Two papers on existential graphs by Charles Peirce

Appendix: Assurance Through Reasoning (MS 669)

MS 669. 25 May to 2 June 1911. Houghton Library.

Deductive Reasoning. The word Deduction will here be used in a generalized sense to include all necessary, or mathematical Reasoning,—every reasoning of which the premisses,—or, as they will here be termed, the Copulate Premiss, having been asserted, the Conclusion cannot consistently be denied without self-contradiction; so that all that the latter asserts has really been already asserted in the former.¹¹ Although all Deduction, as I use the word, is thus Necessary Reasoning, it will, nevertheless, be convenient to speak of “Probable Deduction” as distinct from “Necessary Deduction”; the term “Probable Deduction” being used to denote any Necessary Reasoning that concludes that under stated conditions a given kind of event would have a stated Probability. Example: That any homogeneous cubical die will, at any given throw, turn up six is a chance against which the odds are 5–1. decidedly improbable. Therefore, it is very improbable that a pair of quite disconnected good dice should turn up sixes at any given throw without referring to any calculation of Probabilities.

The study of Necessary Deduction is much facilitated by expressing the Copulate Premiss, not in ordinary grammatical form, but in a sort of “diagrammatic syntax” of which an explanation shall forthwith be described, along with the terminology required in this description. A piece of paper having been taken for this use, and noun or verb written upon it is to be understood as asserting that the object, action, or state signified by what is written is actualized somewhere at some time, past, present or future. If two such words are written, both assertions are made. But if one or more of the words is surrounded by a fine line, that which would have been asserted if it had not been so enclosed is to be understood as being thereby, precisely and as a whole, denied. For example, Fig. 1 would assert that it does not both rain and thunder. Strictly it ought to mean that it either never rains or never thunders, but unless the reasoning turns on such an interpretation, it will not be worth while to be particular on such points, and when nothing to the contrary is asserted it is to be understood that all that is on any one of the sheet refers to some one place and some one time. Common sense on the part of the interpreter is supposed in this special respect, although in others a free rein must not be accorded to that useful servant to him who holds it with a firm hand.

A heavy dot before a verb will denote the perfectly indefinite subject of the verb. A similar dot after the verb will denote with the same indefiniteness its direct object; and such a dot close under the verb will usually denote the dative indirect object. But such conventions must yield to convenience. Similarly, places about other words may be appropriated, each to denote “something” in a particular relation to that which the word signifies. Any such heavy dot may be prolonged into a heavy line; and when this is done the whole line continues to denote the same identical individual. If such line joins another, the junction asserts the identity of two “somethings”. But if it be desired to draw one such “line of identity” across another without joining them, there are two ways of doing so. The first is to make a little bridge, as in Fig. 2 which may be read “Somebody is husband of somebody that loves him”. The other way is to do away with the line and use one of the usual marks of reference, such as, *, $\mathbf \dag$, $\mathbf \ddag$, ${\mathbf \Vert }$, §, ¶, as in Fig. 3.

In either case, it is to be carefully observed that this syntax is endoporeutic. This means that a “line of identity” is to be understood as lying in the outermost of the “areas” within which any part of it lies, meaning by an “area” (here and everywhere), all of the surface that lies inside and outside of precisely the same “cuts”, or fine oval lines.

Thus, Fig. 4 means “Somebody does not love some husband of hers”; Fig. 5, “Somebody is husband of whoever loves him” $=$ “Somebody is loved, if at all, only by those (or some of them), of whom he is husband”; Fig. 6, “Somebody does not love anything that is not husband to her” $=$ “Somebody has for her husband whatever there may be that she loves” $=$ “Somebody if she love anybody, it is some husband of hers that she loves”.

These two features, the finely drawn oval cut that denies as a whole, whatever it includes and the heavily drawn ligature that expresses the identity of whatever it abuts upon or is continuous with make the sum total of the essence of this Syntax. There is, therefore, no other difficulty in using it except that of knowing precisely what it is that one desires to express, without which one cannot think or do anything at all, to good purpose. There are, to be sure, a few signs besides those two, that are occasionally convenient; but they are never indispensable, and are not often wanted. The merit of this syntax is that when, by means of it, one has expressed one’s premisses, with sufficient distinctness (i.e. analytically enough), it only remains to make, according to three general permissions, suitable insertions, followed by suitable deletions, the effect of such insertions and deletions amounting only to the omission of a part of what has been asserted, and one will be able to read in what will then be on the sheet, whatever sound deductive conclusion from those premisses that one may have aimed at in the insertions and deletions. A false logic has caused the habit of speaking of the conclusion from given premisses, as if there were but one. The truth is that the number of conclusions deducible from any proposition is strictly infinite.¹² It is, therefore, necessary to determine what sort of conclusion one desires to draw or by what sort of operation one proposes to proceed before one can deduce definite conclusion.

Before going further it will be well to define a few technical expressions that have been found almost indispensable in describing the properties of this Syntax. A full glossary shall be appended to this chapter.¹³

The Phemic Sheet, or the Sheet simply, is the surface on which the premisses are to be expressed, and from which, after insertions and deletions have been made, the deductive conclusion can be read off.

To scribe is to embody an infima species of pheme, or assertional sign, by writing or drawing.

A Cut is a fine oval line. It is called a Cut because, being the only kind of sign used in this system of syntax that does not, of itself express an assertion, and is in other ways sui generis, it is not convenient to speak of it as “scribed”; and besides imagining it to be cut through the sheet, we further imagine and speak of the part of the sheet within the “Cut” as if it were turned over so that what were exposed to view were the Verso side, unless the Cut in question be itself enclosed within another, or within any other odd number of others, in which case, of course, the even number of reversals will be imagined and spoken of as exposing the Recto again within it.

By an Area is meant so much of the exposed surface as lies wholly within and without the same identical Cuts. The Area within a Cut is called the Area of that Cut; while the Area in which the Cut has been made is called the Place of that Cut.

An Area that is enclosed in an even number of Cuts, or in none at all, is said to be “evenly enclosed”. Any other is “oddly enclosed”, since no Cut is allowed to intersect another.

The word “Graph” was introduced by the still lamented William K. Clifford to mean a diagram expressive of relations by means of lines abutting upon spots, after the fashion of those employed in organic chemistry. The syntax I am describing employs Graphs modified by Cuts. They are called Existential Graphs to distinguish them from another system of logical graphs¹⁴ called Entitative Graphs. But ordinarily it will be convenient in the present essay, for the sake of abbreviating the long name Existential Graphs, the adjective is dropped. A Graph, then, as the word is used by the present when it is plain that an Existential Graph is meant, is not a sign or mark or any other existent or actual individual, but is a kind of sign

which if scribed on the Phemic Sheet (i.e. if an Instance of it stood on the Sheet) would make an assertion. The individual sign that results from the scribing of a Graph has been called an “Instance” of the Graph. This word “Instance” might conveniently be introduced into ordinary parlance. For example, only two words in our language are called articles; but one of these, the definite article, the, will commonly occur, on an average page of novel or essay, over twenty times. They are reckoned by the editor who asks for an article of so many thousand “words” as distinct words; but in fact they are only twenty or more instances of the same word; and if the editor takes any pleasure in speaking accurately he should call for an article of so many thousand “word-instances”. At any rate, it would be highly inconvenient to call “Graph-instances” Graphs.

The “line of identity” is a graph. For just as Fig. 7 asserts that “Fulbert does not love Abelard”, so Fig. 8 asserts that “Fulbert is not identical with Héloise”. But each Instance of a Graph must be either affirmative or negative, and consequently must lie wholly in one area. For that reason we must call the “ligature” of Fig. 8 not a Graph-instance, and consequently, not an Instance of the “Line of Identity”, but as a composite of three graph-instances. This gives rise to a subtile and difficult doctrine about ligatures, with which common-sense finds it hard to have patience, because in its eyes a ligature is the simplest thing in the world. Namely, only the outermost, or least enclosed, part of it signifies anything, and the rest only serves to point out the two individuals objects, each of which the identity it signifies is affirmed or denied, according as that outermost part is evenly or oddly enclosed (i.e. enclosed in an even or an odd number of cuts, zero being, of course, divisible by two and so even). We may pat common sense on the back for this facile method of interpretation; yet there is no difficulty at all in the more formally logical view that a ligature consists of as many separate graph-instances as are the areas into which parts of it enter, these different graph-instances being connected by the dot or dots where they cross a cut or cuts. At such a dot—theoretically, a mathematical point—there can be no predication, since there can be neither affirmation nor denial of that which is neither within nor without the cut. It must be borne in mind that Figs. 9, 10 are both absurd. For Fig. 9 asserts that something is not identical with anything at all—not even with itself.¹⁵ Now Fig. 11 denies, not A, B, and C, but either A or B or C. For it only denies the truth of A, B and C as one copulate. Figure 7 may therefore be read, “Fulbert is something that is something, that is something etc. that is either nothing (not even when it is), unless it be nothing except something that is nothing etc. but something that does not love anything that is something that is Abelard”; which comes to this that “Fulbert does not love Abelard”. The number of times the ligature crosses the cut is immaterial, since at a crossing it merely transmits the identity from an outer line of identity to an inner one.

It will be well to give examples of the most frequently occurring forms; and the reader is counselled to reason out for himself the interpretation of each. Figure 12 reads “something is a dog”; Fig. 13: “Nothing is a phoenix”.

Figure 14: “If it lightens, it thunders”. Strictly this should mean “If it ever lightens, it sometime thunders”. The truth is that it is very seldom requisite, in the critic of Deductive reasoning to observe distinctions of time. In order to do so, however, it would only be requisite to agree that a point on the upper part of the periphery of any graph of action or other change should denote an instant during such action or other change. Then Fig. 15 will read “It sometimes lightens and sometimes does not”.

Figure 16: “If it ever lightens it will shortly after thunder”. For were the outer cut of the figure not there, the interpretation would be “It sometime lightens without thundering at any shortly subsequent time”. Figure 17: “Every multicellular animal dies”. Figure 18: “There is a man who loves nothing but women”. Figure 19: “There is a man who loves every woman”. Figure 20: “Nothing but a man loves any woman”. Figure 21: “The only positive integer that is not higher than One is One itself”. Figure 22: “Philip is identical with whatever is Philip”; i.e. the name Philip denotes a definite Individual”¹⁶ (“Some man” is an indefinite individual. “Any man there may be” is an indeterminate individual according to the terminology herein adopted.)¹⁷

The Syntax of Existential Graphs has thus been described. The elementary signs this syntax may require are, in the first place, such as, thunders, $\bullet$man, $\bullet$lover$\bullet$,

,

, $\bullet$beauty, $\bullet$water, $\bullet$possessor$\bullet$ (as of a character or quality),

(in the dyadic relation),

(in triadic relation), $\bullet$a collection that

and $\bullet$is a collection, $\bullet$is something that includes$\bullet$, $\bullet$is something that excludes$\bullet$, $\bullet$its members are in one to one correspondence to$\bullet$.

In addition to signs of this sort, which may be multiplied indefinitely and can cannot¹⁸ be considered as constituent parts of the syntax, the only signs it so constantly requires that they may be individually regarded as almost inseparable parts of the syntax, are the following: First, the cut, or perhaps more properly, the “Scroll”, which is a pair of cuts, the one enclosing the other, so that a scroll has two areas, its “outer close” and its “inner close”. In the order of the actual mental evolution of the syntax of existential graphs, the Scroll was first adopted as a sign required before all others because it represented a necessary Reasoning, as in Fig. 23,¹⁹ which reads: “If the Copulate Premiss is true, the Conclusion is true”.

The Cut came to be thought of because of the immense frequency of occasions on which it was necessary to express the assertion “If $X$ be true, then every assertion is true”. It was forced upon the logician’s attention that a certain development of reasoning was possible before, or as if before, the concept of falsity had ever been framed, or any recognition of such a thing as a false assertion had ever taken place. Probably every human being passes through such a grade of intellectual life, which may be called the state of paradisiacal logic, when reasoning takes place but when the idea of falsity, whether in assertion or in inference, has never been recognized. But it will soon be recognized that not every assertion is true; and that once recognized, as soon as one notices that if a certain thing were true, every assertion would be true, one at once rejects the antecedent that lead to that absurd consequence. Now that conditional proposition “If A is true, every proposition is true”, is represented, in the model of Fig. 24, “If A is true, C is true” by blackening the entire inner close, as if there were no room, in reason, for any additional consequence. This gives Fig. 25: “If A be true whatever can be asserted is true”, which is as much as to say that “A is not true and the inner close being cut very small”, we get, first Fig. 26 and finally Fig. 27, in which the idea of flat falsity is first matured.

Beside the Syntax of Existential Graphs involves no other sign as essential to it except the Line of Identity, and the signs that grow out of that, such as the Point of Teridentity, where, as in Fig. 28,²⁰ a line of identity furcates, the Peg, or heavy dot which indicates that place on the periphery of a graph that denotes an individual kind of Subject of that graph (since, in this syntax, we have to recognize various other kinds of “Subjects” than the “subject nominative”, as for example, the Object Accusative, which this Syntax shows to be as much a Subject as the other, if “subject” is to have any real meaning; the Object Dative; the Instrumental Object; the Locative; and others usually in the Aryan languages expressed by adverbial phrases). There is also the place on the cut where a Ligature crosses it.

All these are so many Graphs, whose general significations are forced upon them by that of the line of identity. The only other is the bridge, which is required simply to save the trouble of pasting the two ends of a paper ribbon on the sheet to make a real bridge.

It will be acknowledged that a simpler Syntax, capable of expressing any proposition, however intricate, would be difficult to imagine. A proposition too intricate for any living human mind to grasp could be set down without ambiguity, by means of this Syntax with a very moderate amount of trouble. A proposition too intricate to be clearly expressed in a single sentence in any living tongue can be expressed without ambiguity in this syntax as soon as it is distinctly apprehended.

It now only remains to state formulate those general permissions to modify what has already been scribed which express the elementary logicality of those several forms of elementary deductive inference, out of which all other deductions can be built up. There are but two of these general illative permissions; but before stating them there is one other thing that has to be said. Namely, it is to be imagined that every graph-instance anywhere on the sheet can be freely moved about upon the sheet; and since a scroll both of whose closes are empty asserts nothing, it is to be imagined that there is an abundant store of empty scrolls on a part of the sheet that is out of sight, whence one of them can be brought into view whenever desired.²¹ What is here said ought to be reckoned as a permission, but it is not an illative permission, i.e. a permission authorizing a species of inference.

First Illative Permission: the Rule of Deletion and Insertion. From any evenly enclosed area any enclosure or other graph-instance may be deleted, even if it involves the rupture of lines of identity on that area; and upon any oddly-enclosed area any enclosure or other graph-instance may may be scribed and any connections made by lines of identity.

Second Illative Permission: the Rule of Iteration and Deiteration. Any graph may be iterated on the same area or within any number of cuts in that area, the graph-instance so inserted having identically the same connections as that which it imitates; and if a Graph occurs twice in the same area or in two areas one within a series of cuts within the area where the other occurs, and two having identically the same connections, then a deiteration may be executed by deleting the innermost of the two or if they are on the same area by deleting either of the two.

These two permissions will suffice to enable any valid deduction to be performed. The few examples that shall forthwith be given might tempt a lively mind to exclaim: Why, this syntax draws conclusions of itself, automatically. This would be extravagant; but one may say that the Syntax together with the application of the two illative permissions does so, provided it be borne in mind not overlooked that such application can only be made by a living intelligence.