I do not think that there is any one ultimate analysis of logical relations which, from a purely logical standpoint, excluding all psychological considerations, can be said to be the true analysis to the exclusion of all others. I think, on the contrary, as in mathematics an imaginary quantity may equally be defined by its modulus and argument, as RG²ⁱ, or by its real and pure imaginary part, as X + Yi, where the number of the coordinates is alone invariable, so in my opinion when the minimum number relations has been assumed as fundamental, there is absolutely no purely logical criterion to determine which set is simpler than all others. This appears to me evident.

My system of existential graphs is, I hold, as good <pb n=20> [1]an analysis as any, and is the only satisfactory analysis that I have seen, except one substantially the same given by myself. In existential graphs, we have the following signs: 1^st, a sheet upon which every proposition scribed (so I say, since the symbols used may not be written rather than drawn,) is understood to be asserted; 2^nd, two states of things expressed by propositions scribed simultaneously on the sheet are understood to be coexistent; 3^rd, a heavy dot scribed upon the sheet, whether alone, or as a part of a heavy line, or attached to a relative term, is understood to assert ‘something exists’; 4^th, a heavy line, however long or short, is understood to assert the identity of the individuals represented by its points, and that of the corresponding subjects of the propositions upon which it abuts. Thus, <pb n=21>

Fig. 1	Fig. 2	Fig. 3

Fig. 1 supposed to be on the sheet of assertion, asserts ‘Something is a man’, Fig. 2 ‘there is something that is a king’, Fig. 3 ‘there is a man that is a king.’ 5^thly, a finely drawn oval on the sheet of assertion denies the truth of what is scribed upon its enclosed area, as a whole; and if a heavy line crosses from the outside to the inside of the oval, or even touches the oval from the inside, the effect is that the denial is limited in its application to the individual denoted by the point or points of that line that reach the oval.

Thus Fig. 4 asserts that there is a king and denies that

Fig. 4	Fig. 5	Fig. 6

there is any man, Fig 5, asserts that there is a king and that there is something that is not a man. Fig. 6 there is something that is a king and is identical with something that is not a man. <pb n=22>

Fig. 7	Fig. 8	Fig. 9

Fig 7 asserts that there are more than one man. Fig. 8 asserts that everything is identical with something and is other than something.

The system as thus far developed is called the Beta development, the Alpha development being the system without the lines of identity. A Gamma development is required. For example, Fig. 8 does not express the principle of identity since it merely asserts, as a fact, that everything is identical with itself. The Beta development is limited to the expression of propositions de inesse.

In order to represent the reasoning of mathematics in all its fullness, two improvements must be made. In the first place the operation of hypostatic abstraction must be rendered practicable. This has been the object of jeers since Molière died [?] in enacting the recipiendaire in the 3^me Intermède of the <pb n=23> Malade Imaginaire

Nevertheless, this operation of clipping the wings of words, as it may be called, if definite terms are called epea  apteroenta and indefinite epea pteroenta. ‘Solomon is wise’. Here ‘Solomon’ is definite in its denotation; ‘wise’ is indefinite, for it is a wise man, some wise man, not in itself specifying which among wise men. But change it to ‘Solomon is a possessor of something that is wisdom, and though possessor is in itself vague, both as to its subject nominative and its genitive, yet wisdom is the name of a single <pb n=24> quality, a proper name of an ens rationis, whatever that may be,—apparently it is something which does not affect the senses but is discovered by thought. I suppose, then, a man is an ens rationis,—I mean the man himself, not his carcase. The operation of hypostatic abstraction consists in substituting for any predication that of possessing a quality, or that of standing in a relation. It [?] is so that in mathematics an operation is itself made the subject of operations. The hypostatically abstract quality or relation is necessarily general, that is, not subject to the principle of excluded middle. The universe of characters is thereby quite outside the universe of existents.

The second needed improvement is that there should be some way of expressing collective wholes. From a collective assertion one can pass to a distributive general, provided the collective assertion were universal. The reverse <pb n=25> transformations can be effected under certain conditions. The collection of Spaniards is an adorer of the whole collection of women, provided every Spaniard adores a woman and every woman is adored by a Spaniard. That every Spaniard adores a woman shows that all Spaniards are adorers of women, but not that all Spaniards are adorers of a woman. It is sometimes desirable to speak of a collection of Spaniards, and at other times, though rarely except in the theory of numbers, to speak of the collection of whatever Spaniards there may be. Usually, a collective is indefinite. It always refers to existents, for possibilities are not units of collections. In the rare cases in which we wish to say, for example, that the number of all Spaniards is divisible by seven, we can say that there is a collection of Spaniards such that there is no exclusive collection of Spaniards which is a <pb n=26> collection of seven collections of Spaniards. The collective certainly refers exclusively to what is, not to what might be, and it is best taken as indefinite, like any ordinary class-name, as a Spaniard.

A collection is an object whose mode of existence is that of an indefinite object, that is not in all respects subject to the principle of contradiction whose existence consists in the existence of certain objects, its members, which are such that the existence of none of them consists in the existence of the collection.

Representation, by which I mean the function of a sign in general, is a combinant, or trifile, relation; since it subsists between the sign, the object represented, and the interpretant or sign of the same object determined by the sign in the mind of the person addressed, or in other field of signification. The object is something external to and independent of the sign which determines in the sign an element corresponding to itself; so that we have to distinguish the quasi-real object from the presented object; or some may say, the external from the internal object, and the extrous object as it is in itself is to be distinguished from the feature of the external object that is represented. The interpretant is created by the sign; and since the sign as such determines the interpretant, it is in some sense represented in the sign, that is, it is called up by the sign. While in itself it is acted on by the sign, it is so acted upon as to represent the sign to be a sign of the object. Thus, while the object is bifiss, the <pb n=29> interpretant is trifiss or trifissible. It is very easy to distinguish the interpretant as actually acted upon from the interpretant as announced in the sign or as representing the sign to be a sign; but it is not so easy sharply to distinguish these two from each other. In the actual interpretant  has to be divided  into the actual interpretant in these features in which it is a determination of the field of representation & the actual interpretant in those features in which it is acted on by the sign. The representative interpretant is either the interpretant as the sign designs [?] it to represent the sign to be related to its object, the interpretant as it actually does represent the sign to be related to its object, and the interpretant is it ought to represent the sign to be related to its object. For example, a proposition or other asserting sign, such as a portrait with a legend telling of whom it is a portrait, which I call a dicisign, in itself as asserting, intends its interpretant to represent its immediate or internal object as being really related to its external object <pb n=30> so that its representation is determined as a brute compulsory effect of the real state of things;–"brute," I mean, in so far as no reason for it is presented. An argument, on the other hand, intends its representative interpretant to represent its immediate object, not now as a brute effect, but as a sign of its external object. The dicisign is accordingly obliged to represent its external object twice over, once (in the subject) to distinguish it in itself from other objects, and again (in the predicate) in its represented characters. The argument is in a similar way obliged to address its representative interpretant in three ways; namely, in the premisses, especially in the minor premiss, as informing it, as a dicisign does, again in the suggested principle of the reasoning, often expressed as the major premiss, to recall to present representation the interpretant’s own representation of the represented characters of the external object, and finally, in the conclusion, in an appeal to it to represent how the premisses ought to be interpreted as a sign. <pb n=31> Of course, such distinctions might be traced out much further but I limit myself to the distinctions which are needed in present study.

Page 23 verso:

I use these same symbols for characters of characters and for characters of characters of characters, but in the diagrams in different colors

Thus

will mean that there are five objects [red ligatures - jz] between which there are three

triadic relations [instantiated as brown ligatures - jz], one of the objects [red] being the second correlate of all three relations, two the third correlate of the first and first correlate of the second and third correlate of the second and first of the third while the first correlate of the first and third of the third are different individuals; and the first triadic relation [brown] has one relation to the second and the second a relation to the third these two dyadic relations [instantiated as the pair of black ligatures on the right of the ^1's - jz]  having a common quality [the black ligature on the left, connected to the "q" - jz].

The notation is now nearly complete. I use a dotted enclosure to denote the single character which consists in the logical possibility of the rhema written within it. <insert>provided a ^ be written whose hooks it comes rend to the ^ to which it might [?] attached [?out side]</insert> It is the name of a single character; analogant to a proper name. <xout>Thus the graph

is equivalent to

</xout>

Note:

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