The object of Logic is to find how propositions are to be proved. As preliminary to this, it has been already shown that the Conceptualist view of propositions, viz. that they assert a relation between two ideas, and the Nominalist, that they assert agreement or disagreement between the meanings of two names, are both wrong as general theories: for that generally the import of propositions is, to affirm or deny respecting a phenomenon, or its hidden source, one of five kinds of facts. There is, however, a class of propositions which relate not to matter of fact, but to the meaning of names, and which, therefore, as names and their meanings are arbitrary, admit not of truth or falsity, but only of agreement or disagreement with usage. These verbal propositions are not only those in which both terms are proper names, but also some, viz. essential propositions, thought to be more closely related to things than any others. The Aristotelians' belief that objects are made what they are called by the inherence of a certain general substance in the individuals which get from it all their essential properties, prevented even Porphyry (though more reasonable than the mediÆval Realists) from seeing that the only difference between altering a non-essential (or accidental) property, which, he says, makes the thing ??????? [Greek: alloion], and altering an essential one, which makes it ???? [Greek: allo] (i.e. a different thing), is, that the latter change makes the object change its name. But even when it was no longer believed that there are real entities answering to general terms, the doctrine based upon it, viz. that a thing's essence is that without which the thing could neither be, nor be conceived to be, was still generally held, till Locke convinced most thinkers that the supposed essences of classes are simply the significations of their names. Yet even Locke supposed that, though the essences of classes are nominal, individuals have real essences, which, though unknown, are the causes of their sensible properties.

An accidental proposition (i.e. in which a property not connoted by the subject is predicated of it) tacitly asserts the existence of a thing corresponding to the subject; otherwise, such a proposition, as it does not explain the name, would assert nothing at all. But an essential proposition (i.e. in which a property connoted by the subject is predicated of it) is identical. The only use of such propositions is to define words by unfolding the meaning involved in a name. When, as in mathematics, important consequences seem to follow from them, such really follow from the tacit assumption, through the ambiguity of the copula, of the real existence of the object named.

Accidental propositions include, 1, those with a proper name for subject, since an individual has no essence (although the schoolmen, and rightly, according to their view of genera and species as entities inhering in the individuals, attributed to the individual the essence of his class); and, 2, all general or particular propositions in which the predicate connotes any attribute not connoted by the subject. Accidental propositions may be called real; they add to our knowledge. Their import may be expressed (according as the attention is directed mainly, either to what the proposition means, or to the way in which it is to be used), either, by the formula: The attributes of the subject are always (or never) accompanied by those signified by the predicate; or, by the formula: The attributes of the subject are evidence, or a mark, of the presence of those of the predicate. For the purposes of reasoning, since propositions enter into that, not as ultimate results, but as means for establishing other propositions, the latter formula is preferable.