Myriam Quatrini

Figures of dialogue: a view from Ludics

In this paper, we study dialogue as a game, but not only in the sense in which there would exist winning strategies and a priori rules. Dialogue is not governed by game rules like for chess or other games, since even if we start from a priori rules, it is always possible to play with them, provided that some invariant properties are preserved. An important discovery of Ludics is that such properties may be expressed in geometrical terms. The main feature of a dialogue is “convergence”. Intuitively, a dialogue “diverges” when it stops prematurely by some disruption, or a violation of the tacit agreed upon conditions of the discourse. It converges when the two speakers go together towards a situation where they agree at least on some points. As we shall see, convergence may be thought of through the geometrical concept of orthogonality. Utterances in a dialogue have as their content, not only the processes (similar to proofs) which lead to them from a monologic view, but also their interactions with other utterances. Finally, any utterance must be seen as co-constructed in an interaction between two processes. That is to say that it not only contains one speaker’s intentions but also his or her expectations from the other interlocutor. From our viewpoint, discursive strategies like narration, elaboration, topicalization may derive from such interactions, as well as speech acts like assertion, question and denegation.

Ludics and Its Applications to Natural Language Semantics

Proofs in Ludics, have an interpretation provided by their counter-proofs , that is the objects they interact with. We shall follow the same idea by proposing that sentence meanings are given by the counter-meanings they are opposed to in a dialectical interaction. In this aim, we shall develop many concepts of Ludics like designs (which generalize proofs), cut-nets , orthogonality and behaviours (that is sets of designs which are equal to their bi-orthogonal). Behaviours give statements their interactive meaning. Such a conception may be viewed at the intersection between proof-theoretic and game-theoretical accounts of semantics, but it enlarges them by allowing to deal with possibly infinite processes instead of getting stuck to an atomic level when decomposing a formula.

First order in Ludics

In Girard (2001), J.-Y. Girard presents a new theory, The Ludics, which is a model of realisibility of logic that associates proofs with designs, and formulas with behaviours. In this article we study the interpretation in this semantics of formulas with first-order quantifications and their proofs. We extend to the first-order quantifiers the full completeness theorem obtained in Girard (2001) for

A Mixed λ-calculus

MALL_2

Incarnation in Ludics and maximal cliques of paths

. A significant part of this article is devoted to the study of a uniformity property for the families of designs that represent proofs of formulas depending on a first-order free variable.

Logic and grammar: essays dedicated to Alain Lecomte on the occasion of his 60th birthday

The aim of this paper is to define a λ-calculus typed in a Mixed (commutative and non-commutative) Intuitionistic Linear Logic. The terms of such a calculus are the labelling of proofs of a linear intuitionistic mixed natural deduction NILL, which is based on the non-commutative linear multiplicative sequent calculus MNL [RuetAbrusci 99]. This linear λ-calculus involves three linear arrows: two directional arrows and a nondirectional one (the usual linear arrow). Moreover, the λ-terms are provided with seriesparallel orders on free variables. We prove a normalization theorem which explicitly gives the behaviour of the order during the normalization procedure.

Ludics and rhetorics

Ludics is a reconstruction of logic with interaction as a primitive notion, in the sense that the primary logical concepts are no more formulas and proofs but cut-elimination interpreted as an interaction between objects called designs. When the interaction between two designs goes well, such two designs are said to be orthogonal. A behaviour is a set of designs closed under bi-orthogonality. Logical formulas are then denoted by behaviours. Finally proofs are interpreted as designs satisfying particular properties. In that way, designs are more general than proofs and we may notice in particular that they are not typed objects. Incarnation is introduced by Girard in Ludics as a characterization of useful designs in a behaviour. The incarnation of a design is defined as its subdesign that is the smallest one in the behaviour ordered by inclusion. It is useful in particular because being incarnated is one of the conditions for a design to denote a proof of a formula. The computation of incarnation is important also as it gives a minimal denotation for a formula, and more generally for a behaviour. We give here a constructive way to capture the incarnation of the behaviour of a set of designs, without computing the behaviour itself. The method we follow uses an alternative definition of designs: rather than defining them as sets of chronicles, we consider them as sets of paths, a concept very close to that of play in game semantics that allows an easier handling of the interaction: the unfolding of interaction is a path common to two interacting designs.

Study of Behaviours via Visitable Paths

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Argumentation and Inference: A Unified Approach

In this paper, we give some illustrations of the expressive power of Ludics with regards to some well known problems often regrouped under the label of Rhetorics. Nevertheless our way of considering Rhetorics encompasses many questions which have been put nowadays in Semantics and Pragmatics.

Dialogues in Ludics

Ludics is a logical theory that J.-Y. Girard developed around 2000. At first glance, it may be considered as a Brouwer-Heyting-Kolmogorov interpretation of Logic as a formula is denoted by the set of its proofs. More primitively, Ludics is a theory of interaction that models (a variant of) second-order multiplicative-additive Linear Logic. A formula is denoted by a set of objects called a behaviour, a proof by an object that satisfies some criteria. Our aim is to analyze the structure of behaviours in order to better understand and refine the usual notion of formulas or types. More precisely, we study properties that guarantee a behaviour to be recursively decomposable by means of multiplicative-additive linear connectives and linear constants.Around 2000, J.-Y. Girard developed a logical theory, called Ludics. This theory was a step in his program of Geometry of Interaction, the aim of which being to account for the dynamics of logical proofs. In Ludics, objects called designs keep only what is relevant for the cut elimination process, hence the dynamics of a proof: a design is an abstraction of a formal proof. The notion of behaviour is the counterpart in Ludics of the notion of type or the logical notion of formula. Formally a behaviour is a closed set of designs. Our aim is to explore the constructions of behaviours and to analyse their properties. In this paper a design is viewed as a set of coherent paths. We recall or give variants of properties concerning visitable paths, where a visitable path is a path in a design or a set of designs that may be traversed by interaction with a design of the orthogonal of the set. We are then able to answer the following question: which properties should satisfy a set of paths for being exactly the set of visitable paths of a behaviour? Such a set and its dual should be prefix-closed, daimon-closed and satisfy two saturation properties. This allows us to have a means for defining the whole set of visitable paths of a given set of designs without closing it explicitly, that is without computing the orthogonal of this set of designs. We finally apply all these results for making explicit the structure of a behaviour generated by constants and multiplicative/additive connectives. We end by proposing an oriented tensor for which we give basic properties.