categories like $np$, $vp$, and $tv$, where we had rules that were specific to these categories. Our new language of categories will make the idea completely general: we’ll have an infinite number of possible syntactic categories, as well as an infinite number of possible semantic types, and each syntactic category will be seen to be associated with a semantic type. At the same time, we’ll be able to pare down the number of rules we have to only two!

--- title: "Overview" bibliography: ../../semantics.bib format: html: css: ../styles.css html-math-method: mathjax mathjax-config: loader: {load: ['[tex]/bussproofs','[tex]/bbox','[tex]/colorbox']} tex: packages: {'[+]': ['bussproofs','bbox','colorbox']} --- ::: {.hidden} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\true{\ct{T}} \def\false{\ct{F}} $$ ::: In the last few weeks, we've introduced syntactic and semantic rules for English that aim to characterize two things at once: (i) the *distributional properties* of English expressions (e.g., that the $np$ *Ziggy* can precede the $vp$ *meows*, in order to make an $s$); and (ii) the *semantic contributions* these expressions make to the meanings of the larger expressions that contain them. Further, our rules are stated to be sensitive to the *syntactic categories* of the expressions whose behavior they characterize. For example, the rule $$\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}}⟩ ⊢ vp$} \BinaryInfC{$⟨x^{⌢}y, ⟦y⟧_{\mathcal{M}}(⟦x⟧_{\mathcal{M}})⟩ ⊢ s$} \end{prooftree}$$ says: - that you can form an $s$ by sticking an $np$ to the left of a $vp$; - that you can get the meaning of the $s$ by applying the meaning of the $vp$ as a function to the meaning of the $np$ as an argument. So, syntactic categories have an important role: they say where things can go, and they also say what things can *do* (semantically) when they go there. These notes further develop this idea, in two directions. First, we introduce a language of *semantic types*. Semantic types tell us what kind of meaning an expression can have---for example, whether it is an entity, a truth value, or some type of function (and if so, what type). Second, we introduce a language of *syntactic categories* (or *syntactic types*, if you like). Syntactic categories tell us what an expression's distributional properties are---that is, what kinds of expressions can appear to its left and to its right. As we'll see, there is a close relation between syntactic categories and semantic types. This relation holds because of the fact that where an expression can go inside some larger expression *determines* what type of meaning it can have. We've already encoded a version of this idea in our grammar with categories like $np$, $vp$, and $tv$, where we had rules that were specific to these categories. Our new *language* of categories will make the idea completely general: we'll have an *infinite* number of possible syntactic categories, as well as an infinite number of possible semantic types, and each syntactic category will be seen to be associated with a semantic type. At the same time, we'll be able to pare down the number of rules we have to only two!