12.7 Equality

Sometimes it is useful to use more than one term to name a single individual. For example, the terms 4×4, 24, 273-257, and 16 may denote the same number. Sometimes, you want to have each name refer to a different individual. For example, you may want a separate name for different courses in a university. Sometimes you do not know whether or not two names denote the same individual - for example, whether the 8 a.m. delivery person is the same as the 1 p.m. delivery person.

This section considers the role of equality, which allows us to represent whether or not two terms denote the same individual in the world. Note that, in the definite clause language presented earlier in the chapter, all of the answers were valid whether or not terms denoted the same individuals.

Equality is a special predicate symbol with a standard domain-independent intended interpretation.

Term t1 equals term t2, written t1=t2, is true in interpretation I if t1 and t2 denote the same individual in I.

Equality does not mean similarity. If a and b are constants and a=b, it is not the case that there are two things that are similar or even identical. Rather, it means there is one thing with two names.

Figure 12.12: Two chairs

Example 12.41: Consider the world of two chairs given in Figure 12.12. In this world it is not true that chair1=chair2, even though the two chairs may be identical in all respects; without representing the exact position of the chairs, they cannot be distinguished. It may be the case that chairOnRight=chair2. It is not the case that the chair on the right is similar to chair2. It is chair2.