190 IV. STRUCTURAL FACTORS IN A LANGUAGES
hold between A and C. For example, if A is before, or after, or above, or more., than B, and B is before, or after, or above, or more., than C, then A is before, or after, or above, or more., than C.
It should be noted that all relations which give origin to series are transitive. But so are many others. In the above examples, the relations were transitive and asymmetrical, but there are numerous relations which are transitive and symmetrical. Among these are relations of equality, of being equally numerous,.
Relations which are not transitive are called non-transitive. For instance, dissimilarity is not transitive. If A is dissimilar to B, and B dissimilar to C, it does not follow that A is dissimilar to C.
Relations which, whenever they hold between A and B, never hold between A and C are called intransitive. 'Father', 'one inch longer', 'one year later'., are intransitive relations.
Relations are classified in several other ways; but, for our purpose, the above will be sufficient.
It is necessary now to compare the relational forms with the subject-predicate form of representation, which structurally underlies the traditional A -system and two-valued 'logic'. The structural question arises whether all relations can be reduced to the subject-predicate forms of language.
Symmetrical relations, which hold between B and A whenever they hold between A and B, seem plausibly expressed in the subject-predicate language. A symmetrical and transitive relation, such as that of 'equality', could be expressed as the possession of a common 'property'. A non-transitive relation, such as that of 'inequality', could also be considered as representing 'different properties'. But when we analyse asymmetrical relations, the situation becomes obviously different, and we find it a structural impossibility to give an adequate representation in terms of 'properties' and subject-predicates.
This fact has very serious semantic consequences, for we have already seen that some of the most important relations we know at present belong to the asymmetrical class. For example, the term 'greater' obviously differs from the term 'unequal', and 'father' from the term 'relative'. If two things are said to be unequal, this statement conveys that they differ in the magnitude of some 'property' without designating the greater. We could also say that they have different magnitudes, because inequality is a symmetrical relation; but if we were to say that a thing is unequal to another, or that the two have different magnitudes, when one of them was greater than the other, we simply should not give an adequate account of the structural facts at hand. If A is greater than