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

Lambek’s non-associative syntactic calculus (NL, [18]) excels in its resource consciousness: the usual structural rules for weakening, contraction, exchange and even associativity are all dropped. Recently, there have been proposals for conservative extensions dispensing with NL’s intuitionistic bias towards sequents with single conclusions: De Groote and Lamarche’s classical NL (CNL, [10]) and Moortgat’s Lambek-Grishin calculus (LG, [21]). We demonstrate Andreoli’s focalization property ([2]) for said proposals: a normalization result for Cut-free sequent derivations identifying to a large extent those differing only by trivial rule permutations. In doing so, we proceed from a ‘uniform’ sequent presentation, deriving CNL from LG through the addition of structural rules. The normalization proof proceeds by the construction of syntactic phase models wherein every ‘truth’ has a focused proof, similar to [23].