Coordinating subject noun phrases

To coordinate subject noun phrases like Po and Tigress in a sentence such as (1), we had to make a novel move.

Po and Tigress slept.

Instead of the usual category $np$, we assigned these noun phrases the category $(s/(np\backslash s))$. This means that they take a verb phrase to their right—something whose syntactic category is $(np\backslash s)$—and together with that verb phrase, they make a sentence (an $s$).

This is a strange reversal of roles: typically, it is the verb phrase that takes the subject as its argument. The verb phrase’s syntactic category even says that that’s what it does—it takes an $np$ to its left, and with that $np$, it makes a sentence. But alas, now there is no longer an $np$ to the left of the verb phrase. What’s there instead is a souped up noun phrase—a taker of verb phrases, if you will. Someone saying, “Hey verb phrase, remember how you were going to take me as your syntactic argument on your left and make a sentence? Well, now it is I who takes you as my argument on my right (don’t worry, I’ll still use you to make a sentence)! (Evil laugh.)”

Importantly, this is perfectly within the bounds of what we are allowed to do using our category system. If some expression has category $X$, then we can also define an expression that takes an $X$ to its right to make a sentence by assigning it the category $(s/X)$. We can still do this even if $X$ happens to be the category of a verb phrase, i.e., $(np\backslash s)$.

The whole reason we want to do this, of course, is in order to allow noun phrases to have a type that “ends in $t$”. Such types—which also include $(e → t)$ and $(e → (e → t))$—allow noun phrases to be coordinated using the scheme in A general rule.

In particular, if noun phrases have the syntactic category $(s/(np\backslash s))$, then they have the semantic type $((e → t) → t)$. They denote functions from characteristic functions of sets of entities to truth values. Put differently, they are characteristic functions of sets of characteristic functions (of sets of entities). (See this discussion again for some review.)

If we forced noun phrases to have the syntactic category $np$, we would also be limiting them to having the semantic type $e$ of entities—things that are neither truth values, nor functions into truth values, nor functions into functions into truth values, etc. The scheme we developed to define the possible meanings of and and or would be unusable.

Lexical entries for subject noun phrases

We can assign Po and Tigress the lexical entries in (2)

1. $⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$
2. $⟨\textit{Tigress}, (λf.f(\ct{ti}))⟩ ⊢ (s/(np\backslash s))$

Each of the meanings in these lexical entries takes a verb phrase meaning—a characteristic function—and obtains a truth value by applying that function to the entity the noun phrase usually denotes (i.e., $\ct{p}$ or $\ct{ti}$). The idea is that the resulting sentence will denote $\true$ just in case the verb phrase characterizes a set that has that entity as one of its members.

Here’s a derivation of Po slept to illustrate:

\[\begin{prooftree} \AxiomC{$⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$} \AxiomC{$⟨\textit{slept}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)$} \RightLabel{$/$}\BinaryInfC{$\textit{Po slept}, (λf.f(\ct{p}))(λx.\ct{sleep}(x))⟩ ⊢ s$} \end{prooftree}\]

The resulting meaning representation can then be evaluated:

\[(λf.f(\ct{p}))(λx.\ct{sleep}(x))\] \[⇒ (λx.\ct{sleep}(x))(\ct{p})\] \[⇒ \ct{sleep}(\ct{p})\]

The coordinator

The meaning for and that we want in order to coordinate noun phrases like that in (1) is given by the lexical entry in (3).

$⟨\textit{and}_{(s/(np\backslash s))}, \ct{and}_{((e → t) → t)}⟩ ⊢ (((s/(np\backslash s))\backslash(s/(np\backslash s)))/(s/(np\backslash s)))$

What is $\ct{and}_{((e → t) → t)}$? If we follow the scheme given in A general rule, we can see that it is defined as:

\[\ct{and}_{((e → t) → t)} = (λf.(λg.(λx.\ct{and}_{t}(f(x))(g(x)))))\]

\[= (λf.(λg.(λx.f(x) = g(x) = \true)))\]

Here, the $f$ and $g$ represent noun phrase meanings (functions from characteristic functions to truth values), and the $x$ represents a verb phrase meaning (a characteristic function).

An example

Here is an example derivation for the sentence in (1):

\[\begin{prooftree} \AxiomC{$⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$} \AxiomC{$⟨\textit{and}_{(s/(np\backslash s))}, \ct{and}_{((e → t) → t)}⟩ ⊢ (((s/(np\backslash s))\backslash(s/(np\backslash s)))/(s/(np\backslash s)))$} \AxiomC{$⟨\textit{Tigress}, (λf.f(\ct{ti}))⟩ ⊢ (s/(np\backslash s))$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{and\(_{(s/(np\backslash s))}$ Tigress}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))⟩ ⊢ ((s/(np\backslash s))\backslash(s/(np\backslash s)))\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po and\(_{(s/(np\backslash s))}$ Tigress}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))⟩ ⊢ (s/(np\backslash s))\)} \AxiomC{$⟨\textit{slept}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{Po and\(_{(s/(np\backslash s))}$ Tigress slept}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))⟩ ⊢ s\)} \end{prooftree}\]

Let’s evaluate the result by first unfolding the definition of $\ct{and}_{((e → t) → t)}$

\[\ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))\] \[= (λf.(λg.(λx.f(x) = g(x) = \true)))((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))\]

and then reducing the resulting expression, one step at a time:

\[(λf.(λg.(λx.f(x) = g(x) = \true)))((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))\] \[⇒ (λg.(λx.(λf.f(\ct{ti}))(x) = g(x) = \true))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))\] \[⇒ (λx.(λf.f(\ct{ti}))(x) = (λf.f(\ct{p}))(x) = \true)((λx.\ct{sleep}(x)))\] \[⇒ (λf.f(\ct{ti}))((λx.\ct{sleep}(x))) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true\] \[⇒ (λx.\ct{sleep}(x))(\ct{ti}) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true\] \[⇒ \ct{sleep}(\ct{ti}) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true\] \[⇒ \ct{sleep}(\ct{ti}) = (λx.\ct{sleep}(x))(\ct{p}) = \true\] \[⇒ \ct{sleep}(\ct{ti}) = \ct{sleep}(\ct{p}) = \true\]

The truth value we get at the end is $\true$ just in case $\ct{sleep}$ is true of both $\ct{ti}$ and $\ct{p}$—that is, if both Tigress and Po slept.

(I would really recommend reading each of these lines carefully and ensuring that you see which particular reduction has taken place!)

--- title: "Coordinating subject noun phrases" 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} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\true{\ct{T}} \def\false{\ct{F}} $$ ::: To coordinate subject noun phrases like *Po* and *Tigress* in a sentence such as (@ex-po-tigress-slept), we had to make a novel move. (@ex-po-tigress-slept) Po and Tigress slept. Instead of the usual category $np$, we assigned these noun phrases the category $(s/(np\backslash s))$. This means that they take a verb phrase to their right---something whose syntactic category is $(np\backslash s)$---and together with that verb phrase, they make a sentence (an $s$). This is a strange reversal of roles: typically, it is the verb phrase that takes the subject as *its* argument. The verb phrase's syntactic category even says that that's what it does---it takes an $np$ to its left, and with that $np$, it makes a sentence. But alas, now there is no longer an $np$ to the left of the verb phrase. What's there instead is a souped up noun phrase---a taker of verb phrases, if you will. Someone saying, "Hey verb phrase, remember how you were going to take me as your syntactic argument on your left and make a sentence? Well, now it is I who takes you as *my* argument on my *right* (don't worry, I'll still use you to make a sentence)! (Evil laugh.)" Importantly, this is perfectly within the bounds of what we are allowed to do using our category system. If some expression has category $X$, then we can also define an expression that takes an $X$ to its right to make a sentence by assigning it the category $(s/X)$. We can still do this even if $X$ happens to be the category of a verb phrase, i.e., $(np\backslash s)$. The whole reason we want to do this, of course, is in order to allow noun phrases to have a type that "ends in $t$". Such types---which also include $(e → t)$ and $(e → (e → t))$---allow noun phrases to be coordinated using the scheme in [A general rule](general_rule.html). In particular, if noun phrases have the syntactic category $(s/(np\backslash s))$, then they have the semantic type $((e → t) → t)$. They denote functions from characteristic functions of sets of entities to truth values. Put differently, they are characteristic functions of sets of characteristic functions (of sets of entities). (See [this discussion](../applicative_cg/sem_types.html#third-example) again for some review.) If we forced noun phrases to have the syntactic category $np$, we would also be limiting them to having the semantic type $e$ of entities---things that are neither truth values, nor functions into truth values, nor functions into functions into truth values, etc. The scheme we developed to define the possible meanings of *and* and *or* would be unusable. ### Lexical entries for subject noun phrases We can assign *Po* and *Tigress* the lexical entries in (@ex-lex-noun-phrases) (@ex-lex-noun-phrases) a. $⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$ b. $⟨\textit{Tigress}, (λf.f(\ct{ti}))⟩ ⊢ (s/(np\backslash s))$ Each of the meanings in these lexical entries takes a verb phrase meaning---a characteristic function---and obtains a truth value by applying that function to the entity the noun phrase usually denotes (i.e., $\ct{p}$ or $\ct{ti}$). The idea is that the resulting sentence will denote $\true$ just in case the verb phrase characterizes a set that has that entity as one of its members. Here's a derivation of *Po slept* to illustrate: $$\begin{prooftree} \AxiomC{$⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$} \AxiomC{$⟨\textit{slept}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)$} \RightLabel{$/$}\BinaryInfC{$\textit{Po slept}, (λf.f(\ct{p}))(λx.\ct{sleep}(x))⟩ ⊢ s$} \end{prooftree}$$ The resulting meaning representation can then be evaluated: $$(λf.f(\ct{p}))(λx.\ct{sleep}(x))$$ $$⇒ (λx.\ct{sleep}(x))(\ct{p})$$ $$⇒ \ct{sleep}(\ct{p})$$ ### The coordinator The meaning for *and* that we want in order to coordinate noun phrases like that in (@ex-po-tigress-slept) is given by the lexical entry in (@ex-lex-and-np). (@ex-lex-and-np) $⟨\textit{and}_{(s/(np\backslash s))}, \ct{and}_{((e → t) → t)}⟩ ⊢ (((s/(np\backslash s))\backslash(s/(np\backslash s)))/(s/(np\backslash s)))$ What is $\ct{and}_{((e → t) → t)}$? If we follow the scheme given in [A general rule](general_rule.html), we can see that it is defined as: $$\ct{and}_{((e → t) → t)} = (λf.(λg.(λx.\ct{and}_{t}(f(x))(g(x)))))$$ $$= (λf.(λg.(λx.f(x) = g(x) = \true)))$$ Here, the $f$ and $g$ represent noun phrase meanings (functions from characteristic functions to truth values), and the $x$ represents a verb phrase meaning (a characteristic function). ### An example Here is an example derivation for the sentence in (@ex-po-tigress-slept): $$\begin{prooftree} \AxiomC{$⟨\textit{Po}, (λf.f(\ct{p}))⟩ ⊢ (s/(np\backslash s))$} \AxiomC{$⟨\textit{and}_{(s/(np\backslash s))}, \ct{and}_{((e → t) → t)}⟩ ⊢ (((s/(np\backslash s))\backslash(s/(np\backslash s)))/(s/(np\backslash s)))$} \AxiomC{$⟨\textit{Tigress}, (λf.f(\ct{ti}))⟩ ⊢ (s/(np\backslash s))$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{and\(_{(s/(np\backslash s))}$ Tigress}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))⟩ ⊢ ((s/(np\backslash s))\backslash(s/(np\backslash s)))\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po and\(_{(s/(np\backslash s))}$ Tigress}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))⟩ ⊢ (s/(np\backslash s))\)} \AxiomC{$⟨\textit{slept}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{Po and\(_{(s/(np\backslash s))}$ Tigress slept}, \ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))⟩ ⊢ s\)} \end{prooftree}$$ Let's evaluate the result by first unfolding the definition of $\ct{and}_{((e → t) → t)}$ $$\ct{and}_{((e → t) → t)}((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))$$ $$= (λf.(λg.(λx.f(x) = g(x) = \true)))((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))$$ and then reducing the resulting expression, one step at a time: $$(λf.(λg.(λx.f(x) = g(x) = \true)))((λf.f(\ct{ti})))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))$$ $$⇒ (λg.(λx.(λf.f(\ct{ti}))(x) = g(x) = \true))((λf.f(\ct{p})))((λx.\ct{sleep}(x)))$$ $$⇒ (λx.(λf.f(\ct{ti}))(x) = (λf.f(\ct{p}))(x) = \true)((λx.\ct{sleep}(x)))$$ $$⇒ (λf.f(\ct{ti}))((λx.\ct{sleep}(x))) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true$$ $$⇒ (λx.\ct{sleep}(x))(\ct{ti}) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true$$ $$⇒ \ct{sleep}(\ct{ti}) = (λf.f(\ct{p}))((λx.\ct{sleep}(x))) = \true$$ $$⇒ \ct{sleep}(\ct{ti}) = (λx.\ct{sleep}(x))(\ct{p}) = \true$$ $$⇒ \ct{sleep}(\ct{ti}) = \ct{sleep}(\ct{p}) = \true$$ The truth value we get at the end is $\true$ just in case $\ct{sleep}$ is true of both $\ct{ti}$ and $\ct{p}$---that is, if both Tigress and Po slept. (I would really recommend reading each of these lines carefully and ensuring that you see which particular reduction has taken place!)