Type Polymorphism, Natural Language Semantics, and TIL

Abstract

Transparent intensional logic (TIL) is a well-explored type-theoretical framework for semantics of natural language. However, its treatment of polymorphic functions, which are essential for the analysis of various natural language phenomena, is still underdeveloped. In this paper, we address this issue and propose an extension of TIL that introduces polymorphism via type variables ranging over types and generalized variables ranging over constructions and types. Furthermore, we offer an analysis of sentences involving non-specific notional attitudes of the general form ‘A considers (believes, desires, wants, seeks, ...) something’.

Appendix

1.1 Constructions

The original definition of TIL constructions was given by Tichý (1988), here we follow Duží et al. (2010) (an alternative formulation can be found in Raclavský et al., 2015):

Definition 7

(Constructions)

• 1. The variable x is a construction that constructs an object O of the respective type dependently on a valuation. We say that it v-constructs O.

• 2. Where X is any object, $^0 X$ is the construction trivialization. It constructs X without any change.

• 3. The composition $[X_0 \, X_1 \ldots X_m]$ is the following construction. If X v-constructs a function f of type $(\alpha \beta _1 \ldots \beta _m)$ and $X_1 \ldots X_m$ v-construct objects $b_1 , \ldots , b_m$ of types $\beta _1 , \ldots , \beta _m$, respectively, then the composition $[X_0 \, X_1 \ldots X_m]$ v-constructs the value (an object of type $\alpha$, if any) of f on the tuple-argument $\langle b_1 , \ldots ,b_m \rangle$. Otherwise, it is a v-improper construction, i.e., construction that does not construct anything.

• 4. The closure $[\lambda x_1 \ldots x_m \, Y]$ is the following construction. Let $x_1 , \ldots x_m$ be pairwise distinct variables v-constructing objects of types $\beta _1 , \ldots , \beta _m$ and Y a construction v-constructing an object of type $\alpha$. Then $[\lambda x_1 \ldots x_m \, Y]$ is the construction closure. It v-constructs the following function f of type $(\alpha \beta _1 \ldots \beta _m)$: let $\langle b_1, \ldots , b_m \rangle$ be a tuple of objects of types $\beta _1 \ldots \beta _m$, respectively, and $v'$ be a valuation that associates $x_i$ with $b_i$ and is identical to v otherwise. Then the value of function f on argument tuple $\langle b_1, \ldots , b_m \rangle$ is the object of type $\alpha$ $v'$-constructed by Y. If Y is $v'$-improper, then f is undefined on $\langle b_1, \ldots , b_m \rangle$.

• 5. The single execution $^1 X$ is the construction that either v-constructs the object v-constructed by X or, if X v-constructs nothing, is v-improper.

• 6. The double execution $^2 X$ is the following construction: let X be any object, the double execution $^2 X$ is v-improper if X is a non-construction or if X does not v-construct a construction or if X v-constructs a v-improper construction. Otherwise, let X v-construct a construction $X'$ and let $X'$ v-construct and object $X''$, then $^2 K$ v-constructs $X''$.

• 7. Nothing other is a construction, unless it follows from 1 to 6.

1.2 Ramified Type Theory

We follow the specification from Tichý (1988):

Definition 8

(Ramified type theory of TIL)

Let B be a base, i.e., a set of atomic types.

• 1.

  ($\hbox {t}_1$i):

  Every member of B is a type of order 1 over B.

  ($\hbox {t}_1$ii):

  If $0<m$ and $\alpha , \beta _1 , \ldots , \beta _m$ are types of order 1 over B, then the collection $(\alpha \beta _1 \ldots \beta _m)$ of all m-ary (total and partial) mappings from $\beta _1 ,\ldots , \beta _m$ to $\alpha$ is also a type of order 1 over B.

  ($\hbox {t}_1$iii):

  Nothing is a type of order 1 over B unless it follows from ($\hbox {t}_1$i) and ($\hbox {t}_1$ii).

• 2.

  ($\hbox {c}_k$i):

  Let $\alpha$ be any type of order k over B. Every variable ranging over $\alpha$ is a construction of order k over B. If X is of (i.e., belongs to) type $\alpha$, then $^0 X$, $^1 X$, and $^2 X$ are constructions of order k over B.

  ($\hbox {c}_k$ii):

  If $0<m$ and $X_0, X_1, \ldots , X_m$ are constructions of order k, then $[X_0 \; X_1 \ldots \, X_m]$ is a construction of order k over B. If $0<m$, $\alpha$ is a type of order k over B, and Y as well as the distinct variables $x_1, \ldots , x_m$ are constructions of order k over B, then $[\lambda _\alpha \, x_1 \ldots x_m \; Y]$ is a construction of order k over B.

  ($\hbox {c}_k$iii):

  Nothing is a construction of order k over B unless it follows from ($\hbox {c}_k$i) and ($\hbox {c}_k$ii).

Let $*_k$ be the collection of constructions of order k over B. The collection of types of order $k+1$ over B is defined as follows:

($\hbox {t}_{k+1}$i):

$*_k$ and every type of order k is a type of order $k+1$.

($\hbox {t}_{k+1}$ii):

If $0<m$ and $\alpha , \beta _1 , \ldots , \beta _m$ are types of order $k+1$ over B, then the collection $(\alpha \beta _1 \ldots \beta _m)$ of all m-ary (total and partial) mappings from $\beta _1 , \ldots , \beta _m$ to $\alpha$ is also a type of order $k+1$ over B.

($\hbox {t}_{k+1}$iii):

Nothing is a type of order $k+1$ over B unless it follows from ($\hbox {t}_{k+1}$i) and ($\hbox {t}_{k+1}$ii).

1.3 Types and Rules

In this paper, we have explored a semantic treatment of type variables in TIL. Alternatively, we could attempt a syntactic rule-based analysis as well. However, since this topic is outside the scope of the present paper, we sketch only the basics of this approach. The key observation is that type variables generally appear in two kinds of judgments in TIL literature:

$$\begin{aligned}&C : *_n \rightarrow \sigma (\alpha )\\&X : \sigma (\alpha ) \end{aligned}$$

which can be read as ‘a construction C belonging to a type $*_n$ is typed to v-construct an object of type $\sigma$’ and ‘a non-construction X belongs to a type $\alpha$’, respectively, where $\sigma$ may contain $\alpha$ as a free type variable. For example²⁸:

$Cardinality_\tau : (\tau (o \tau ))$

$Cardinality_\alpha : (\tau (o \alpha ))$

$Tr_\alpha : (*_n \alpha )$

$X : *_n \rightarrow ((\alpha \tau ) \omega )$

$z : *_1 \rightarrow \alpha$

Furthermore, note that these judgments are composed of three parts: a typed object (i.e., either a construction C or a non-construction X), a typing relation ‘ : ’ and a type term (i.e., either $*_n \rightarrow \sigma (\alpha )$ in case of constructions or $\sigma (\alpha )$ in case of non-constructions).

Now, as we mentioned above, type terms might depend on some variable $\alpha$ of type $\textsf {Type}$ (the type of all types). We can explicitly capture this dependency by abstracting type terms from $\alpha$ via ${\uplambda }$ abstractor and get ${\uplambda }$$\alpha : \textsf {Type} . *_n \rightarrow \sigma (\alpha )$ and ${\uplambda }$$\alpha : \textsf {Type} . \sigma (\alpha )$. Consequently, every free occurrence of $\alpha$ in $\sigma$ becomes bound in ${\uplambda }$$\alpha : \textsf {Type} . *_n \rightarrow \sigma (\alpha )$ and ${\uplambda }$$\alpha : \textsf {Type} . \sigma (\alpha )$. The obvious complement to abstraction is, of course, application. Thus, the rules we obtain are as follows:²⁹

where $\sigma [\kappa /\alpha ]$ is the result of substituting $\kappa$ for all occurrences of $\alpha$ in $\sigma$.

Note that the premises of abs-C and abs-X rules are hypothetical judgments, i.e., judgments made in a certain context. Thus, we can read $\alpha : \mathsf {Type} \vdash C : *_n \rightarrow \sigma (\alpha )$ as ‘a judgment $C : *_n \rightarrow \sigma (\alpha )$ is assertable given that we have some type term $\alpha$ of type $\textsf {Type}$’. Analogously for the hypothetical judgment $\alpha : \mathsf {Type} \vdash X : \sigma (\alpha )$.

These rules can help us to explain syntactically the general process of instantiation of type terms containing type variables to some specific type, a process which was investigated semantically in this paper via the use of expanded valuation arrays.