Week 1 [2023-04-06 Thu]
We observed that there was a certain regulary between syntactic feature bundles and semantic types.
Consider the lexical items below and their semantic types.
| Lexeme | Features | Type |
|---|---|---|
| p | \(\textsf{s}\) | \(t\) |
| not | \(\hbox{ =}\textsf{s}.\textsf{s}\) | \(t\rightarrow t\) |
| and | \(\hbox{ =}\textsf{s}.\textsf{s}\!\!\hbox{ =}.\textsf{s}\) | \(t\rightarrow t\rightarrow t\) |
We see that to each syntactic selection feature there corresponds a type in a semantic input position, and that to the syntactic category feature there corresponds a type in the semantic output position.
We can express this observation more concisely in terms of a function mapping a syntactic feature bundle to a semantic type. In mathematical notation, we might write:
\[
\begin{array}{r@{\mathrel{=}}l} \hat{\tau}(\textsf{s}) & t\\ \hat{\tau}(\hbox{ =}\textsf{s}.\alpha) & t \rightarrow \hat{\tau}(\alpha) \end{array}
\]
This definition conflates two distinct things:
- what semantic type each syntactic feature is associated with (i.e. that \(\textsf{s}\) and \(t\) go together)
- the fact that category features are associated with the output type, and that selection features are associated with input types
We can seperate these two kinds of information by defining a feature-to-type correspondence independently of a feature bundle-to-type correspondence. A feature-to-type correspondence associates each syntactic feature type with a semantic type. In our particular example, we have that \(\tau(\textsf{s}) = t\)—here \(\tau\) (without a hat) is the function mapping features to types. Now we can define a map from feature bundles to semantic types in terms of a feature-to-type correspondence:
\[
\begin{array}{r@{\mathrel{=}}l} \hat{\tau}(x) & \tau(x)\\ \hat{\tau}(\hbox{ =}x.\alpha) & \tau(x) \rightarrow \hat{\tau}(\alpha) \end{array}
\]