The Logic of Optimality Theory
Michael Hammond · 2000
This paper is an attempt to determine the underpinnings of Optimality Theory (henceforth OT; Prince and Smolensky, 1993; McCarthy and Prince, 1993), and what logic might have to do with it.1 This is important because, without this kind of work, very strange misunderstandings of the theory crop up. The work reported here is preliminary, but nothing of this sort has previously been attempted for OT, even though the theory has been around since 1993 and is the predominant framework for phonological research in North America. The goals of the paper are as follows. First, I develop a logical statement of Optimality Theory. Next, I go on to prove some theorems. Third, I go on to show how the framework allows us to reason about partial derivations, and provides promise as the basis of a theory of acquisition and a theory of parsing. The organization of the paper is as follows. First, I provide a review of OT. Next, I introduce the basic logical formulation for a pruned-down version of OT with only one constraint. In the following section, I extend the formulation to treat real OT, where there is more than one constraint. I go on in the next section to prove several theorems. Some of these are simply to show that the formalization is doing the right thing, but some of these are quite important in their own right. Finally, I go on to show how the formalization allows us to reason about derivations given incomplete information. There are a number of results of this paper, but one real important one is that I establish that OT allows for multiple winners. That is, a tableau can have several winning candidates. This may seem obvious to the reader familiar with OT, but many presentations of OT imply or even state that this isn’t so. For example: