# Seminar on Computational Semantics with Haskell

## Extension and Intension

June 17, 2011

### Montague's Typing in PTQ

The mapping $\alpha \mapsto \overline{\alpha},$ which replaces each occurrence of $$e$$ and $$t$$ by $$s \rightarrow e$$ and $$s \rightarrow t$$, and the associated intensionalization and extensionalization operators ($${}^{\cap}$$ and $${}^{\cup}$$) is not familiar to linguists. In linguistics, a common practice nowadays is to use the fewest instances of $$s$$ as are necessary for adequate semantic analysis, rather than systematically replacing each occurrence of an atomic type by its intensional counterpart.

In Montague's original work, however, there was a systematic placement of $$s$$ in the semantic types associated with syntactic categories. In PTQ, syntactic categories are built from basic categories $$e$$ and $$t$$ by means of two connectives $$/$$ and $$/\!/$$. The semantic type $$f(A)$$ associated with a syntactic category $$A$$ was defined by the following recursion: \begin{align*} f(e) & = e\\ f(t) & = t\\ f(A/B) & = f(A/\!/B) = (s \rightarrow f(B)) \rightarrow f(A) \end{align*} This gives rise to the following association between an extensional semantic type $$\alpha$$ and its intensional counterpart $$g(\alpha)$$: \begin{align*} g(e) & = e\\ g(t) & = t\\ g(\alpha \rightarrow \beta) & = (s \rightarrow g(\alpha)) \rightarrow g(\beta). \end{align*} This mapping $$\alpha \mapsto g(\alpha)$$ looks quite different from the above mapping $$\alpha \mapsto \overline{\alpha}$$. For example, if $$\alpha = (e \rightarrow t) \rightarrow t$$, then we have \begin{align*} \overline{\alpha} & = ((s \rightarrow e) \rightarrow s \rightarrow t) \rightarrow s \rightarrow t\\ g(\alpha) & = (s \rightarrow ((s \rightarrow e) \rightarrow t)) \rightarrow t \end{align*} Note that the number of occurrences of $$s$$ in the two types is different: it is three for $$\overline{\alpha}$$ and two for $$g(\alpha)$$.

Nevertheless, there is a systematic correspondence between the two approaches. First, note that $$f(A)$$ is the type of the extension of an expression of syntactic category $$A$$. The type of the intension of an expression of syntactic category $$A$$ is $$s \rightarrow f(A)$$. Thus, what we should really be comparing to $$\overline{\alpha}$$ is not $$g(\alpha)$$, but $$s \rightarrow g(\alpha)$$. It is easy to see that the number of occurrences of $$s$$ in $$\overline{\alpha}$$ and in $$s \rightarrow g(\alpha)$$ is the same for all $$\alpha$$. Indeed, we can go from one type to the other by repeatedly applying the operation of changing the order of arguments: $\beta \rightarrow \gamma \rightarrow \delta \quad\leadsto\quad \gamma \rightarrow \beta \rightarrow \delta$ With $$\alpha = (e \rightarrow t) \rightarrow t$$, \begin{align*} \overline{\alpha} & \quad = \quad ((s \rightarrow e) \rightarrow s \rightarrow t) \rightarrow s \rightarrow t\\ & \quad \leadsto \quad (s \rightarrow (s \rightarrow e) \rightarrow t) \rightarrow s \rightarrow t\\ & \quad \leadsto \quad s \rightarrow (s \rightarrow (s \rightarrow e) \rightarrow t) \rightarrow t\\ & \quad = \quad s \rightarrow g(\alpha). \end{align*}

### Bennett's Simplified Typing

Bennet (1974) took $$t, \mathit{CN}, \mathit{IV\,}$$ to be basic syntactic categories and defined the correspondence between syntactic categories and semantic types as follows: \begin{align*} f'(t) & = t\\ f'(\mathit{CN\,}) & = f'(\mathit{IV\,}) = e \rightarrow t\\ f'(A/B) & = f'(A/\!/B) = (s \rightarrow f'(B)) \rightarrow f'(A) \end{align*} The type of the intension of a determiner will then be $s \rightarrow f'((t/\mathit{IV\,})/\mathit{CN\,}) = s \rightarrow (s \rightarrow e \rightarrow t) \rightarrow (s \rightarrow e \rightarrow t) \rightarrow t$ Since $$e$$ is not a basic syntactic category, it follows that the type of an individual concept $$s \rightarrow e$$ never occurs anywhere.

### Exercise 2

1. Using $C_{\beta,\gamma,\delta} = \lambda xyz.xzy : (\beta \rightarrow \gamma \rightarrow \delta) \rightarrow \gamma \rightarrow \beta \rightarrow \delta,$ define two $$\lambda$$-terms \begin{align*} M_{\alpha} & : \overline{\alpha} \rightarrow s \rightarrow g(\alpha)\\ N_{\alpha} & : (s \rightarrow g(\alpha)) \rightarrow \overline{\alpha} \end{align*} for each extensional type $$\alpha$$ such that \begin{align*} \lambda x.N_{\alpha}(M_{\alpha}\: x) & \mathrel{=_{\beta\eta}} \lambda x.x : \overline{\alpha} \rightarrow \overline{\alpha},\\ \lambda x.M_{\alpha}(N_{\alpha}\: x) & \mathrel{=_{\beta\eta}} \lambda x.x : (s \rightarrow g(\alpha)) \rightarrow s \rightarrow g(\alpha). \end{align*} (In technical jargon, this means that the types $$\overline{\alpha}$$ and $$s \rightarrow g(\alpha)$$ are isomorphic.)
2. Give defining equations for the versions of intensionalization and extensionalization operators for Montague's approach: \begin{align*} \cap^{\mathrm{PTQ}}_{\alpha} & : (s \rightarrow \alpha) \rightarrow s \rightarrow g(\alpha)\\ \cup^{\mathrm{PTQ}}_{\alpha} & : (s \rightarrow g(\alpha)) \rightarrow s \rightarrow \alpha \end{align*} Make sure that the required equation holds: $\lambda x.\cup^{\mathrm{PTQ}}_{\alpha}(\cap^{\mathrm{PTQ}}_{\alpha} \: x) \mathrel{=_{\beta\eta}} \lambda x.x : (s \rightarrow \alpha) \rightarrow s \rightarrow \alpha.$
3. Rewrite the code in this chapter using Montague's typing.
4. Try to relate Montague's approach with Bennett's.
5. Compare Bennett's typing with Ben-Avi and Winter's (2007).