A Computable Solution to Partee’s Temperature Puzzle

Partee’s temperature puzzle [33, p. 267] is a touchstone for any formal semantics for natural language. This puzzle regards the incompatibility of our intuitions about the validity of the inference from (1) (i.e. invalid) with predictions about the validity of this inference in extensional semantics (cf. [8, 32]) (i.e. valid).

Montague-style formal semantics (e.g. [13, 17, 29, 33]) solve this puzzle by distinguishing two readings of the DP the temperature: a function-reading (cf. (1b)), on which the DP is interpreted as an individual concept (i.e. as a function from indices/world-time pairs to individuals; type¹ se), and a value-reading (cf. (1a)), on which the DP is interpreted as the extension of this concept at the current index, @ (i.e. as an individual; type e). The different readings prevent the replacement of the occurrence of the DP the temperature from (1b) by the name ninety (s.t. the conclusion of (1) cannot be derived from the premises) (cf. (2)).²

Montague’s solution to the temperature puzzle is inspired by Carnap’s theory of intensions (cf. [7]) and is supported by the fact that Montague semantics already uses indices in the semantic analysis of declarative sentences, which are interpreted as functions from indices to truth-values (cf. also [26]). Because of its ready availability, Montague’s solution has been adopted by many contemporary theories of formal semantics.³ However, there are a number of problems with this solution. These include the non-computability of natural language interpretations in this solution, (ii) the high type-complexity of natural language interpretations in this solution, and (iii) the disregard of relevant contextual parameters in this solution. The latter are described below:

We solve the above problems by interpreting natural language in a model⁴ that is inspired by the Kleene-Kreisel model of countable-continuous functionals [21, 25] (cf. [30, Ch. 2.3.1]). In this model, continuous functionals are represented by lower-type objects called associates.

Following Kleene [21] and Kreisel [25], we hereafter use finite types over the natural numbers. The latter are the smallest set of strings that contains the type for natural numbers, 0, and the types for function spaces over natural numbers, \((\rho \rightarrow \tau )\) (with \(\rho , \tau \) finite types) (cf. [36]). To ease notation, we abbreviate the type for functions over natural numbers, \((0 \rightarrow 0)\), as ‘1’, abbreviate the type for functionals over sequences of natural numbers, \(((0 \rightarrow 0) \rightarrow 0)\) (\(\equiv (1 \rightarrow 0)\)), as ‘2’, and abbreviate \((n \rightarrow 0)\) as ‘\(n+1\)’. Our considerations will make special use of coded finite sequences of natural numbers (type 0). To distinguish natural numbers which do from natural numbers which do not code such sequences, we denote the former by ‘\(0^{*}\)’.

Our solution to the temperature puzzle briefly works as follows: By representing the DP the temperature from (1b) as (a code for) a finite sequence of natural numbers (type \(0^{*}\)) and by approximating the continuous functional denoted by rise by an associate of type \(1 \equiv (0^{*} \rightarrow 0)\), we ‘lower’ the types of many expressions from (1) (cf. Problem 2). In particular, our solution interprets the DP’s occurrence from (1a) as a natural number (type 0) and the DP’s occurrence from (1b) as a (coded) sequence of natural numbers (type \(0^{*}\)). Since distinguishing between types 0 and \(0^{*}\) is decidable, we obtain a computable solution to the temperature puzzle (cf. Problem 1). Because associates are introduced through the use of a context-dependent variable, the domain of application of the verb rise is restricted to a specific, contextually salient, temporal interval (cf. Problem 3). As to the computability of our solution, it suffices for now to point out that the Kleene-Kreisel model can be defined inside Martin-Löf type theory and has been implemented in the associated programming language Agda [14, 45‐47].

Note the integrative nature of our solution to the above problems: Since associates are computable, lower-type representations of continuous functionals that approximate these functionals with regard to a contextually determined parameter, our solution(s) to the above problems are all sides of the same (three-sided) coin. This contrasts with other solutions to the temperature puzzle (e.g. [3, 20, 27, 41]) which still assume more complex types, are not effective, and/or rely on the use of other methods to render the interpretation of the sentences from (1) context-sensitive.

We describe our solution in some detail below. To this end, we first show how the Montagovian interpretation of the verb rise corresponds to a continuous functional (in Sect. 3.1). Following the informal introduction of associates (in Sect. 3.2), we then outline our associates-approach to the temperature puzzle (in Sect. 3.3). This approach receives a compositional implementation in Sect. 4. The empirical domain of our associates-approach and the computational properties of associates are discussed in Sects. 5 and 3.4.

To obtain our associates-solution to the temperature puzzle, we compositionally interpret natural language in a model, inspired by the Kleene-Kreisel model of countable-continuous functionals, which contains continuous functionals and their associates. This interpretation proceeds via the translation of the relevant subset of the linguistic fragment from [33] into the language of the simply typed lambda logic \(\pmb {\lambda }^{0}_{\rightarrow }\)([8]; cf. [36, Ch. 1.1]). This is a logic with a single atomic type, 0, from which all other types are built up through the type constructor \(\rightarrow \) (see the definition of finite types from Sect. 3). The language and models of \(\pmb {\lambda }^{0}_{\rightarrow }\)are specified in [2, 8].

To identify the \(\pmb {\lambda }^{0}_{\rightarrow }\)-interpretation of the sentences from (1), we first specify the particular language  (abbreviated ‘\(\mathcal {L}\)’) and frame  (abbreviated ‘\(\mathcal {F}\)’) whose elements translate resp. interpret the syntactic constituents of these sentences. The members of \(\mathcal {L}\) are specified in Table 1. Our conventions for the use of \(\pmb {\lambda }^{0}_{\rightarrow }\)variables are introduced in Table 2.

In the list of non-logical \(\pmb {\lambda }^{0}_{\rightarrow }\)constants, \(\alpha _\mathrm{rise}\) enables the translation of the verb rise as an associate of the continuous functional denoted by \(rise \) (formerly, \(\varphi _{\mathrm{rise}}\)).

The interpretation function \(\mathcal {I}_{\mathcal {F}}: \mathcal {L} \rightarrow \mathcal {F}\) respects the way in which different content words are conventionally related. Thus, this function identifies the interpretation of the generalized \(\pmb {\lambda }^{0}_{\rightarrow }\)-translation, \(\lambda P. P(\pmb {ninety})\), of the DP ninety as a subset of the interpretation of the \(\pmb {\lambda }^{0}_{\rightarrow }\) translation, \(\lambda P \exists \gamma . \textit{temp} (\gamma ) \wedge P(\gamma )\), of the DP a temperature (s.t. ninety is a temperature under this interpretation). To ensure the ‘right’ interpretation of the syntactic constituents of (1a) to (1c), we demand that the function \(\mathcal {I}_{\mathcal {F}}\) further satisfies a number of semantic constraints.

The constraint (C1) demands that the interpretation of the type-0 constant ninety be the output of the functional now on input \(\pmb {ninety}\) (cf. [33, rule T1.(d), MP1]). The constraints (C2) and (C3) demand that the constant \(\textit{rise}\) be interpreted as a continuous functional (cf. (C2)) resp. that \(\alpha _\mathrm{rise}\) behaves as an associate of this functional (cf. (C3)).

Admittedly, (C2) and (C3) are additional requirements on our semantic models which are not postulated for the models of Montague’s Intensional Logic (cf. [33]). However, since these requirements reflect natural assumptions about the domain of interpretation of the verb rise (cf. Sect. 3.1) – and since continuous functionals can be represented via their associates (cf. Sect. 3.2) –, these requirements are rather innocent.

This completes our specification of the interpretation function \(\mathcal {I}_{\mathcal {F}}\). We next turn to the compositional translation of Partee’s temperature puzzle: To enable this translation, we first translate the lexical elements of the sentences from (1). In these translations, \(\rightsquigarrow \) is the smallest relation between syntactic trees and \(\pmb {\lambda }^{0}_{\rightarrow }\)terms which conforms to the rules from [22]:

As expected, Definition 4 specifies the translation of the verb rise as an associate of the continuous functional denoted by the \(\pmb {\lambda }^{0}_{\rightarrow }\)constant \(\textit{rise}\) (cf. (C2), (C3)). The translations of the copula is, of the DP ninety, and of the definite determiner follow the translations of these expressions from [33, cf. rules T1.(b), (d), T2].⁸ In particular, our translation of is follows Montague’s translation of the copula as the designator of a relation between the extensions of (generalized quantifiers over) individual concepts (here: as the designator of a relation between natural numbers, rather than between sequences of numbers).

The above translations enable the compositional \(\pmb {\lambda }^{0}_{\rightarrow }\)translation of the sentences from (1). We start with the translation of (1a):Sentences (1b) and (1c) are translated as follows: The resulting \(\pmb {\lambda }^{0}_{\rightarrow }\)formulas are exactly the formulas from (8).

Our previous discussion has been restricted to the example of the verb rise. However, the associates-approach generalizes to all degree achievement verbs and change-of-state verbs ([28]; cf. [1, 5, 11]) whose interpretation corresponds to a continuous functional. The latter constitute a sizable⁹ class of verbs with the following members:

The above-listed verbs all take individual concepts as their arguments (i.e. they are co-classified with the verb rise) (cf. [10]). The intensional interpretation of these verbs is motivated by their particular, non-instantaneous, evaluation procedure: To judge whether John is blushing (cf. class 2), it does not suffice to observe his red face at a particular point in time.¹⁰ Instead, we need to observe John’s facial complexion at different neighboring points in time. We can only conclude that John is blushing if he has a normal (non-red) skin color at the earliest observed time-point and an increasingly redder complexion at the later time-points (cf. [27]).

Note that, in contrast to their counterparts from class 1, the ‘continuous functional’-interpretations of the verbs from classes 2 to 5 are not restricted to input sequences of natural numbers (see blush), may describe non-temporal change [10, 18] (see the extent reading of verbs like narrow and darken)¹¹, and do not presuppose an established scale or unit of measurement (i.e. they describe non-discrete change). For example, in contrast to rising, blushing and narrowing are not properties of sequences of numbers, but of sequences of temporal states of an individual (viz. of his/her face) resp. of spatial states of an object. Further, there is no established unit of measurement of a person’s facial redness (or of a window cracking, a storm arriving, a person recovering from illness, etc.).

The above-described absence of a numerical/measurement structure does not compromise the applicability of our associates-approach to the verbs from clas-ses 2 to 5. This is due to the possibility of labelling temporal stages of individuals (or of other physical objects) by natural numbers, of identifying a contextually salient unit and scale (here: dominant wavelength or visible change in hue) for the measurement of the relevant property, and of selecting the value of the measurement (under the selected scale and unit of measurement) of the individual’s relevant attribute for that property. In particular, the continuous functional-interpretation, blush, of the verb blush will return ‘1’ on input a given sequence of temporal ‘John’-stages if the values of the measurement (under the contextually presupposed measurement unit) of John’s facial complexion at these stages are increasing, and will return ‘0’ otherwise.

Our associates-approach distinguishes itself from existing solutions to the temperature puzzle. This is due to the proximity of our approach to Montague’s original solution from [33] (cf. Sect. 3.3) and to its focus on improving the computational properties of this solution (cf. Sect. 2.2):

Firstly, in contrast to the solutions from [3, 20, 41], and to solutions from event semantics, our solution is not based on an alternative interpretation of (1a) that uses a locative interpretation of the copula (i.e. ‘is at ninety’), a measurement-explicit interpretation of the DP ninety (i.e. ‘is ninety degrees Fahrenheit’), or an event-based interpretation of the verb rise (s.t. ‘rise’ describes a rising event).

Secondly, in contrast to the solutions from [12, 27, 29, 40], our solution is not directed at a variant of the temperature puzzle (i.e. Gupta’s problem; cf. (12)) that arises from the double index-dependence of intensional nouns like temperature; viz. from the dependence of temperature-values on the index-argument of a particular individual concept [i.e. inner index-dependence] and the dependence of noun-interpretations on the index of evaluation¹² [i.e. outer index-dependence] (cf. [40]). As a result of this double dependence, Montague semantics blocks the intuitively valid inference from (12):It should come as no surprise that the different solutions to Gupta’s problem can be integrated into our associates-approach to Partee’s temperature puzzle. However, our approach even provides its own solution to the puzzle, which also involves computability considerations. We will detail this solution in a sequel to this paper.

We have presented a computable, low-type, context-sensitive solution to Partee’s temperature puzzle which uses the countable approximation of continuous functionals via their associates. The success of our solution is challenged by the restriction of associates to continuous functionals. This restriction prevents the application of our approach to expressions that are traditionally interpreted as discontinuous functionals (e.g. mostly above 90).

Its exclusion of discontinuous intensional verbs hampers the generality of the presented approach. However, in natural language, discontinuous expressions are rather rare: of the 369 intensional intransitive verbs listed in [28] (see Sect. 5 for a selection), only 5 are discontinuous. Their scarcity notwithstanding, discontinuous verbs can be accommodated in Bezem’s model \(\mathscr {M}\) of strongly majorizable functionals (cf. [23, Ch. 3, 11]). The weak continuity functional ([4, Sect. 5, p. 171]) in this model serves a similar role to the fan functional in the Kleene-Kreisel model: it produces a lower-type correlate of its input functional. However, whereas the associate of a continuous functional is an accurate representation of the continuous functional (in the sense that no information is lost), the output of the weak continuity functional only partially represents the input functional in Bezem’s model. The detailed development of this account is a project for future work.