Symmetric relations, intensional individuals, and variable binding

Abstract

A symmetric relation such as ... are adjacent or ... are related is characterized by not distinguishing among two or more of its arguments. Such a relation may efficiently be represented as a relation that takes a set as its argument, or as one of its arguments. The semantics of such a representation is, in part, determined by the instantiation (matching or unification) rule used by the reasoning system operating on the representation. Two such rules are discussed. One interprets the relation to be reflexive, the other does not. Since many of these relations are not reflexive, we prefer the latter rule, which forbids two distinct variables from matching the same term. It is argued that this apparently strange restriction is actually reasonable if the rules of the system are interpreted as fully intensional. Under that interpretation, an even stronger version of the instantiation rule emerges, which we name the Unique Variable Binding Rule (UVBR). Considering the behavior of the UVBR when reasoning about reflexive relations and about nonreflexive relations used reflexively casts light on the implications of a fully intensional knowledge representation scheme. These ideas are illustrated by the output of an intensional, rule-based knowledge representation system that has been modified to allow the choice of using the UVBR instead of standard unification.