Logistic

The logical sum of two classes is the class including each of them and included in every class including them.

In these definitions we have assumed the unanalysed concept of the relation of one class to another which contains it. This is analogous to the relation of part to whole. We may then say, somewhat metaphorically, that the logical product of two classes (x and y) is the largest class including them both; whereas their logical sum is the smallest class including them both. It is not necessary to assume that the elements of the logical sum should be exclusive. Following the mathematical analogies, we may rep resent the logical product of x and y by "xXy" or by "xy"; and their logical sum by "x+y." Bearing in mind that these operations are to be completely formal, therefore completely general, we see that the symbols x and y must represent any classes. Also "xy" and "x+y" must also represent classes. But these may be classes the elements of which have no objects in common. Hence, since no objects are found in both classes, their logical product will be a class with no members. For example, the product of "squares" and "circles" is the class "squared circles." But there are no squared circles; hence, this is a class with no members. Or again, the product of "living beings" and "temples" is a class with no members. It follows, therefore, that in order to secure complete generality, we must admit the notion of an empty class, i.e., a class with no members. This is called the "null-class." Let us represent the null-class by o, standing for nothing. Clearly "x+o =x." Hence, from the above definition of the logical sum it follows that the null-class is contained in every class. Again, "x X o = o." That is, whatever x may be, the class "x X o" is the class containing both x and nothing. But what is both x and nothing is nothing. It is easy to see that the only class that remains unaltered no matter what class is selected from it is the class with no members, i.e., the null-class.

Again following the mathematical analogies, we see that "x X I =x." From the above definition of the logical product of

two classes, we see that, if x represents any class, "x X i" must represent that class which contains the individuals common to x and to 1. Hence, 1 represents the universe of conceivable objects, since this is the only class in which all the individuals in any class are contained. That is to say, any class x is contained in i.

Inclusion and Implication.

So far we have been concerned only with the calculus of classes, and we have assumed the rela tion of "inclusion in." Boole constructed his calculus by means of the relation "equals." But we shall see that equality is not fundamental, for it can be defined in terms of inclusion in. The calculus of classes can be interpreted as a calculus of propositions if the literal symbols, x, y, etc., are regarded as the class of moments at which a proposition is true, or is false. Thus "x=o" would mean "x is false"; "x= I" would mean "x is true." There are, however, certain formal differences between the calculus of classes and the calculus of propositions, which makes their com plete identification impossible. These differences cannot be dis cussed here. All that we can do is to show the bearing of this two fold interpretation upon the relations between propositions. We take as the fundamental relation "inclusion in." This is a primitive idea ; hence, it is indefinable. Its meaning, therefore, can only be indicated. The relation "is included in" is frequently represented by <, owing to its formal analogy with "is less than." It is im portant to remember that < as used in the logical calculus is an ideographic symbol and must not be translated as "less than." If now, we take x, y, etc., as representing classes, then "x