Vito Michele Abrusci

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

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".

La teoria della ricorsivita come rigorizzazione della nozione intuitiva di funzione calcolabile. Sviluppo della teoria della ricorsivita con riferimento alla teoria della complessita implicita attraverso la nozione di funzione elementare. L’aritmetizzazione della sintassi. Dimostrazione dei risultati fondamentali della teoria (teoremi di Kleene, di enumerazione, Smn, di ricorsione). Decidibilita. Indecidibilita della fermata e teorema di Rice.

INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable --- it is quite problematic in applications like linguistics or computer science ---, and actually the desire of a non-commutative logic goes back to the very beginning of LL [9]. Previous works on non-commutativity deal essentially with non-commutative fragments of LL, obtained by removing the exchange rule at all. At that point, a simple remark on the status of exchange in the sequent calculus is necessary to be clear: there are two presentations of exchange in commutative LL, either sequents are finite sets of occurrences of formulas and exchange is obviously implicit, or sequents are fini

In this paper we show that the dynamic interpretation techniques of Janssen (assignment modalities), Groenendijk and Stokhof (dynamic binding), and Hendriks (exibly scoping rules) enable a rigorous formulation of the semantics of intersentential anaphoric relationships, as well as of telescoping and periscoping phenomena in natural language. Keywords: Natural Language Semantics, Dynamic Interpretation, Flexible Type Theory, Scope and Binding. 1 Introduction In this paper we want to show that the potential of techniques of dynamic interpretation as developed by Janssen, Groenendijk and Stokhof, and Hendriks in the second half of the eighties has not yet been fully recognized. Not only do these techniques allow a rigorous formulation of the semantics of anaphoric relationships across sentential or clausal borders (including `donkey-anaphora'), but they are also the right ones for formulating the semantics of what Craige Roberts has dubbed `telescoping' and that of what is ca...

Abstract: 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 givesomehintsforadifferentviewbasedonprooftheory,whichonmanyaspects gets closer to the language itself. We stress the importance of Hilbert’s operator 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. Résumé Cet article dresse un rapide panorama de l’approche commune de la quantification généralisée ou non en linguistique formelle et en philosophie du langage. Nous montrons que ce cadre général est va parfois à l’encontre des données linguistiques, et nous donnons quelq...

This paper 1 presents a logical formalization of Tree-Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of a tree inside another, surrounding the subtree at the adjunction node. This seems to be contradictory with standard logical ability. We prove that some logic, namely a fragment of non-commutative intuitionistic linear logic (N-ILL), can serve this purpose. Brieey speaking, linear logic is a logic considering facts as resources. N-ILL can then be considered either as an extension of Lambek calculus, or as a restriction of linear logic. We model the TAG formalism in four steps: trees (initial or derived) and the way they are constituted, the operations (substitution and adjunction), and the elementary trees, i.e. the grammar. The sequent calculus is a restriction of the standard se-quent calculus for N-ILL. Trees (initial or derived) are then obtained as the cl...

This paper presents a logical formalization of Tree Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of one tree inside another, surrounding the subtree at the adjunction node. This seems to contradict standard logical ability. We prove that some logical formalisms, namely a fragment of the Lambek calculus, can handle adjunction.We represent objects and operations of the TAG formalism in four steps: first trees (initial or derived) and the way they are constituted, then the operations (substitution and adjunction), and finally the elementary tree, i.e., the grammar. Trees (initial or derived) are obtained as the closure of the calculus under two rules that mimic the grammatical ones. We then prove the equivalence between the language generated by a TAG grammar and the closure under substitution and adjunction of its logical representation. Besides this nice prope...

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".

First, I will introduce the distinction between analytic and synthetic as a partition of propositions, proofs, programs and definitions, as an extension of the distinction between analytic categorical propositions and synthetic categorical propositions in ancient Logic. Secondly, I will discuss some logical questions concerning the distinction between analytic and synthetic, and I will present main results obtained in mathematical logic as answers to the logical questions concerning the relationships between analytic and synthetic.

L’assiomatizzazione di Peano dell’aritmetica al secondo ordine, e l’aritmetica di Peano al primo ordine. Modelli dell’aritmetica di Peano al primo ordine. Rappresentabilita delle funzioni ricorsive nell’aritmetica di Peano al primo ordine. Dimostrazione della incompletezza dell’aritmetica del primo ordine e della logica del secondo ordine.

This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci (On residuation, 2014) it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation , we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach laws and Switching laws.

We study increasing $F$-sequences, where $F$ is a dilator: an increasing $F$-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step $x$ where $F(x)$ is reached (at every step $x + 1$ we use the same process as in decreasing $F$-sequences, cf. [2], but with "+ 1" instead of "-

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.

ABSTRACT We study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with “+ 1” instead of “−1”). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator F the point where the increasing F-sequence terminates.We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id)(ω)-sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in ID1.For a general presentation of the results stated in this paper, see [1].We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25–31].

Noncommutative Logic (NL) has been introduced by Abrusci and Ruet (Non-Commutative Logic I, Annals of Mathematical Logic, 2001). NL is a refinement of Linear Logic (LL) and a conservative extension of Lambek Calculus (LC). Therefore, NL is a constructive logic (i.e. proofs are programs).

ABSTRACT In this paper we explore the residuation laws that are at the basis of the Lambek calculus, and more generally of categorial grammar. We intend to show how such laws are characterized in the framework of a purely non-commutative fragment of linear logic, known as cyclic multiplicative linear logic.

The question we want to investigate was expressed by Girard in [3]: “Assume that I am given a program P [a proof-net II], and that I cut it in two parts arbitrarily. I create two ... modules, linked together by their border. Can I express that my two modules are complementary [orthogonal], in other terms that I can branch them

ABSTRACT Semantic Web is the initiative supported by the W3C that aims to make the WorldWideWeb a place of interaction among machines – or at least among their “representatives” known as autonomous agents [3] – thanks to the exchange of “labelled” data. It is clearly something more complex and more interesting than today’s Web, which allows only for the exchange of files and “raw” data that machines simply display on a monitor.

ABSTRACT This paper presents a logical formalization of Tree-Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of a tree inside another, surrounding the subtree at the adjunction node. This seems to be contradictory with standard logical ability. We prove that some logic, namely a fragment of non-commutative intuitionistic linear logic (N-ILL), can serve this purpose. Briefly speaking, linear logic is a logic considering facts as resources. NILL can then be considered either as an extension of Lambek calculus, or as a restriction of linear logic. We model the TAG formalism in four steps: trees (initial or derived) and the way they are constituted, the operations (substitution and adjunction), and the elementary trees, i.e. the grammar. The sequent calculus is a restriction of the standard sequent calculus for N-ILL. Trees (initial or derived) are then obtained as the closure of the calculus under two rules that mimic the grammatical ones. We then prove the equivalence between the language generated by a TAG grammar and the closure under substitution and adjunction of its logical representation. Besides this nice property, we relate parse trees to logical proofs, and to their geometric representation: proofnets. We briefly present them and give examples of parse trees as proofnets. This process can be interpreted as an assembling of blocks (proofnets corresponding to elementary trees of the grammar).

In this paper we explore the residuation laws that are at the basis of the Lambek calculus, and more generally of categorial grammar. We intend to show how such laws are characterized in the framework of a purely non-commutative fragment of linear logic, known as cyclic multiplicative linear logic.