as the landing-place of Cæsar all material circumstances must be shown to agree. If the modern Wroxeter is the ancient Uriconium, there must be the like agreement of all features of the country not subject to alteration by time.
Such identities must be expressed in the form A = B. We may say
In these and similar propositions we assert identity of single qualities or causes of sensation. In the same form we may also express identity of any group of qualities, as in
A multitude of propositions involving singular terms fall into the same form, as in
In mathematical and scientific theories we often meet with simple identities capable of expression in the same form. Thus in mechanical science “The process for finding the resultant of forces = the process for finding the resultant of simultaneous velocities.” Theorems in geometry often give results in this form, as
The more profound and important laws of nature are often expressible in the form of simple identities; in addition to some instances which have already been given, I may suggest,
All definitions are necessarily of this form, whether the objects defined be many, few, or singular. Thus we may say,
It is an extraordinary fact that propositions of this elementary form, all-important and very numerous as they are, had no recognised place in Aristotle’s system of Logic. Accordingly their importance was overlooked until very recent times, and logic was the most deformed of sciences. But it is impossible that Aristotle or any other person should avoid constantly using them; not a term could be defined without their use. In one place at least Aristotle actually notices a proposition of the kind. He observes: “We sometimes say that that white thing is Socrates, or that the object approaching is Callias.”1 Here we certainly have simple identity of terms; but he considered such propositions purely accidental, and came to the unfortunate conclusion, that “Singulars cannot be predicated of other terms.”
Propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole; as when we say,
When Blackstone asserts that “The only true and natural foundation of society are the wants and fears of individuals,” we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society. But many propositions which might seem to be collective are but groups of singular propositions or identities. When we say “Potassium and sodium are the metallic bases of potash and soda,” we obviously mean,
It is the work of grammatical analysis to separate the various propositions often combined into a single sentence. Logic cannot be properly required to interpret the forms and devices of language, but only to treat the meaning when clearly exhibited.
Partial Identities.
A second highly important kind of proposition is that which I propose to call a partial identity. When we say that “All mammalia are vertebrata,” we do not mean that mammalian animals are identical with vertebrate