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Generalised quantifiers, which include Henkin's branching quantifiers, have been introduced by Mostowski and Lindstr\"om and developed as a substantial topic application of logic, especially model theory, to linguistics with work by Barwise, Cooper, Keenan. In this paper, we mainly study the proof theory of some non-standard quantifiers as second order formulae . Our first example is the usual pair of first order quantifiers (for all / there exists) when individuals are viewed as individual concepts handled by second order deductive rules. Our second example is the study of a second order translation of the simplest branching quantifier: ``A member of each team and a member of each board of directors know each other", for which we propose a second order treatment.

After reminding what coherences spaces are and how they interpret linear logic, we define a modality "flag" in the category of coherence spaces (or hypercoherences) with two inverse linear (iso)morphisms: "duplication" from (flag A) to ((flag A) < (flag A)) and "contraction" in the opposite direction -- where < is the self dual and non-commutative connective known as "before" in pomset logic and known as "seq(ential)" in the deep inference system (S)BV. In addition, as expected, the coherence space A is a retract of its modal image (flag A). This suggests an intuitive interpretation of (flag A) as "repeatedly A" or as "A at any instant" when "before" is given a temporal interpretation. We hope the semantic construction of flag(A) will help to design proof rules for "flag" and we briefly discuss this at the end of the paper., Comment: In Proceedings Linearity&TLLA 2020, arXiv:2112.14305

In this chapter, we introduce a new dialogical system for first order classical logic which is close to natural language argumentation, and we prove its completeness with respect to usual classical validity. We combine our dialogical system with the Grail syntactic and semantic parser developed by the second author in order to address automated textual entailment, that is, we use it for deciding whether or not a sentence is a consequence of a short text. This work-which connects natural language semantics and argumentation with dialogical logic-can be viewed as a step towards an inferentialist view of natural language semantics., Comment: Natural Language Processing in Artificial Intelligence, In press

Thirty years ago, I introduced a non-commutative variant of classical linear logic, called "pomset logic", issued from a particular categorical interpretation of linear logic known as coherence spaces. In addition to the usual commutative multiplicative connectives of linear logic, pomset logic includes a non-commutative connective, "$\triangleleft$" called "before", associative and self-dual: $(A\triangleleft B)^\perp=A^\perp \triangleleft B^\perp$. The conclusion of a pomset logic proof is a Partially Ordered MultiSET of formulas. Pomset logic enjoys a proof net calculus with cut-elimination, denotational semantics, and faithfully embeds sequent calculus. The study of pomset logic has reopened with recent results on handsome proof nets, on its sequent calculus, or on its following calculi like deep inference by Guglielmi and Strassburger. Therefore, it is high time we published a thorough presentation of pomset logic, including published and unpublished material, old and new results. Pomset logic (1993) is a non-commutative variant of linear logic (1987) as for Lambek calculus (1958!) and it can also be used as a grammatical formalism. Those two calculi are quite different, but we hope that the algebraic presentation we give here, with formulas as algebraic terms and with a semantic notion of proof (net) correctness, better matches Lambek's view of what a logic should be.

Aristotle considered particular quantified sentences in his study of syllogisms and in his famous square of opposition. Of course, the logical formulas in Aristotle work were not modern formulas of mathematical logic, but ordinary sentences of natural language. Nowadays natural language sentences are turned into formulas of predicate logic as defined by Frege, but, it is not clear that those Fregean sentences are faithful representations of natural language sentences. Indeed, the usual modelling of natural language quantifiers does not fully correspond to natural language syntax, as we shall see. This is the reason why Hilbert's epsilon and tau quantifiers (that go beyond usual quantifiers) have been used to model natural language quantifiers. Here we interpret Aristotle quantified sentences as formulas of Hilbert's epsilon and tau calculus. This yields to two potential squares of opposition and provided a natural condition holds, one of these two squares is actually a square of opposition i.e. satisfies the relations of contrary, contradictory, and subalternation.

This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis of the logical models and algorithms currently used in natural language semantics, defined as the mapping of a statement to logical formulas - formulas, because a statement can be ambiguous. We argue that as long as possible world semantics is left out, one can compute the semantic representation(s) of a given statement, including aspects of lexical meaning. We also discuss the algorithmic complexity of this process.

This report first shows the equivalence bewteen several formulations of classical logic in intuitionistic logic (tertium non datur, reductio ad absurdum, Pierce's law). Then it establishes the correctness of the G\"odel-Kolmogorov translation, whose restriction to the propositional case is due to Glivenko. This translation maps a formula $F$ of first order logic to a formula $F^{\lnot\lnot}$ in such a way that $F$ is provable in classical logic if and only if $F^{\lnot\lnot}$ is provable in intuitionistic logic. All formal proofs are presented in natural deduction. These questions are well-known proof theoretical facts, but in textbooks, they are often ignored or left to the reader. Because of the combinatorial difficulty of some of the needed formal proofs, we hope that this report may be useful, in particular to students and colleagues from other areas.

The semantics of determiner phrases, be they definite de- scriptions, indefinite descriptions or quantified noun phrases, is often as- sumed to be a fully solved question: common nouns are properties, and determiners are generalised quantifiers that apply to two predicates: the property corresponding to the common noun and the one corresponding to the verb phrase. We first present a criticism of this standard view. Firstly, the semantics of determiners does not follow the syntactical structure of the sentence. Secondly the standard interpretation of the indefinite article cannot ac- count for nominal sentences. Thirdly, the standard view misses the linguis- tic asymmetry between the two properties of a generalised quantifier. In the sequel, we propose a treatment of determiners and quantifiers as Hilbert terms in a richly typed system that we initially developed for lexical semantics, using a many sorted logic for semantical representations. We present this semantical framework called the Montagovian generative lexicon and show how these terms better match the syntactical structure and avoid the aforementioned problems of the standard approach. Hilbert terms rather differ from choice functions in that there is one polymorphic operator and not one operator per formula. They also open an intriguing connection between the logic for meaning assembly, the typed lambda calculus handling compositionality and the many-sorted logic for semantical representations. Furthermore epsilon terms naturally introduce type-judgements and confirm the claim that type judgment are a form of presupposition.

This paper establishes the normalisation of natural deduction or lambda calculus formulation of Intuitionistic Non Commutative Logic --- which involves both commutative and non commutative connectives. This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Regarding Intuitionistic Non Commutative Logic with full entropy does not have up to now a proof net syntax, and that for linguistic applications, sequent calculi which are only more or less equivalent to natural deduction, are not convenient because they lack the standard Curry-Howard isomorphism.

We propose a lexical account of action nominals, in particular of deverbal nominalisations, whose meaning is related to the event expressed by their base verb. The literature about nominalisations often assumes that the semantics of the base verb completely defines the structure of action nominals. We argue that the information in the base verb is not sufficient to completely determine the semantics of action nominals. We exhibit some data from different languages, especially from Romance language, which show that nominalisations focus on some aspects of the verb semantics. The selected aspects, however, seem to be idiosyncratic and do not automatically result from the internal structure of the verb nor from its interaction with the morphological suffix. We therefore propose a partially lexicalist approach view of deverbal nouns. It is made precise and computable by using the Montagovian Generative Lexicon, a type theoretical framework introduced by Bassac, Mery and Retor\'e in this journal in 2010. This extension of Montague semantics with a richer type system easily incorporates lexical phenomena like the semantics of action nominals in particular deverbals, including their polysemy and (in)felicitous copredications., Comment: A revised version will appear in the Journal of Logic, Language and Information

We seize the opportunity of the publication of selected papers from the \emph{Logic, categories, semantics} workshop in the \emph{Journal of Applied Logic} to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning., Comment: Survey on the occasion of a special issue of the journal of applied logic

We developed a type-theoretical framework for natural lan- guage semantics that, in addition to the usual Montagovian treatment of compositional semantics, includes a treatment of some phenomena of lex- ical semantic: coercions, meaning, transfers, (in)felicitous co-predication. In this setting we see how the various readings of plurals (collective, dis- tributive, coverings,...) can be modelled.

We propose a cognitively and linguistically motivated set of sorts for lexical semantics in a compositional setting: the classifiers in languages that do have such pronouns. These sorts are needed to include lexical considerations in a semantical analyser such as Boxer or Grail. Indeed, all proposed lexical extensions of usual Montague semantics to model restriction of selection, felicitous and infelicitous copredication require a rich and refined type system whose base types are the lexical sorts, the basis of the many-sorted logic in which semantical representations of sentences are stated. However, none of those approaches define precisely the actual base types or sorts to be used in the lexicon. In this article, we shall discuss some of the options commonly adopted by researchers in formal lexical semantics, and defend the view that classifiers in the languages which have such pronouns are an appealing solution, both linguistically and cognitively motivated.

As the etymology of the word shows, logic is intimately related to language, as exemplified by the work of philosophers from Antiquity and from the Middle-Age. At the beginning of the XX century, the crisis of the foundations of mathematics invented mathematical logic and imposed logic as a language-based foundation for mathematics. How did the relations between logic and language evolved in this newly defined mathematical framework? After a survey of the history of the relation between logic and linguistics, traditionally focused on semantics, we focus on some present issues: 1) grammar as a deductive system 2) the transformation of the syntactic structure of a sentence to a logical formula representing its meaning 3) taking into account the context when interpreting words. This lecture shows that type theory provides a convenient framework both for natural language syntax and for the interpretation of any of tis level (words, sentences, discourse)., Comment: Transcription d'une "le\c{c}on de math\'ematique d'aujourd'hui" donn\'ee le 7 juillet 2011, in French

In addition to their limpid interface with semantics, categorial grammars enjoy another important property: learnability. This was first noticed by Buskowsky and Penn and further studied by Kanazawa, for Bar-Hillel categorial grammars. What about Lambek categorial grammars? In a previous paper we showed that product free Lambek grammars where learnable from structured sentences, the structures being incomplete natural deductions. These grammars were shown to be unlearnable from strings by Foret and Le Nir. In the present paper we show that Lambek grammars, possibly with product, are learnable from proof frames that are incomplete proof nets. After a short reminder on grammatical inference \`a la Gold, we provide an algorithm that learns Lambek grammars with product from proof frames and we prove its convergence. We do so for 1-valued also known as rigid Lambek grammars with product, since standard techniques can extend our result to $k$-valued grammars. Because of the correspondence between cut-free proof nets and normal natural deductions, our initial result on product free Lambek grammars can be recovered. We are sad to dedicate the present paper to Philippe Darondeau, with whom we started to study such questions in Rennes at the beginning of the millennium, and who passed away prematurely. We are glad to dedicate the present paper to Jim Lambek for his 90 birthday: he is the living proof that research is an eternal learning process., Comment: A revised version will appear in a volume in honour of Lambek 90th birthday

The integration of lexical semantics and pragmatics in the analysis of the meaning of natural lan- guage has prompted changes to the global framework derived from Montague. In those works, the original lexicon, in which words were assigned an atomic type of a single-sorted logic, has been re- placed by a set of many-facetted lexical items that can compose their meaning with salient contextual properties using a rich typing system as a guide. Having related our proposal for such an expanded framework \LambdaTYn, we present some recent advances in the logical formalisms associated, including constraints on lexical transformations and polymorphic quantifiers, and ongoing discussions and research on the granularity of the type system and the limits of transitivity.

This paper presents and extends our type theoretical framework for a compositional treatment of natural language semantics with some lexical features like coercions (e.g. of a town into a football club) and copredication (e.g. on a town as a set of people and as a location). The second order typed lambda calculus was shown to be a good framework, and here we discuss how to introduced predefined types and coercive subtyping which are much more natural than internally coded similar constructs. Linguistic applications of these new features are also exemplified.

The variation of word meaning according to the context leads us to enrich the type system of our syntactical and semantic analyser of French based on categorial grammars and Montague semantics (or lambda-DRT). The main advantage of a deep semantic analyse is too represent meaning by logical formulae that can be easily used e.g. for inferences. Determiners and quantifiers play a fundamental role in the construction of those formulae. But in our rich type system the usual semantic terms do not work. We propose a solution ins- pired by the tau and epsilon operators of Hilbert, kinds of generic elements and choice functions. This approach unifies the treatment of the different determi- ners and quantifiers as well as the dynamic binding of pronouns. Above all, this fully computational view fits in well within the wide coverage parser Grail, both from a theoretical and a practical viewpoint.

This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's oper- ator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to natural language, in particular with quantified noun phrases as individual terms. We also define guidelines for the design of the proof rules corresponding to generalised quantifiers.

We present a framework, named the Montagovian generative lexicon, for computing the semantics of natural language sentences, expressed in many sorted higher order logic. Word meaning is depicted by lambda terms of second order lambda calculus (Girard's system F) with base types including a type for propositions and many types for sorts of a many sorted logic. This framework is able to integrate a proper treatment of lexical phenomena into a Montagovian compositional semantics, including the restriction of selection which imposes the nature of the arguments of a predicate, and the possible adaptation of a word meaning to some contexts. Among these adaptations of a word's sense to the context, ontological inclusions are handled by an extension of system F with coercive subtyping that is introduced in the present paper. The benefits of this framework for lexical pragmatics are illustrated on meaning transfers and coercions, on possible and impossible copredication over different senses, on deverbal ambiguities, and on "fictive motion". Next we show that the compositional treatment of determiners, quantifiers, plurals,... are finer grained in our framework. We then conclude with the linguistic, logical and computational perspectives opened by the Montagovian generative lexicon.

This paper proposes a way to compute the meanings associated with sentences with generic noun phrases corresponding to the generalized quantifier most. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can thereafter be interpreted in your favourite models. To do so, we depart significantly from the dominant Fregean view with a single untyped universe. Indeed, our proposal adopts type theory with some hints from Hilbert \epsilon-calculus (Hilbert, 1922; Avigad and Zach, 2008) and from medieval philosophy, see e.g. de Libera (1993, 1996). Our type theoretic analysis bears some resemblance with ongoing work in lexical semantics (Asher 2011; Bassac et al. 2010; Moot, Pr\'evot and Retor\'e 2011). Our model also applies to classical examples involving a class, or a generic element of this class, which is not uttered but provided by the context. An outcome of this study is that, in the minimalism-contextualism debate, see Conrad (2011), if one adopts a type theoretical view, terms encode the purely semantic meaning component while their typing is pragmatically determined.

This paper proposes to compute the meanings associated to sentences with generic NPs corresponding to the most of generalized quantifier. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can be thereafter interpreted in your favorite models. We rather depart from the dominant Fregean single untyped universe and go for type theory with hints from Hilbert epsilon calculus and from medieval philosophy. Our type theoretic analysis bears some resemblance with on going work in lexical semantics. Our model also applies to classical examples involving a class (or a generic element of this class) which is provided by the context. An outcome of this study is that, in the minimalism-contextualism debate, if one adopts a type theoretical view, terms encode the purely semantic meaning component while their typing is pragmatically determined., Comment: Genius III (2011)

In order to model in compositional framework some phenomena of lexical pragmatics and in particular the ones studied by Nicholas Asher several contributions developed in our team did use the system F of Jean-Yves Girard to construct logical formulae expressing the meaning of sentences --- while other authors prefer to use Per Martin-L\"of's type theory In this note we explain the motivations supporting our preference for system F., Comment: (24/08/2011)

We firstly show that the standard interpretation of natural quantification in mathematical logic does not provide a satisfying account of its original richness. In particular, it ignores the difference between generic and distributive readings. We claim that it is due to the use of a set theoretical framework. We therefore propose a proof theoretical treatment in terms of proofs and refutations. Thereafter we apply these ideas to quantifiers that are not first order definable like "the majority of"., Comment: 14TH Congress of Logic, Methodology and Philosophy of Science, Nancy : France (2011)

We first recall some basic notions on minimalist grammars and on categorial grammars. Next we shortly introduce partially commutative linear logic, and our representation of minimalist grammars within this categorial system, the so-called categorial minimalist grammars. Thereafter we briefly present \lambda\mu-DRT (Discourse Representation Theory) an extension of \lambda-DRT (compositional DRT) in the framework of \lambda\mu calculus: it avoids type raising and derives different readings from a single semantic representation, in a setting which follows discourse structure. We run a complete example which illustrates the various structures and rules that are needed to derive a semantic representation from the categorial view of a transformational syntactic analysis.