to another, the correlate of the one has the relation to the correlate of the other, and vice versa. A figure will make this clearer. Let and be two terms having the relation . Then there are to be two terms , , such that has the relation to , has the relation to , and has the relation to . If this happens with every pair of terms such as and , and if the converse happens with every pair of terms such as and , it is clear that for every instance in which the relation holds there is a corresponding instance in which the relation holds, and vice versa; and this is what we desire to secure by our definition. We can eliminate some redundancies in the above sketch of a definition, by observing that, when the above conditions are realised, the relation is the same as the relative product of and and the converse of , i.e. the -step from to may be replaced by the succession of the -step from to , the -step from to , and the backward -step from to . Thus we may set up the following definitions:— A relation is said to be a "correlator" or an "ordinal correlator" of two relations and if is one-one, has the field of for its converse domain, and is such that is the relative product of and and the converse of . Two relations and are said to be "similar," or to have "likeness," when there is at least one correlator of and . These definitions will be found to yield what we above decided to be necessary. It will be found that, when two relations are similar, they share all properties which do not depend upon the actual terms in their fields. For instance, if one implies diversity, so does the other; if one is transitive, so is the other; if one is connected, so is the other. Hence if one is serial, so is the other. Again, if one is one-many or one-one, the other is one-many [Pg 54] or one-one; and so on, through all the general properties of relations. Even statements involving the actual terms of the field of a relation, though they may not be true as they stand when applied to a similar relation, will always be capable of translation into statements that are analogous. We are led by such considerations to a problem which has, in mathematical philosophy, an importance by no means adequately recognised hitherto. Our problem may be stated as follows:— Given some statement in a language of which we know the grammar and the syntax, but not the vocabulary, what are the possible meanings of such a statement, and what are the meanings of the unknown words that would make it true?