208 Mr DE MORGAN, ON THE SYLLOGISM, No. Ill, 



Aggregation of the impossible does not destroy the notion ; composition of the impossible 

 does. (§ X.) 



The universe is the whole sphere of thought within which the matter in hand is contained : 

 usually not the whole possible universe of thought, but a limited portion of it. In the last 

 syllogism of ^ IV, the universe is speed of transmission of news. The universe being U, the 

 class X introduces with itself the class not-X (or x). Let X and x be called contraries 

 or contradictories (I make no distinction between these words). It is understood that every 

 class is only part of the universe. The symbol (X, x) is equivalent to U: in extension, 

 it contains everything; in intension, it belongs to everything. Thus A, if in the universe, 

 is in (X, x) ; and A is A-(X, x), aggregate of A-X and A-x. The universe is the 

 maximum of extension, and the minimum of intension. (S IX.) 



The contrary of an aggregate is the compound of the contraries of the aggregants : 

 the contrary of a compound is the aggregate of the contraries of the components. Thus 

 (A, B) and AB have ab and (a, b) for contraries. 



XXIII. When two objects, qualities, classes, or attributes, viewed together by the 

 mind, are seen under some connexion, that connexion is called a relation. To make very 

 perfect parallelism, we should say that relation may be either of the four: that a boat 

 towed by a ship, for instance, has the tow-rope for an object of relation. But relation, for 

 all useful logical purposes, is a word of second intention, used only of class and attribute. 



A proposition is the presentation of two names under a relation. A judgment is the 

 sentence of the mind upon a proposition, true, false, more or less probable. The distinc- 

 tion of judgments, other than the simple true or false, is referred to the theory of proba- 

 bilities, as a matter of practical convenience. The absolute exclusion of this distinction 

 from logic is an error: the difference between certainty and uncertainty is of the form of 

 thought ; the amount of uncertainty is of the matter of this form. Full belief is a logical 

 whole, which is divided into parts in the theory of probabilities : and the division of a logical 

 whole into parts is of logic, whether it be convenient or not to treat it in the same book 

 which treats the syllogism. 



The distinction of subject and predicate is the distinction between the notion in relation 

 and the notion to which it is in relation. 



Every relation has its counter-relation, or converse relation : thus if X be in the relation 

 A to Y, Y is therefore in some relation B to X : and A and B are converse relations, and 

 the propositions are converse propositions. Every proposition has its converse, of meaning 

 identical with itself. 



When a relation is its own converse, the proposition is said to be convertible : meaning 

 that the converse exhibits no change of relation. It is the terms (subject and predicate) which 

 are convertible, strictly speaking. 



When X has a relation (A) to that which has a relation (B) to Y, X has to Y a combined 

 relation : the combinants are A and B. Relations have both extension and intension. Thus, 

 to take one of those relations which have appropriated the word in common life, the relation 

 of first-cousin is the aggregate of son of uncle, daughter of uncle, son of aunt, daughter of 

 aunt. The relation of the minister to the crown is the compound of subordinate and adviser. 



