Meaningful Games: Exploring Language with Game Theory

2. My Fall from Platonic Heaven

2 My Fall from Platonic Heaven The theory outlined in chapter 1, the Mentalese theory, is a formidable one. Its intellectual roots run deep. One sees it anticipated in Plato and Kant. It has absorbed ideas from the philosophy of mathematics and logic, computation theory, and artificial intelligence. No one should take it lightly; without this theory, linguistics as we know it today would look radically di¤erent. My formulation may be a bit oversimplified, but I think it’s fair to say that many linguists believe some version of it. I fervently believed it as a graduate student and defended it and taught it when I became a faculty member. The theory takes the mind (or brain) as a computational device. What exactly does that mean? At the very least, a computational device is a system that has a set of symbols and operations defined to manipulate those symbols. In chapter 1, I imagined that the human capacity for language was one kind of computational system. It would have an internal vocabulary that could be used to specify a grammar, namely, a set of rules that would tell the system how to construct and parse sentences. Phrase Structure Grammar Let’s take a simple example of a grammar and work out the relation between the rules of grammar and their meaning in Mentalese. Figure 2.1 shows a very simple grammar called a context-free phrase structure grammar for a few sentences of English. Each line in the figure is a single rule. The arrow symbol, !, is either an instruction to replace the symbol on the left-hand side of the arrow with the string on the righthand side, or an instruction that allows the symbol on the left-hand side of the arrow to be replaced by a single choice from the options listed between the curly braces, f and g. The symbols on either side of the arrow are the symbols of the computational system, and the operation is speci- fied by the arrow; it is either the concatenation the stringing together of symbols or the choice of a single symbol from a set of possibilities. In order to construct a sentence from this grammar, we start with the symbol S (for sentence): S The system says that we can replace the S symbol with the string ‘‘NP VP’’ (for noun phrase and verb phrase): NP VP The rules in figure 2.1 allow us to replace NP with the string ‘‘Det Noun’’ (Det indicates determiner; see figure 2.1 for examples): Det Noun VP We are allowed to replace Det by the: the Noun VP and replace Noun by monkey: the monkey VP VP can be replaced by VIntrans, where Intrans is short for intransitive and means there is no object of the action named in the verb: the monkey VIntrans Finally, VIntrans can be replaced by snored to yield the monkey snored S ! NP VP NP ! Det Noun VP ! VIntrans VP ! VTrans NP Det ! fthe, a, every, some, no, allg Noun ! ftiger, monkey, humang NP ! fAlice, Bill, John, Maryg VIntrans ! fslept, walked, snoredg VTrans ! fsaw, licked, ate, killedg Figure 2.1 A Very Simple Grammar 22 Chapter 2 which, while not exactly Shakespeare, still counts as a grammatical sentence of English. Usually, linguists prefer to show the derivation (or parse) of a sentence in terms of a tree, which is neutral between building the sentence and assigning the sentence a parse. The root of the tree is the symbol we started with, S, and under each symbol is the string that replaces the symbol . The root of the tree is at the top and the tree grows down. So the tree for the monkey snored is the following: At every level the tree corresponds to steps in the construction of the sentence by the computational system the grammar that was specified in figure 2.1. Of course, a more adequate grammar would be much more complex, but the simple grammar su‰ces to make a few points. Recalling the example for kill that ended chapter 1, readers can verify that the simple grammar in figure 2.1 allows the system to build the tree in (1): (1) Grammar and Compositionality The central idea of chapter 1 was that sentences of English can be translated into expressions of Mentalese (assume that the computational system of the mind/brain knows how...