Abstract
In compositional model-theoretic semantics, researchers assemble truth-conditions or other kinds of denotations using the lambda calculus. It was previously observed [26] that the lambda terms and/or the denotations studied tend to follow the same pattern: they are instances of a monad. In this paper, we present an extension of the simply-typed lambda calculus that exploits this uniformity using the recently discovered technique of effect handlers [22]. We prove that our calculus exhibits some of the key formal properties of the lambda calculus and we use it to construct a modular semantics for a small fragment that involves multiple distinct semantic phenomena.
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Notes
- 1.
This kind of distinction is the same distinction as the one between a mathematical function and a function in a programming language, which might have all kinds of side effects and therefore not be an actual function.
- 2.
Side effects are to programming languages what pragmatics are to natural languages: they both study how expressions interact with the worlds of their users. It might then come as no surprise that phenomena such as anaphora, presupposition, deixis and conventional implicature yield a monadic description.
- 3.
Also known as monad transformers in functional programming.
- 4.
Pronounced “banana”. See [20] for the introduction of banana brackets.
- 5.
The two types \(\alpha \) and \(\beta \) are to be seen as the operation’s input and output types, respectively.
- 6.
These are similar to recursors/paramorphisms. See [20] for the difference. Catamorphisms are also known as folds and the common higher-order function fold found in functional programming languages is actually the iterator/catamorphism for lists.
- 7.
1 is the unit type whose only element is written as \(\star \).
- 8.
Other solutions to this problem include separating the language of logical forms and the metalanguage used in the semantic lexical entries to manipulate logical forms as objects [13].
- 9.
Our \(\mathcal {C}\) has been inspired by an operator of the same name proposed in [9]: de Groote introduces a structure that specializes applicative functors in a similar direction as monads by introducing the \(\mathcal {C}\) operator and equipping it with certain laws; our \(\mathcal {C}\) operator makes the \(\mathcal {F}_E\) type constructor an instance of this structure.
- 10.
In our limited fragment, we will only see it sneak out of a quantifier.
- 11.
Multiple occurrences of the same \(\mathtt {op}_i\) are alright, since those are not metavariables.
- 12.
The definition of (non-trivial) overlap is the same one as the one used when defining critical pairs. See [16] for the precise definition.
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Maršík, J., Amblard, M. (2016). Introducing a Calculus of Effects and Handlers for Natural Language Semantics. In: Foret, A., Morrill, G., Muskens, R., Osswald, R., Pogodalla, S. (eds) Formal Grammar. FG FG 2015 2016. Lecture Notes in Computer Science(), vol 9804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53042-9_15
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