A gentle introduction to Type Logical Gram- mar, the Curry-Howard correspondence, and cut-elimination

This paper aims to give an introduction to Type Logical Grammar (TLG) for the general linguist who may not have a background in logic or proof theory. It is shorter than book-length treatments such as Hepple 1990, Morrill 1994, Carpenter 1996, Jäger 2001, and Moot 2002, and less technical than surveys such as Moortgat 1997. What does it mean to use logic as a grammatical formalism? The short answer goes like this: what a logical system does is take a list of assumptions and derive some conclusions from those assumptions. For instance, from the assumptions A and A → B (read “A implies B”), the conclusion B follows in most logical systems. Analogously, given a sequence of assumptions consisting of an NP John of category np and a verb phrase left of category np→ s, a Type-Logical grammar might conclude that John left as a whole must be of category s—i.e., a well-formed sentence. Well, using logic to derive sentences is a nice trick, but what makes this approach appealing? I will concentrate below on two of the more popular answers. First, Type-Logical grammars have a particularly attractive style of semantic compositionality. As explained below, thanks to the Curry-Howard correspondence, each syntactic derivation has a definite, natural, automatically-generated semantic interpretation. Second, in TLG, certain generalizations about complex linguistic behavior that must be stipulated in most theories (e.g., type lifting, function composition, and more) turn out to be quite literally theorems of the basic system. This paper will briefly describe the logical heritage of TLG, sketch a simple TLG analysis, convey the spirit of the Curry-Howard result, discuss Gentzen’s famous cut-elimination result, and motivate multimodal extensions of Lambek’s logic.