The Logic of Scope

Chris Barker

2013/2014 Annual Logic Lecture

28 Mar, 1:30pm-3:30pm, Class of 1947 room

Scope-taking is one of the most dramatic, as well as one of the most characteristic, phenomena in natural language. In scope-taking, a deeply embedded constituent controls (take scope over) the interpretation of surrounding material. For instance, when we gloss the sentence “Mary called everyone yesterday” as `for every person x, Mary called x yesterday’, we are claiming that the embedded direct object “everyone” controls the interpretation of the entire surrounding sentence.

50 years ago, Lambek provided a substructural logic called NL for reasoning about ordinary function-argument combination (`merge’) in natural language. He analyzed argument\function combination and function/argument combination as the left and right adjoints of string concatenation. In Linear Logic terms, his merge is a (noncommutative) multiplicative conjunction (tensor).

In order to extend Lambek’s logic to scope-taking, we need to residuate not on concatenation, but on the part-whole relation. The adjoints then are subpart\whole and whole/subpart. This characterizes a syntactic relationship not of left or right adjacency, but of being-surrounded-by, and of surrounding—exactly what is needed for characterizing scope-taking.

I will present a substructural logic called NL_lambda in which the relationship between the merge mode and the scope-taking mode is characterized by a single structural inference rule. Reporting on joint work with Chung-chieh Shan, I will show that the logic is sound and complete with respect to the usual class of relational models. I will also show that the logic is conservative with respect to Lambek’s original logic. That is, a sequent in the language of NL is a theorem in NL_lambda iff it is a theorem in NL. In addition, I will show that NL_lambda is decidable.

Illustrative applications of the logic to natural language phenomena will include not only ordinary scope-taking and scope ambiguity, but more exotic phenomena such the parasitic scope analysis for words such as “same” and “different”.