We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

Building on Ben-Avi and Winter’s (2007) work, this paper provides a general “intensionalization” procedure that turns an extensional semantics for a language into an intensionalized one that is capable of accommodating “truly intensional” lexical items without changing the compositional semantic rules. We prove some formal properties of this procedure and clarify its relation to the procedure implicit in Montague’s (1973) PTQ.

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of linear context-free rewriting systems. This (...) suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)