The mathematical form of thought builds upon the notions of con tained and excluded the metaphysical form upon the notions of essential and repugnant. The metaphysical form should be cultivated because, right or wrong, human beings are made to be metaphysicians, children most of all, uneducated persons more than educated. The must be and the cannot be are the strong points of our mental constitu tions. We know all about can and cannot from our cradles : we never feel the same assurance about is and is not. A philosopher, in a dark age, may determine to set out with a knowledge of the naturally possible and impossible ; but not even a philosopher ever pretended to set out with a knowledge of the existent and non-existent. Logic excludes actual metaphysics by declining to enter into the inquiry whether this or that use of the metaphysical form of thought be true or false in matter. But logic claims to interfere to give a logical cor rectness in the use of terms and relating notions : to prevent, for instance, a person who, rightly or wrongly, makes reason an essential of man, from pronouncing it to be therefore repugnant to the nature of brute.
Additional intent may diminish extent, but never increases it : additional extent may diminish intent, but never increases it. By con tinual introduction of new attributes we may, if speaking of things existing, not of possibilities of thought, curtail the widest class until we have brought it down to an individual. Thus being includes every thing ; material being is body ; living body is animal; human animal is man ; man of old time is ancient ; go on thus until we come to ancient Roman general who conquered Gauls, and wrote his own campaigns, and we have the individual Julius Caesar.
If we read our cumular propositions by the notion of enumeration of attributes, we have the following arithmetical reading by enumera tion of attributes, just as " every x is Y," &c., show arithmetical reading by enumeration of the smallest * possible classes, that is, individuals.
x )) Y. All the attributes of 1' are attributes of x.
x ( ( Y. Some attributes of 1' are not attributes of x.
x) ( Y. Everything wants either some attribute of v, or some attri bute of x.
x ( ) v. Some things want neither any attribute of Y, nor any attri bute of x.
x ( ) v. Everything has either all the attributes of v, or all the attributes of x.
) ( v. Some things want either some attributes of Y, or some attrihutes of x.
x ( ( v. Some attributes of Y are all the attributes of x.
x) ) v. Any attributes of v are not some attributes of x.
The complete inversion of the quantities will be perceived. We
now give a metaphysical nomenclature.
x ) ) Y. X a dependent of Y ; 1' an essential of x.
x ( ( y. x an independent of Y; Y an inessential of x.
) ( Y. x a repugnant of 1' ; and v of x.
x ( ) Y. X an irrcpugnant of Y ; and y of x.
x ( ) Y. X an alternative of Y ; and Y of x.
x ) ( Y. x an inalternative of Y ; and Y of x.
x ( ( y. x an essential of Y ; v a dependent of x.
x) ) v. x an inessential of Y ; v an independent of x.
The syllogisms are now described in a way which will suggest itself : thus ( ) ) ) giving ( ), which read mathematically exhibits the com bination "A complement of a deficient is a partient," will, when read metaphysically, give " an alternative of an inessential is an irre pugnant." In the article PREDICABLES we shall add something on the extension which must follow the admission of contrary terms. In RELATION we shall enter farther upon the doctrine of figure, which is of no import ance in any system in which only convertible relations are used, and quantities and terms are converted together. In PROBABILITY we enter upon the subject of belief.
When reduced to so brief an abstract as we have here given, a system of logic appears like a string of names. But the more names the more ideas ; provided only that every name has meaning whieh is not merely repeated in any other. So far as the invention of words is analysis of ideas, or the completion of such analysis, and prevents it from being lost, so far it is augmentation of power over ideas. Dr. Thomson well observes that " as the distinctions between the relations of objects grow more numerous, involved, and subtle, it [language] becomes more analytic, to be able to express them : and, inversely, those who are born to be the heirs of a highly analytic language, must needs learn to think up to it, to observe and distinguish all the relations of objects, for which they find the expressions already formed, so that we have an instructor for the 'thinking powers in that speech which we are apt to deem no more than their handmaid and minister." It is very common to discuss the utility of the subject, in an article treating of it in general; but of all arguments, that about the utility of any branch of knowledge is the most useless. Allowing an in frequent exception here and there, those who have mastered the first difficulties of any branch of knowledge are all convinced of its utility ; and with those who have not, and who have the boldness to form an opinion, there is no basis for an argument.