3°: Symbols which also independently determine their interpretants, and thus the minds to which they appeal, by premising a proposition or propositions which such a mind is to admit. These are arguments.
And it is remarkable that, among all the definitions of the proposition, for example, as the oratio indicativa, as the subsumption of an object under a concept, as the expression of the relation of two concepts, and as the indication of the mutable ground of appearance, there is, perhaps, not one in which the conception of reference to an object or correlate is not the important one. In the same way, the conception of reference to an interpretant or third, is always prominent in the definitions of argument.
In a proposition, the term which separately indicates the object of the symbol is termed the subject, and that which indicates the ground is termed the predicate. The objects indicated by the subject (which are always potentially a plurality,—at least, of phases or appearances) are therefore stated by the proposition to be related to one another on the ground of the character indicated by the predicate. Now this relation may be either a concurrence or an opposition. Propositions of concurrence are those which are usually considered in logic; but I have shown in a paper upon the classification of arguments that it is also necessary to consider separately propositions of opposition, if we are to take account of such arguments as the following:—
Whatever is the half of anything is less than that of which it is the half;
A is half of B:
∴ A is less than B.
The subject of such a proposition is separated into two terms, a “subject nominative” and an “object accusative.”
In an argument, the premises form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premises may afford a likeness, index, or symbol of the conclusion. In deductive argument, the conclusion is represented by the premises as by a general sign under which it is contained. In hypotheses, something like the conclusion is proved, that is, the premises form a likeness of the conclusion. Take, for example, the following argument:—
M is, for instance, P′, P″, P‴ and Piv;
S is P′, P″, P‴, and Piv:
∴ S is M.
Here the first premise amounts to this, that “P′, P″, P‴, and Piv” is a likeness of M, and thus the premises are or represent a likeness of the conclusion. That it is different with induction another example will show.
S′, S″ S″, and Siv are taken as samples of the collection M;
S′, S″, S‴, and Siv are P:
∴ All M is P.
Hence the first premise amounts to saying that “S′, S″, S‴, and Siv” is an index of M. Hence the premises are an index of the conclusion.
The other divisions of terms, propositions, and arguments arise from the distinction of extension and comprehension. I propose to treat this subject in a subsequent paper. But I will so far anticipate that, as to say that there is, first, the direct reference of a symbol to its objects, or its denotation; second, the reference of the symbol to its ground, through its object, that is, its reference to the common characters of its objects, or its connotation; and third, its reference to its interpretants through its object, that is, its reference to all the synthetical propositions in which its objects in common are subject or predicate, and this I term the information it embodies. And as every addition to what it denotes, or to what it connotes, is effected by means of a distinct proposition of this kind, it follows that the extension and comprehension of a term are in an inverse relation, as long as the information remains the same, and that every increase of information is accompanied by an increase of one or other of these two quantities. It may be observed that extension and comprehension are very often taken in other senses in which this last proposition is not true.
This is an imperfect view of the application which the conceptions which, according to our analysis, are the most fundamental ones find in the sphere of logic. It is believed, however, that it is sufficient to show that at least something may be usefully suggested by considering this science in this light.
1. This agrees with the author of De Generi bus et Speciebus, Ouvrages Inédits d’Abélard, p. 528.
2. Herbart says: “Unsre sämmtlichen Gedanken lassen sich von zwei Seiten betrachten; theils als Thätigkeiten unseres Geistes, theils in Hinsicht dessen, was durch sie gedacht wird. In letzterer Beziehung heissen sie Begriffe, welches Wort, indem es das Begriffene bezeichnet, zu abstrahiren gebietet von der Art und Weise, wie wir den Gedanken empfangen, produciren, oder reproduciren mögen.” But the whole difference between a concept and an external sign lies in these respects which logic ought, according to Herbart, to abstract from.
Upon the Logic of Mathematics
P 33: Presented (by title) 10 September 1867
PART I
The object of the present paper is to show that there are certain general propositions from which the truths of mathematics follow syllogistically, and that these propositions may be taken as definitions of the objects under the consideration of the mathematician without involving any assumption in reference to experience or intuition. That there actually are such objects in experience or pure intuition is not in itself a part of pure mathematics.
Let us first turn our attention to the logical calculus of Boole. I have shown in a previous communication to the Academy, that this calculus involves eight operations, viz. Logical Addition, Arithmetical Addition, Logical Multiplication, Arithmetical Multiplication, and the processes inverse to these.
Definitions
1. Identity,
2. Logical Addition,
3. Logical Multiplication. a,b denotes only whatever is both a and b.
4. Zero denotes nothing, or the class without extent, by which we mean that if a is any member of any class,
5. Unity, denotes being, or the class without content, by which we mean that, if a is a member of any class, a is a,1.
6. Arithmetical Addition, a + b, if
7. Arithmetical Multiplication, ab represents an event when a and b are events only if these events are independent of each other, in which case