17.5.5 Representing Beliefs

There are a fair number of words and expressions that have what might be called a world creating ability. By this, we mean that their meaning representations contain logical formulas that are not intended to be taken as true in the real world, but rather as part of some kind of hypothetical world. In addition, these meaning representations often denote a relation from the speaker, or some other entity, to this hypothetical world. Examples of words that have this ability are believe, want, imagine and know. World-creating words generally take various sentence-like constituents as arguments.

Consider the following example:

(17.76) I believe that Mary ate British food

Applying our event-oriented approach we would say that there are two events underlying this sentence: a believing event relating the speaker to some specific belief, and an eating event that plays the role of the believed thing. Ignoring temporal information, a straightforward application of our reified event approach would produce the following kind of representation:

$$ \exists u,v ISA(u,Believing)\land ISA(v,Eating) $$

$$ \land Believer(u,Speaker)\land BelievedProp(u,v) $$

$$ \land E a t e r(v,M a r y)\land E a t e n(v,B r i t i s h F o o d) $$

This seems relatively straightforward, all the right roles are present and the two events are tied together in a reasonable way. Recall, however, that in conjunctive representations like this all of the individual conjuncts must be taken to be true. In this case, this results in a statement that there actually was an eating of British food by Mary. Specifically, by breaking this formula apart into separate formulas by conjunction elimination, the following formula can be produced:

$ \exists v \text{ ISA}(v, Eating) $

$$ \land E a t e r(v,M a r y)\land E a t e n(v,B r i t i s h F o o d) $$

This is clearly more than we want to say. The fact that the speaker believes this proposition does not make it true; it is only true in the world represented by the speaker's beliefs. What is needed is a representation that has a structure similar to this, but where the Eating event is given a special status.

Note that reverting to the simpler predicate representations we used earlier in this chapter does not help. A common mistake using such representations would be to represent this sentence with the following kind of formula:

Believing(Speaker, Eating(Mary, BritishFood))

The problem with this representation is that it is not even valid FOPC. The second argument to the Believing predicate should be a FOPC term, not a formula.

This syntactic error reflects a deeper semantic problem. Predicates in FOPC hold between the objects in the domain being modeled, not between the relations that hold among the objects in the domain. Therefore, FOPC lacks a meaningful way to assert relations about full propositions, which is unfortunately exactly what words like believe, want, imagine and know want to do.

The standard method for handling this situation is to augment FOPC with operators that allow us to make statements about full logical formulas. Let's consider how this approach might work in the case of example (17.76). We can introduce an operator called Believes that takes two FOPC formulas as its arguments: a formula designating a believer, and a formula designating the believed proposition. Applying this operator would result in the following meaning representation:

Believes(Speaker, ∃vISA(v, Eating)

Under this approach, the contribution of the word believes to this meaning representation is not a FOPC proposition at all, but rather an operator that is applied to the believed proposition. Therefore, as we discuss in Ch. 18, these world creating verbs play quite a different role in the semantic analysis than more ordinary verbs like eat.

As one might expect, keeping track of who believes what about whom at any given point in time gets rather complex. As we will see in Ch. 21, this is an important task in interactive systems that must track users' beliefs as they change during the course of a dialogue.

Operators like Believes that apply to logical formulas are known as modal operators. Correspondingly, a logic augmented with such operators is known as a modal logic. Modal logics have found many uses in the representation of commonsense knowledge in addition to the modeling of belief, among the more prominent are representations of time and hypothetical worlds.

Not surprisingly, modal operators and modal logics raise a host of complex theoretical and practical problems that we cannot even begin to do justice to here. Among the more important issues are the following:

• How inference works in the presence of specific modal operators.

• The kinds of logical formula that particular operators can be applied to.

• How modal operators interact with quantifiers and logical connectives.

The influence of these operators on the equality of terms across formulas.

The last issue in this list has consequences for modeling agent's knowledge and beliefs in dialogue systems and deserves some elaboration here. In standard FOPC systems, logical terms that are known to be equal to one another can be freely substituted without having any effect on the truth of sentences they occur in. Consider the following examples:

(17.77) Snow has delayed Flight 1045.

(17.78) John's sister's flight serves dinner.

Assuming that these two flights are the same, substituting Flight 1045 for John's sister's flight has no effect on the truth of either sentence.

Now consider, the following variation on the first example:

(17.79) John knows that snow has delayed Flight 1045.

(17.80) John knows that his sister's flight serves dinner.

Here the substitution does not work. John may well know that Flight 1045 has been delayed without knowing that his sister's flight is delayed, simply because he may not know the number of his sister's flight. In other words, even if we assume that these sentences are true, and that John's sister is on Flight 1045, we can not say anything about the truth of the following sentence:

(17.81) John knows that snow has delayed his sister's flight.

Settings like this where a modal operator like Know is involved are called referentially opaque. In referentially opaque settings, substitution of equal terms may or may not succeed. Ordinary settings where such substitutions always work are said to be referentially transparent.