Abstract
According to the received view of type-logical semantics (suggested by Montague and adopted by many of his successors), the correct prediction of entailment relations between lexically complex sentences requires many different types of semantic objects. This paper argues against the need for such a rich semantic ontology. In particular, it shows that Partee’s temperature puzzle – whose solution is commonly taken to require a basic type for indices or for individual concepts – can be solved in the more parsimonious type system from [11], which only assumes basic individuals and propositions. We generalize this result to show the soundness of the PTQ-fragment in the class of models from [11]. Our findings support the robustness of type-theoretic models w.r.t. their objects’ codings.
I would Like to thank two anonymous referees for LENLS 11 for their comments and suggestions. Thanks also to Ede Zimmermann, whose comments on my talk at Sinn und Bedeutung 18 have inspired this paper. The research for this paper has been supported by the Deutsche Forschungsgemeinschaft (grant LI 2562/1-1), by the LMU-Mentoring program, and by Stephan Hartmann’s Alexander von Humboldt-professorship.
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- 1.
We follow the computer science-notation for function types. Thus, \(\sigma \rightarrow t\) corresponds to Montague’s type \(\langle \sigma , t\rangle \) (or, given Montague’s use of the index-type s, to \(\langle s, t\rangle \)).
- 2.
For example, since linguists often assign degree modifiers (e.g. very) the type for degrees \(\delta \) (rather than the type for properties of individuals, \((\iota \rightarrow o) \rightarrow o\)), gradable adjectives (e.g. tall) are interpreted in the type \(\delta \rightarrow (\iota \rightarrow o)\), rather than in the type \(((\iota \rightarrow o) \rightarrow o)\) \(\rightarrow (\iota \rightarrow o)\).
- 3.
In [11], Montague uses a direct interpretation of natural language into logical models, which does not proceed via the translation of natural language into the language of some logic. As a result, [11] does not identify a logical language with EFL-typed expressions. However, since such a language is easily definable (cf. Definition 7), we hereafter refer to any EFL-typed language as an ‘EFL-language’.
- 4.
As a result, this reading is sometimes called the function reading (cf. [8]). The reading of the phrase the temperature from the first premise is called the value reading.
- 5.
These are terms which are associated with PTQ-expressions.
- 6.
The latter are occurrences of index- and truth-value types which are not a constituent of the propositional type \(\sigma \rightarrow t\). The need for the distinction between propositional and non-propositional occurrences of the types \(\sigma \) and t is discussed below.
- 7.
We will see that, since no other syntactic category of the PTQ-fragment receives an interpretation in a construction involving the type \(o \rightarrow \iota \), semantic types involving this type still motivate the syntactic categories. This contrasts with the coding of degrees as equivalence classes of individuals (in [2]), which assigns adjectives (originally, type \(\delta \rightarrow (\iota \rightarrow t)\)) the type for verbal modifiers, \((\iota \rightarrow t) \rightarrow (\iota \rightarrow t)\).
- 8.
- 9.
Following Muskens, we write ‘
’ in postfix notation, such that ‘
’ denotes 
. - 10.
These type-assignments incorporate the type-\(\iota \) interpretation of names and the mea-ning postulates from [12, pp. 263–264]. The latter are given in square brackets.
- 11.
- 12.
Since we only stipulate that \(\varepsilon \in \textsf {PropType}\), clause (iv) describes
as a non-uniquely typed constant, which applies to pairs of arguments of all propositional \(\mathrm{TY}_{1}\) types. To avoid an extension of the \(\mathrm{TY}_{1}\) type system via polymorphic types, we assume a schematic (or abbreviatory) polymorphism of types. The latter is a syntactic device whereby a metatheoretical symbol is used to abbreviate a range of (monomorphic) types. Thus, in (iv), \(\varepsilon \) may be instantiated by any of the elements in PropType. The constant 
then represents a family, 
, of distinct identical-looking constants, one for each type. - 13.
These constraints are formulated in the \(\mathrm{TY}_{1}\) metatheory, \(\mathrm{TY}_{2}\) (cf. Sect. 3.3).
- 14.
This is reminiscent of the translation of dynamic to typed terms from [13, p. 9].
- 15.
For reasons of space, we only translate some representative elements. Expressions of the same lexical (sub-)category receive an analogous translation.
- 16.
To perspicuate the compositional properties of our PTQ-translations, we assign lexi-cal PTQ-elements variants of their \(\mathrm{TY}_{2}\) types from Table 1. Thus, the translation of extensional nouns as type-
constants facilitates the application of translations of determiners to the translations of these expressions. To enable a compositional translation of other complex expressions (e.g. the application of verb- to name-translations), we use a permutation operation on the translations’ lambdas. - 17.
Here, the type of the argument is underlined.
- 18.
Since this operator is restricted to the types of proper names and determiners, it cannot be used to provide an intensional translation of the first premise from (\(\star \)) (and, hence, to ‘allow’ Partee’s temperature puzzle). I owe this observation to Ede Zimmermann.
’ in postfix notation, such that ‘
’ denotes 
.
as a non-uniquely typed constant, which applies to pairs of arguments of all propositional 
then represents a family, 
, of distinct identical-looking constants, one for each type.
constants facilitates the application of translations of determiners to the translations of these expressions. To enable a compositional translation of other complex expressions (e.g. the application of verb- to name-translations), we use a permutation operation on the translations’ lambdas.References
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Liefke, K. (2015). Codability and Robustness in Formal Natural Language Semantics. In: Murata, T., Mineshima, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2014. Lecture Notes in Computer Science(), vol 9067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48119-6_2
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