Polarized Classical Non-associative Lambek Calculus and Formal Semantics

While initially motivated for studying natural language syntax, the intuitionistic bias underlying traditional Lambek calculi renders them particularly suitable to a Montagovian formal semantics through the Curry-Howard correspondence. Several recent proposals, however, have departed from the intuitionistic tradition, seeking instead to formulate 'classical' Lambek calculi. We show that this classical turn need not come at the cost of the tight connection with formal semantics, concentrating on De Groote and Lamarche's Classical Non-Associative Lambek calculus (CNL). Our work is founded in Girard's and Andreoli's research into polarities and focused proofs, suggesting the definition of polarized CNL, its connection to De Groote and Lamarche's original proposal explicated through the use of phase spaces. We conclude with a discussion of related literature, particularly Moortgat's Lambek-Grishin calculus.