A first approximation of English

As our second case study, let’s begin developing a syntax and semantics for a small fragment of English. Recall that we are regarding English—just like Arabic numeral arithmetic—as a language (in the formal sense discussed here). Thus one of the things we would like to do is define a particular set of strings—the “grammatical” strings of (some small fragment of) English. We can call this set (i.e., language) ‘$L_{\textit{Eng.}}$’.

As usual, to define a language, we need an alphabet. For now, let’s define our alphabet to be the one in (1).

$Σ_{\textit{Eng.}} = \{\textit{Ziggy}, \textit{Bella}, \textit{runs}, \textit{run}, \textit{jumps}, \textit{jump}, \textit{doesn't}, \textit{and}\}$

Conveniently, the alphabet can also act as our set of atomic expressions—so that when we define the interpretation function $I_{\textit{Eng.}}$, we can say that the domain of this function is simply the set $Σ_{\textit{Eng.}}$.

Without saying anything more for now about our model for English—$\mathcal{M}_{\textit{Eng.}}$—let’s say a little about the lexicon. Let us work with the lexicon in (2).

$⟨\textit{Ziggy}, I_{\textit{Eng.}}(Ziggy)⟩ ⊢ np$
$⟨\textit{Bella}, I_{\textit{Eng.}}(Bella)⟩ ⊢ np$
$⟨\textit{runs}, I_{\textit{Eng.}}(runs)⟩ ⊢ vp$
$⟨\textit{run}, I_{\textit{Eng.}}(runs)⟩ ⊢ bvp$
$⟨\textit{jumps}, I_{\textit{Eng.}}(jumps)⟩ ⊢ vp$
$⟨\textit{jump}, I_{\textit{Eng.}}(jump)⟩ ⊢ bvp$
$⟨\textit{doesn't}, I_{\textit{Eng.}}(doesn't)⟩ ⊢ aux$
$⟨\textit{and}, I_{\textit{Eng.}}(and)⟩ ⊢ con$

That is: $\textit{Ziggy}$ and $\textit{Bella}$ are our noun phrases; $\textit{runs}$ and $\textit{jumps}$ are our basic verb phrases; $\textit{run}$ and $\textit{jump}$ are our basic bare verb phrases; $\textit{doesn't}$ is our one auxiliary; and $\textit{and}$ is our one connective (for now).

Let us now provide our full set of grammatical rules. These rules will allow us, all at once, (i) to prove that particular strings are expressions of English, and (ii) to show, for any of these expressions, what its meaning is.

\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \RightLabel{Rule 1}\BinaryInfC{$⟨x^{⌢}y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ s$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ aux$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \RightLabel{Rule 2}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦y⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ vp$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ con$} \AxiomC{$⟨z, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \RightLabel{Rule 3}\TrinaryInfC{$⟨x^{⌢}y^{⌢}z, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ bvp$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ con$} \AxiomC{$⟨z, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \RightLabel{Rule 4}\TrinaryInfC{$⟨x^{⌢}y^{⌢}z, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ vp$} \end{prooftree}\]

--- title: "A first approximation of English" 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} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\True{\ct{T}} \def\False{\ct{F}} $$ ::: As our second case study, let's begin developing a syntax and semantics for a small fragment of English. Recall that we are regarding English---just like Arabic numeral arithmetic---as a *language* (in the formal sense discussed [here](../formal_preliminaries/languages.html)). Thus one of the things we would like to do is define a particular set of strings---the "grammatical" strings of (some small fragment of) English. We can call this set (i.e., language) '$L_{\textit{Eng.}}$'. As usual, to define a language, we need an *alphabet*. For now, let's define our alphabet to be the one in (@ex-english-alphabet). (@ex-english-alphabet) $Σ_{\textit{Eng.}} = \{\textit{Ziggy}, \textit{Bella}, \textit{runs}, \textit{run}, \textit{jumps}, \textit{jump}, \textit{doesn't}, \textit{and}\}$ Conveniently, the alphabet can also act as our set of atomic expressions---so that when we define the interpretation function $I_{\textit{Eng.}}$, we can say that the *domain* of this function is simply the set $Σ_{\textit{Eng.}}$. Without saying anything more for now about our model for English---$\mathcal{M}_{\textit{Eng.}}$---let's say a little about the lexicon. Let us work with the lexicon in (@ex-english-lexicon). (@ex-english-lexicon) $⟨\textit{Ziggy}, I_{\textit{Eng.}}(Ziggy)⟩ ⊢ np$ $⟨\textit{Bella}, I_{\textit{Eng.}}(Bella)⟩ ⊢ np$ $⟨\textit{runs}, I_{\textit{Eng.}}(runs)⟩ ⊢ vp$ $⟨\textit{run}, I_{\textit{Eng.}}(runs)⟩ ⊢ bvp$ $⟨\textit{jumps}, I_{\textit{Eng.}}(jumps)⟩ ⊢ vp$ $⟨\textit{jump}, I_{\textit{Eng.}}(jump)⟩ ⊢ bvp$ $⟨\textit{doesn't}, I_{\textit{Eng.}}(doesn't)⟩ ⊢ aux$ $⟨\textit{and}, I_{\textit{Eng.}}(and)⟩ ⊢ con$ That is: $\textit{Ziggy}$ and $\textit{Bella}$ are our noun phrases; $\textit{runs}$ and $\textit{jumps}$ are our basic verb phrases; $\textit{run}$ and $\textit{jump}$ are our basic bare verb phrases; $\textit{doesn't}$ is our one auxiliary; and $\textit{and}$ is our one connective (for now). Let us now provide our full set of grammatical rules. These rules will allow us, all at once, (i) to prove that particular strings are *expressions* of English, and (ii) to show, for any of these expressions, what its *meaning* is. (@ex-english-semantics) \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \RightLabel{Rule 1}\BinaryInfC{$⟨x^{⌢}y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ s$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ aux$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \RightLabel{Rule 2}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦y⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ vp$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ con$} \AxiomC{$⟨z, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ bvp$} \RightLabel{Rule 3}\TrinaryInfC{$⟨x^{⌢}y^{⌢}z, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ bvp$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ con$} \AxiomC{$⟨z, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}}⟩ ⊢ vp$} \RightLabel{Rule 4}\TrinaryInfC{$⟨x^{⌢}y^{⌢}z, ⟦y⟧_{\mathcal{M}_{\textit{Eng.}}}(⟦x⟧_{\mathcal{M}_{\textit{Eng.}}}, ⟦z⟧_{\mathcal{M}_{\textit{Eng.}}})⟩ ⊢ vp$} \end{prooftree}