LINEAR LOGIC IN LINGUISTICS AND COGNITIVE SCIENCE: LAMBEK CALCULUS, PREGROUPS AND THE GEOMETRY OF COGNITION

In this talk we explore the contributions of linear logic to linguistics and cognition, paying particular attention to the notions of interaction and geometrical representation of information. A kernel contribution to linguistic analysis in this respect is given by the Syntactic Calculus (SC) introduced by Lambek (1958, 1961). As shown and proved by Abrusci (1991, 1995, 2002), Buszkowski (2001, 2002) and Lambek himself (1995), SC is a fragment of Non-commutative Intuitionistic Linear Logic (NILL); also a classical formulation of this calculus has been provided: Non-commutative Classical Linear Logic (NCLL), or Bilinear Logic in Lambek’s definition, in which interaction and geometrical representation of linguistic domains are obtained by means of two negations and the logical properties of the multiplicative connectives “par” and “times” (Casadio 2001, Casadio & Lambek 2002).       

The calculus of Pregroups introduced by Lambek in (1999) is a revision and “simplification” of NCLL, with two negations and just one connective, a multiplicative self-dual conjunction: a pregroup   {G . 1 l r →} is a partially ordered monoid in which each element has a left adjoint and a right adjoint, the dot . (usually omitted) stands for multiplication with unit 1, and the arrow denotes the partial order (Lambek 1999, ff.). In linguistic applications the symbol 1 stands for the empty string of types and multiplication is interpreted as concatenation (in the last ten years numerous applications to spoken languages have been produced by e.g. M. Barr, D. Bargelli, W. Buszkowski, A. Kislak, T. Kusalik, G. Kobele, A. Preller, M. Sadrzadeh, E. Stabler; a survey in Casadio & Lambek 2008). The calculus of Pregroups preserves the geometrical properties and parallel computation of linear logic (is very close to the NCLL calculus) but is also innovative from the point of view of computational time and complexity (Bechet et al. (2008), Preller 2007b). In the talk examples will be given concerning unbounded dependencies, feature agreement and quantifiers interpretation.  
      

In the last part of the talk the question of introducing interaction inside linguistics is extended to the more general question of introducing interaction inside cognition. Recent formulations of linear logic and ludics pay particular attention to the intrinsic positive and negative polarity of logical operators (connectives and quantifiers) and the sentences in which they are embedded (Abrusci 2008). This distinction appears to have interesting applications to cognition, and particularly to the analysis of reasoning, given that negation sends from one polarity to the other, and alternating polarities characterize proof search, where the positive polarity plays a creative-irreversible role, whilst the negative polarity a descriptive-reversible role.  We conclude with a brief presentation of a research on logic, reasoning and neuroscience, developed in this perspective.    
 
KEYWORDS: interaction, negation, polarity, language, cognition.