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).


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Makoto Kanazawa

Last modified: 2017-04-06 19:07:58 JST