To be completely formal is to be completely general. Consequently there are various degrees of approximation to this ideal of form. Aristotle's theory of the syllogism is the earliest attempt to exhibit the formal principles of deduction. It suffers from three main defects : (I) its restric tion to a single mode of deduction, the syllogism; (2) its failure to symbolize the relations involved; (3) its defective analysis of these relations. The recognition of non-syllogistic modes of deduc tion, combined with the attempt to use symbols as the basis of operations, led to rapid elaboration of symbolic systems ; whilst the distinction and analysis of the different relations employed in deduction made possible that logistic development of mathematics that marks the culmination of the ideal of form.
It is not difficult to see that the familiar syllogism : "All men are mortal, Socrates is a man, therefore Socrates is mortal" is not a logically simple form. Nor did even the traditional logic assert the conclusion but only the fact that the premises jointly imply the conclusion. It must then, first, be restated in the form : "If all men are mortal and Socrates is a man, then Socrates is mortal." Now it is clear that the validity of the reasoning does not depend upon the fact that Socrates is mentioned. It would do just as well if we substituted Plato, or Newton, or any other individual. Thus what we are asserting is a relation between "being a man" and "being a mortal." We can, therefore, replace "Socrates" by a symbol, x, denoting an undetermined individual. Such a symbol is called a variable. Let us write for the second premise "2 is a man." Here x denotes an empty place to be filled with an appropriate individual. Such an expression as "x is a man" is called a proposi tional function. It becomes a proposition when the name of an individual is substituted for 2; it will be a true proposition if the individual is human. A similar analysis applies to "Socrates is mortal" which is of the same form. But "all men are mortal" is not of the same form, since it asserts a relation between all the individuals that are human and all those that are mortal. It is thus a compound proposition and requires analysis into "
is a man' implies is mortal' no matter what x may be." Thus we see that the copula "are" in the major premise denotes a different relation from the copula "is" in the minor premise. The difference may be provisionally stated by saying that "all men are mortal" asserts a relation between the class men and the class mortals, whereas "Socrates is a man" asserts that a given individual belongs to a class. Confusion between these relations is hardly to be avoided so long as we rely for our analysis upon the verbal expression of these propositions. The separation between these two forms can
not be clearly effected until an adequate symbolism is devised. Such a symbolism will, at the same time, afford a basis of operations.
The canonical forms of the Aristote lian syllogism reveal an attempt to employ symbols to test the validity of the reasoning involved. We have seen that these sym bols were inadequate so that the forms were not completely analysed. The purpose of a calculus is to state the premises in ideographic symbols in such a manner that all the conclusions implied can be drawn in accordance with rules of transformation analogous to those of mathematical operations. Thus thought is economized and accuracy ensured. In the first attempts to develop an adequate symbolism certain formal analogies with mathematical operations were stressed, sometimes with unfortunate results. The foundations of such an algebra of logic were laid down by George Boole (1815-64) ; it has been subsequently developed by Ernst Schroeder (1841-1902), Charles Saunders Peirce
Louis Couturat (1868-1914) and Mrs. Ladd-Franklin. Its trans formation into an instrument for the logical analysis of mathe matical concepts was not achieved until Peano invented a non algebraic notation. This development cannot be followed in detail. All that can be attempted here is to give some account of certain fundamental notions employed in the subsequent logical analysis. We begin with the unanalysed concept of a class as a set of indi vidual objects. Let us select from the universe of conceivable objects all those that are lawyers. This is the class "lawyers." In the same way, select the class "knights." It is assumed that the combination of two classes will itself be a class. There are two essential modes of combining classes, represented by the conjunc tions "and" and "or." Thus we may form the class of those ob jects that are "lawyers and knights." This is the class "knighted lawyers." Obviously this mode of combination is analogous to mathematical multiplication. Hence, "knighted lawyers" is called the logical product of "knights" and "lawyers." Again, we may form the class of those objects that are either "knights" or "law yers"; or of those who are either "solicitors" or are "barristers" and so on. Thus for example we obtain the class "solicitors or barristers." This mode of operation is obviously analogous to mathematical addition. Hence the class "solicitors or barristers" is called the logical sum of "solicitors" and "barristers." These two operations may be defined as follows: The logical product of two classes is the class included in each of them and including every class included in each.