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Lambek's non-associative syntactic calculus (NL) excels in its resourceconsciousness: the usual structural rules for weakening, contraction, exchangeand even associativity are all dropped. Recently, there have been proposals forconservative extensions dispensing with NL's intuitionistic bias towardssequents with single conclusions: De Groote and Lamarche's classicalnon-associative Lambek calculus (CNL) and the Lambek-Grishin calculus (LG) ofMoortgat and associates. We demonstrate Andreoli's focalization property forsaid proposals: a normalization result for Cut-free sequent derivationsidentifying to a large extent those differing only by trivial rulepermutations. In doing so, we proceed from a `uniform' sequent presentation,deriving CNL from LG through the addition of structural rules. Thenormalization proof proceeds by the construction of syntactic phase modelswherein every `truth' has a focused proof, similar to work of Okada and ofHerbelin and Lee.

Focalization and phase models for classical extensions of non-associative Lambek calculus