Natural Language Semantics and Logic

We started out trying to capture the meaning of (1a) by translating it into a query in another language, SQL, which the computer could interpret and execute. But this still begged the question whether the translation was correct. Stepping back from database query, we noted that the meaning of and seems to depend on being able to specify when statements are true or not in a particular situation. Instead of translating a sentence S from one language to another, we try to say what S is about by relating it to a situation in the world. Let's pursue this further. Imagine there is a situation s where there are two entities, Margrietje and her favorite doll, Brunoke. In addition, there is a relation holding between the two entities, which we will call the love relation. If you understand the meaning of (3), then you know that it is true in situation s. In part, you know this because you know that Margrietje refers to Margrietje, Brunoke refers to Brunoke, and houdt van refers to the love relation.

We have introduced two fundamental notions in semantics. The first is that declarative sentences are true or false in certain situations. The second is that definite noun phrases and proper nouns refer to things in the world. So (3) is true in a situation where Margrietje loves the doll Brunoke, here illustrated in Figure 10-1.

Once we have adopted the notion of truth in a situation, we have a powerful tool for reasoning. In particular, we can look at sets of sentences, and ask whether they could be true together in some situation. For example, the sentences in (5) can be both true, whereas those in (6) and (7) cannot be. In other words, the sentences in (5) are consistent, whereas those in (6) and (7) are inconsistent.

(5) a. Sylvania is to the north of Freedonia. b. Freedonia is a republic.

Figure 10-1. Depiction of a situation in which Margrietje loves Brunoke.

(6) a. The capital of Freedonia has a population of 9,000. b. No city in Freedonia has a population of 9,000.

(7) a. Sylvania is to the north of Freedonia. b. Freedonia is to the north of Sylvania.

We have chosen sentences about fictional countries (featured in the Marx Brothers' 1933 movie Duck Soup) to emphasize that your ability to reason about these examples does not depend on what is true or false in the actual world. If you know the meaning of the word no, and also know that the capital of a country is a city in that country, then you should be able to conclude that the two sentences in (6) are inconsistent, regardless of where Freedonia is or what the population of its capital is. That is, there's no possible situation in which both sentences could be true. Similarly, if you know that the relation expressed by to the north of is asymmetric, then you should be able to conclude that the two sentences in (7) are inconsistent.

Broadly speaking, logic-based approaches to natural language semantics focus on those aspects of natural language that guide our judgments of consistency and inconsistency. The syntax of a logical language is designed to make these features formally explicit. As a result, determining properties like consistency can often be reduced to symbolic manipulation, that is, to a task that can be carried out by a computer. In order to pursue this approach, we first want to develop a technique for representing a possible situation. We do this in terms of something that logicians call a "model."

A model for a set W of sentences is a formal representation of a situation in which all the sentences in W are true. The usual way of representing models involves set theory. The domain D of discourse (all the entities we currently care about) is a set of individuals, while relations are treated as sets built up from D. Let's look at a concrete example. Our domain D will consist of three children, Stefan, Klaus, and Evi, represented respectively as s, k, and e. We write this as D = {s, k, e}. The expression boy denotes the set consisting of Stefan and Klaus, the expression girl denotes the set consisting of Evi, and the expression is running denotes the set consisting of Stefan and Evi. Figure 10-2 is a graphical rendering of the model.

Figure 10-2. Diagram of a model containing a domain D and subsets of D corresponding to the predicates boy, girl, and is running.

Later in this chapter we will use models to help evaluate the truth or falsity of English sentences, and in this way to illustrate some methods for representing meaning. However, before going into more detail, let's put the discussion into a broader perspective, and link back to a topic that we briefly raised in Section 1.5. Can a computer understand the meaning of a sentence? And how could we tell if it did? This is similar to asking "Can a computer think?" Alan Turing famously proposed to answer this by examining the ability of a computer to hold sensible conversations with a human (Turing, 1950). Suppose you are having a chat session with a person and a computer, but you are not told at the outset which is which. If you cannot identify which of your partners is the computer after chatting with each of them, then the computer has successfully imitated a human. If a computer succeeds in passing itself off as human in this "imitation game" (or "Turing Test" as it is popularly known), then according to Turing, we should be prepared to say that the computer can think and can be said to be intelligent. So Turing side-stepped the question of somehow examining the internal states of a computer by instead using its behavior as evidence of intelligence. By the same reasoning, we have assumed that in order to say that a computer understands English, it just needs to behave as though it did. What is important here is not so much the specifics of Turing's imitation game, but rather the proposal to judge a capacity for natural language understanding in terms of observable behavior.

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