Relation-numbers are important for the following reason. In addition to the propositions which can be proved by logic (considered in Chapter XVII.), there are other propositions which can be enunciated by logic, though they cannot be proved or disproved except by empirical evidence. Such, for example, is the proposition: "There are classes which are not[Pg 251] finite." This is a proposition which is purely logical in content, but there is no a priori way of knowing whether it is true or false. (Many such have been proposed, but they are all fallacious.) Then, again, there are propositions which contain some particular constituent, but would be capable of enunciation in logical terms if that constituent were turned into a variable. Take, e.g.: "Before is a transitive relation." This is not a statement which pure logic can enunciate, because before is an empirical relation. But " is a transitive relation," where is variable, can be enunciated by pure logic. We will say that a proposition containing a certain constituent attributes a "logical property" to if, when is replaced by a variable , the result is a propositional function which can be expressed by logic. The test of a logical property is very simple: apart from the constant , there must be no constants involved—except such purely formal constants as "incompatibility" and "for all values of " which are not constituents of the propositions in whose verbal or symbolic expression they occur. It will be seen that transitiveness, e.g., is a logical property of a relation; so is asymmetry or symmetry; so is having terms in its field; so is, in the case of a three-term relation (between), the property of generating a Euclidean space; so is, in the case of a four-term relation (separation of couples), the property of generating a projective space; and so on. We can now state the proposition on account of which structure is important. When two relations have the same structure (or relation-number), all their logical properties are identical. [Pg 252] Logical properties include all those which can be expressed in mathematical terms. Moreover, the inferences from perceptions to their causes, assuming such inferences to be valid, are concerned mainly, if not exclusively, with logical properties. This latter proposition is one which we must now examine.