SYMBOLIC LOGIC ordinary syllogism. Thus, the premises of Baroko, ‘ Any M is P ' and ‘ Some S is not P, may be written { M[P~\ f S (P). Then, as just seen, we can write { M |F] [ B (P). Then, by iteration, {M [P (P)] j B (P). Breaking the line under even enclosures, we get {[P(P)]J/j B(P). But we have already shown that [P (P)] can be written unen¬ closed. Hence it can be struck out under one enclosure ; and the unenclosed (P) can be erased. Thus we get {Af\ S, or £ Some S is not M! The great number of steps into which syllogism is thus analysed shows the perfec¬ tion of the method for purposes of analysis. In taking account of relations, it is neces¬ sary to distinguish between the different sides of the letters. Thus let Z be taken in such a sense that X—Z— Y means (X loves Y'.’ Then will mean * Y loves X.’ Then, if m—- means ‘ Something is a man,’ and —w means ‘ Something is a woman,’ m—Z—w will mean ‘ Some man loves some woman m [(J-Z-J)”Lm;] will mean ‘ Some man loves all women ’ ; [(m.—l^ÿw] will mean ‘ Every woman is loved by some man,’ &c. Since enclosures signify negation, by en¬ closing a part of the line of identity, the relation of otherness is represented. Thus, will assert ‘Some A is not some P.’ Given the premises ‘ Some A is B ’ and ‘ Some C is not P,’ they can be written A—BC(B). By Rule III, this can be written A {[P]} 0(B). By iteration, this gives A {[P (P)]} C (P). The lines of identity are to be conceived as passing through the space between the braces outside of the brackets. By breaking the lines under even enclosures, we get JTp [P (Bf\ ^}C (P). As we have already seen, oddly enclosed [P (P)] can be erased. This, with erasure of the detached (P), gives A{~j} C. Joining the lines under odd enclosures, we get J~{~| )C, ‘ Some A is not some C! For all considerable steps in ratiocination, the reasoner has to treat qualities, or collec¬ tions (they only differ grammatically), and especially relations, or systems, as objects of relation about which propositions are asserted or and inferences drawn. It is, therefore, neces¬ sary to make a special study of the logical relatives ‘-is a member of the collection -,’ and ‘- is in the relation-to The key to all that amounts to much in symbolical logic lies in the symbolization of these relations. But we cannot enter into this extensive subject in this article. The system of which the slightest possible sketch has been given is not so iconoidal as the so-called Euler’s diagrams; but it is by far the best general system which has yet been devised. The present writer has had it under examination for five years with continually increasing satisfaction. However, it is proper to notice some other systems that are now in use. Two systems which are merely extensions of Boole’s algebra of logic may be mentioned. One of these is called by no more proper designation than the ‘ general algebra of logic.’ The other is called ‘ Peirce’s algebra of dyadic relatives.’ In the former there are two operations—aggregation, which Jevons (to whom its use in algebra is due) signifies by a sign of division turned on its side, thus • | • (I prefer to join the two dots, in order to avoid mistaking the single character for three) ; and composition, which is best signified by a somewhat heavy dot, \ Thus, if A and P are propositions, A • | • P is the proposition which is true if A is true, is true if P is true, but is such that if A is false and P is false, it is false. A'B is the proposition which is true if A is true and P is true, but is false if A is false and false if P is false. Considered from an algebraical point of view, which is the point of view of this system, these expressions A • | • P and A *P are mean functions', for a mean function is defined as such a symmetrical function of several vari¬ ables, that when the variables have the same value, it takes that same value. It is, there¬ fore, wrrong to consider them as addition and multiplication, unless it be that truth and falsity, the two possible states of a proposi¬ tion, are considered as logarithmic infinity and zero. It is therefore well to let o represent a false proposition and oo (meaning logarithmic infinity, so that + oo and — oo are different) a true proposition. A heavy line, called an ‘ obelus,’ over an expression negatives it. The letters i,j, k, &c., written below the line after letters signifying predicates, denote individuals, or supposed individuals, of udiich the predicates are true. Thus, Zÿ may mean that i loves j. To the left of the expression a series of letters n and 2 are written, each 649