Type-theoretical natural language semantics: on the system F for meaning assembly

Roughly speaking, the compositional semantics analysis of natural language consists in mapping a sentence to a logical formula which depicts its meaning. To do so, a lexicon endow each word with a partial formula (a λ-term over the base type t for propositions and e for individuals), and the (binary) parse tree of the sentence specifies for each node which subtree is the function and which one is the argument. Hence a λ-term corresponding to the whole parse tree, and by reduction one obtains a λ-term of type t which corresponds to a formula of higher order logic. This classical process which uses Church’s representation of formulae as simply typed λ-terms is the basis of the so-called Montague semantics, see e.g. [7, Chapter 3]. But it is more accurate to have many individual base types rather than just e. This way, the application of a predicate to an argument only happens when it makes sense. For instance sentences like “The chair barks.” or “Their five is fast.” are easily ruled out when there are several types for individuals by saying that “barks” and “fast” respectively require arguments of type “dog” and physical or animated object. Nevertheless, such a type system needs to incorporate some flexibility. Indeed, in the context of a football match, the second sentence makes sense, because “their five” may be understood as a player. Accounts of these meaning transfers received a lot of attention since the 80’s and some formal accounts were proposed in particular by Asher [1]. We also proposed an account using the system F of Girard (1971) [3] and we studied the compositional properties of such a system in particular for quantifiers, plurals and generic elements, as well as the related lexical issues like meaning transfers, copredication, fictive motion,... see e.g. [2, 8]. Our system works as follows: the lexicon provides each word with a main λ-term, the ”usual one” which specifies the argument structure of the word, by using refined types (e.g. ”sleeps: λxsleeps(x)” requires an animated subject). In addition, the lexicon may endow each word with a finite number of λ-terms (possibly none) that implement meaning transfers. For instance a “book” may be turned into a physical object φ or into an informational content I, by constants of respective types book→ I and book→φ (sometimes the λ-terms are more complex than simple constants). That way a sentence like “This book is heavy but interesting.” can be properly analysed. Some meaning transfers are declared to be rigid in the lexicon: rigidity prohibits the use of other meaning transfers thus we can block “* Liverpool defeated Chelsea and decided to build new docks.” because the meaning transfer from a town to a football club is declared in the lexicon as rigid. In such a setting it is very convenient to quantify over types, for instance quantifiers ∀, ∃ may be given a type Λα.(α→t)→t It also allows a factorised treatment of conjunction: each time an object x of type ξ can be viewed both as an object of type α (via an optional term f0 : ξ → α) to which the property P applies and as an object of type β (via an optional term g0 : ξ→β) to which the property Q applies, one can express that x enjoys P ∧ Q. This polymorphic “and” simply is: ΛαΛβλPλQΛξλxλf λg . (∧ (P (f x))(Q (g x))). To sum up the logical system, we also have two layers but they slightly vary from the ones of Montague