Bibliography

Logical Tracts. No. 2. On Existential Graphs, Euler's Diagrams, and Logical Algebra

A Diagram is a representamen which is predominantly an icon of relations and is aided to be so by conventions. Indices are also more or less used. It should be carried out upon a perfectly consistent system of representation, one founded upon a simple and easily intelligible basic idea.

A rhema which has one blank is called a monad; a rhema of two blanks, a dyad; a rhema of three blanks, a triad; etc.

It is a subsidiary recommendation of a mode of diagrammatization, but one which ought to be accorded some weight, that it is one that the nature and habits of our minds will cause us at once to understand, without our being put to the trouble of remembering a rule that has no relation to our natural and habitual ways of expression. Certainly, no convention of representation could possess this merit in a higher degree than the plan of writing both of two assertions in order to express the truth of both. It is so very natural, that all who have ever used letters or almost any method of graphic communication have resorted to it. It seems almost unavoidable, although in my first invented system of graphs, which I call entitative graphs, propositions written on the sheet together were not understood to be independently asserted but to be alternatively asserted. The consequence was that a blank sheet instead of expressing only what was taken for granted had to be interpreted as an absurdity. One system seems to be about as good as the other, except that unnaturalness and aniconicity haunt every part of the system of entitative graphs, which is a curious example of how late a development simplicity is.

An existential graph is a logical graph governed by a. system of representation founded upon the idea that the sheet upon which it is written, as well as every portion of that sheet, represents one recognized universe, real or fictive, and that every graph drawn on that sheet, and not cut off from the main body of it by an enclosure, represents some fact existing in that universe, and represents it independently of the representation of another such fact by any other graph written upon another part of the sheet, these graphs, however, forming one composite graph.

A graph is a superficial diagram composed of the sheet upon which it is written or drawn, of spots or their equivalents, of lines of connection, and (if need be) of enclosures. The type, which it is supposed more or less to resemble, is the structural formula of the chemist.

An icon is a representamen of what it represents and for the mind that interprets it as such, by virtue of its being an immediate image, that is to say by virtue of characters which belong to it in itself as a sensible object, and which it would possess just the same were there no object in nature that it resembled, and though it never were interpreted as a sign. It is of the nature of an appearance, and as such, strictly speaking, exists only in consciousness, although for convenience in ordinary parlance and when extreme precision is not called for, we extend the term icon to the outward objects which excite in consciousness the image itself. A geometrical diagram is a good example of an icon. A pure icon can convey no positive or factual information; for it affords no assurance that there is any such thing in nature. But it is of the utmost value for enabling its interpreter to study what would be the character of such an object in case any such did exist. Geometry sufficiently illustrates that.

A pure icon can convey no positive or factual information; for it affords no assurance that there is any such thing in nature. But it is of the utmost value for enabling its interpreter to study what would be the character of such an object in case any such did exist. Geometry sufficiently illustrates that. Of a completely opposite nature is the kind of representamen termed an index. This is a real thing or fact which is a sign of its object by virtue of being connected with it as a matter of fact and by also forcibly intruding upon the mind, quite regardless of its being interpreted as a sign. It may simply serve to identify its object and assure us of its existence and presence. But very often the nature of the factual connexion of the index with its object is such as to excite in consciousness an image of some features of the object, and in that way affords evidence from which positive assurance as to truth of fact may be drawn. A photograph, for example, not only excites an image, has an appearance, but, owing to its optical connexion with the object, is evidence that that appearance corresponds to a reality.

The purpose of reasoning is to proceed from the recognition of the truth we already know to the knowledge of novel truth. This we may do by instinct or by a habit of which we are hardly conscious. But the operation is not worthy to be called reasoning unless it be deliberate, critical, self-controlled. In such genuine reasoning we are always conscious of proceeding according to a general rule which we approve. It may not be precisely formulated, but still we do think that all reasoning of that perhaps rather vaguely characterized kind will be safe. This is a doctrine of logic. We never can really reason without entertaining a logical theory. That is called our logica utens.

A logical graph is a graph representing logical relations iconically, so as to be an aid to logical analysis.

The logical universe is that object with which the utterer and the interpreter of any proposition must be well-aquainted and mutually understand each other to be well acquainted, and must understand that all their discourse refers to it.

A rhema with no blank is called a medad, and is a complete proposition.

A rhema which has one blank is called a monad; a rhema of two blanks, a dyad; a rhema of three blanks, a triad; etc.

Let a heavy dot or dash be used in place of a noun which has been erased from a proposition. A blank form of proposition produced by such erasures as can be filled, each with a proper name, to make a proposition again, is called a rhema, or, relatively to the proposition of which it is conceived to be a part, the predicate of that proposition.

A rhema of more than one blank is a relative.

Let a heavy dot or dash be used in place of a noun which has been erased from a proposition. A blank form of proposition produced by such erasures as can be filled, each with a proper name, to make a proposition again, is called a rhema, or, relatively to the proposition of which it is conceived to be a part, the predicate of that proposition.

Every rhema whose blanks may be filled by signs of ordinary individuals, but which signifies only what is true of symbols of those individuals, without any reference to qualities of sense, is termed a rhema of second intention. For second intention is thought about thought as symbol. Second intentions and certain entia rationis demand the special attention of the logician.

… in any case in which the lines of identity become too intricate to be perspicuous, it is advantageous to replace some of them by signs of a sort that in this system are called selectives. A selective is very much of the same nature as a proper name; for it denotes an individual and its outermost occurrence denotes a wholly indesignate individual of a certain category (generally a thing) existing in the universe, just as a proper name, on the first occasion of hearing it, conveys no more. But, just as on any subsequent hearing of a proper name, the hearer identifies it with that individual concerning which he has some information, so all occurrences of the selective other than the outermost must be understood to denote that identical individual. If, however, the outermost occurrence of any given selective is oddly enclosed, then, on that first occurrence the selective will refer to any individual whom the interpreter may choose, and in all other occurrences to the same individual.

What we have to do […] is to form a perfectly consistent method of expressing any assertion diagrammatically. The diagram must then evidently be something that we can see and contemplate. Now what we see appears spread out as upon a sheet. Consequently our diagram must be drawn upon a sheet. We must appropriate a sheet to the purpose, and the diagram drawn or written on the sheet is to express an assertion. We can, then, approximately call this sheet our sheet of assertion.

A certain sheet, called the sheet of assertion, is appropriated to the drawing upon it of such graphs that whatever may be at any time drawn upon it, called the entire graph, shall be regarded as expressing an assertion by an imaginary person, called the graphist, concerning a universe, perfectly definite and entirely determinate, but the arbitrary creation of an imaginary mind, called the grapheus.

Every symbol is an ens rationis, because it consists in a habit, in a regularity; now every regularity consists in the future conditional occurrence of facts not themselves that regularity.

A symbol is a representamen whose special significance or fitness to represent just what it does represent lies in nothing but the very fact of there being a habit, disposition, or other effective general rule that it will be so interpreted. Take, for example, the word “man.” These three letters are not in the least like a man; nor is the sound with which they are associated. Neither is the word existentially connected with any man as an index. It cannot be so, since the word is not an existence at all. The word does not consist of three films of ink. If the word “man” occurs hundreds of times in a book of which myriads of copies are printed, all those millions of triplets of patches of ink are embodiments of one and the same word. I call each of those embodiments a replica of the symbol. This shows that the word is not a thing. What is its nature? It consists in the really working general rule that three such patches seen by a person who knows English will effect his conduct and thoughts according to a rule.

A rhema which has one blank is called a monad; a rhema of two blanks, a dyad; a rhema of three blanks, a triad; etc.