# Existential Graphs

### MS 514 by Charles Sanders Peirce

with commentary by John F. Sowa

Peirce wrote MS 514 as a tutorial on existential graphs, their rules of inference, and related topics in logic. The original manuscript MS 514 is located in the Houghton Library, Harvard University, and this version was transcribed by Michel Balat. The published version can be found in Peirce's New Elements of Mathematics, edited by Carolyn Eisele, vol 3, pp. 162 to 169.

In this commentary, Peirce’s words (either from MS 514 or in quotations from other sources) are printed in black, and the commentary is printed in red. The numbers highlighted in blue have hyperlinks to the bibliography. The text is essentially unchanged, some of the more obvious errors in spelling and transcription have been corrected, and some long stretches of text have been broken into paragraphs. The transcription begins at the bottom of page 9 of MS 514; page breaks in the manuscript are marked in brackets, such as [p.10].

For other views of Peirce’s contributions to logic, see “Peirce the Logician” by Hilary Putnam or articles by Quine (1995), Dipert (1995), and Hintikka (1997). Peirce begins with a summary of his own work:

One of my earliest works was an enlargement of Boole’s idea so as to take into account ideas [p.10] of relation, — or at least of all ideas of existential relation. By an existential relation I mean any relation, R, such that anything that is R to x (where x is some particular kind of object) is nonexistent in case x is nonexistent. Thus lovers of women of bright green complexions are nonexistent in case there are no such women.

George Boole (1847, 1854) applied his algebra to propositions, sets, and monadic predicates. The expression p×q, for example, could represent the conjunction of two propositions, the intersection of two sets, or the conjunction of two monadic predicates. With his algebra of dyadic relations, Peirce (1870) made the first major breakthrough in extending symbolic logic to predicates with two arguments (or subjects, as he called them). With that notation, he could represent expressions such as "lovers of women with bright green complexions". That version of the relational algebra was developed further by Ted Codd (1970, 1971), who earned his PhD under Arthur Burks, the editor of volumes 7 and 8 of Peirce’s Collected Papers. At IBM, Codd promoted relational algebra as the foundation for database systems, a version of which was adopted for the query language SQL, which is used in all relational database systems today. Like Peirce’s version, Codd’s relational algebra and the SQL language leave the existential quantifier implicit and require a double negation to express universal quantification.

I invented several different systems of signs to deal with relations. One of them is called the general algebra of relations, and another the algebra of dyadic relations. I was finally led to prefer what I call a diagrammatic syntax. It is a way of setting down on paper any assertion, however intricate, and if one so sets down any premises, and then (guided by 3 simple [p.11] rules) makes erasures and insertions, he will read before his eyes a necessary conclusion from premises.

In other writings, Peirce discussed his concept of diagrammatic reasoning, which is best illustrated by the rules of inference and model theory that he developed for existential graphs:
By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have the same results, and expresses this in general terms. (NEM 4:47-48)

From 1880 to 1885, Peirce developed his general algebra of relations, which Giuseppe Peano (1889) adopted as the basis for the modern notation of predicate calculus. Gottlob Frege (1879) had developed an equivalent notation for first-order logic, which he called the Begriffsschrift (concept writing), but no one else ever used it.

Peirce experimented with relational graphs as early as 1882, but those graphs couldn’t express all possible combinations of Boolean operators, quantifiers, and their scope. His entitative graphs of 1896, which were based on disjunction, negation, and universal quantifiers, were the first graphs that had the full expressive power of the algebraic notation for first-order logic with equality. In 1897, he switched to the dual form, existential graphs, which were based on conjunction, negation, and existential quantifiers. Peirce continued to develop versions of existential graphs for modal logic and higher-order logic until his death in 1914. In this commentary, "EG" will be used as an abbreviation for existential rather than entitative graphs.

This syntax is so simple that I will describe it. Every word makes an assertion. Thus, —man means "there is a man" in whatever universe the whole sheet offers it. The dash before "man" is the "line of identity".

this means "Some man eats a man".

There are two lines of identity in this graph: the curve on the left, and the straight line on the right. In the algebraic notation, each line of identity corresponds to an existential quantifier, which Peirce represented by the Greek letter Σ. The graph could therefore be represented by the following formula in Peirce’s notation of 1885:
```Σx Σy (manx × many × eatsx,y).
```
The modern notation for predicate calculus is based on Peirce’s notation. Since Peano wanted to use logic to represent mathematics, he replaced Peirce’s symbols with new symbols that could be freely mixed with mathematical symbols in the same formulas. To form the logical symbols, Peano began the practice of turning letters upside down or backwards. The following table lists Peirce’s symbols and Peano’s replacements:

Peirce’s Notation Peano’s Notation
Operation Symbol Explanation Symbol Explanation
Disjunction + Logical sum v for vel
Conjunction × Logical product Upside down v
Negation −1=0 and −0=1 ~ Curly minus sign
Implication −< Equal or less than  C for consequentia
Existential Quantifier Σ Iterated sum E for existere
Universal Quantifier Π Iterated product (  ) O for omnis

For his logical algebra, Boole used 1 for truth and 0 for falsehood, and he chose the symbols +, ×, and − to represent disjunction, conjunction, and negation. The bottom three lines of the table are Peirce’s innovations:

• Implication: Peirce observed that if p implies q, then q must always be true when p is true, but q might also be true for some reason independent of p. Therefore, the truth value of p is always less than or equal to the truth value of q. Instead of using the symbol ≤, which combines two operators, Peirce invented the claw symbol −< because it suggests a single indivisible operation.

• Existential quantifier: In Boolean algebra, 1+1=1. Therefore, Peirce adopted Σ to indicate a logical summation of any number of terms, which would be true if at least one of the terms happened to be true.

• Universal quantifier: He aslo adopted Π to indicate a logical product of any number of terms, which would only be true if every one of the terms happened to be true.
In 1880, Peirce began to use the symbols Π and Σ, which he called quantifiers, shortly after Frege (1879) had independently developed his Begriffsschrift (concept writing). A few years later, Peirce (1885) published complete rules of inference for first-order logic by adding the rules for quantifiers to the usual rules for Boolean algebra. For the next 30 years, Frege’s work was largely ignored. In Germany, Ernst Schröder (1890-95) adopted Peirce’s notation, which was used for most work on logic for over 20 years. In Italy, the logicians followed Peano, who declared Frege’s notation to be unreadable. In England, Bertrand Russell praised Frege, but adopted the Peirce-Peano notation, which came to be called Peano-Russell notation.

In the commonly used Peirce-Peano-[Russell] notation, Peirce’s example "Some man eats a man" would be expressed in the following formula:

```(∃x)(∃y)(man(x) ∧ man(y) ∧ eats(x,y)).
```
Since most readers are likely to be more familiar with Peano’s symbols, they will be used in the remainder of this commentary.
To deny that there is any phoenix, we shade that assertion which we deny as a whole:

Thus what I have just scribed means "It is false that there is a phoenix".

Without the shading, the graph —phoenix would assert that there exists a phoenix. Shading has the effect of negating the shaded graph. Figure 1 may be translated to the following formula:
```~(∃x)phoenix(x).
```
To indicate negation in his original version of EGs, Peirce used an unshaded oval enclosure, which he called a cut because it separated the sheet of assertion into a positive (outer) area and a negative (inner) area. In this version, he added shading to highlight the distinction between positive and negative areas: any area inside an odd number of ovals is shaded (negative), and any area inside an even number of ovals (possibly zero) is unshaded (positive).
But the following:

only means "There is something that is not identical with any phoenix".

In Figure 1, the entire graph was negated, but in Figure 2, part of the line of identity is outside the negation. When a line of identity crosses one or more negations, the corresponding existential quantifier is asserted in the outermost area in which the line occurs. Unlike Figure 1, which asserts the negation before the quantifier, Figure 2 asserts the quantifier before the negation:
```(∃x)~phoenix(x).
```
This formula says that there exists something x, which is not a phoenix. When an oval is drawn inside another oval, the doubly nested area is positive (unshaded), as in Figure 3:

Fig. 3 denies fig. 4, which asserts that it thunders without lightening. For a denial shades the unshaded and unshades the shaded. Consequently fig. 3 means "If it thunders, it lightens".

Figure 4 may be translated to the following formula:
```(∃x)(thunder(x) ∧ ~lightening(x)).
```
This formula says that there exists something x, which is thunder and not lightening. Figure 3 adds one more negation to the front of the formula:
```~(∃x)(thunder(x) ∧ ~lightening(x)).
```
This formula says that it is false that there exists something x, which thunders and not lightens. By Peirce’s 1885 rules for quantifiers, the negation at the front of the formula can be moved after the existential quantifier provided that the existential quantifier (Σ in Peirce’s notation or ∃ in Peano’s notaion) is converted to a universal quantifier (Π in Peirce’s notation, ( ) in Peano’s notation, or ∀, which the German logicians introduced to represent alle):
```(∀x)~(thunder(x) ∧ ~lightening(x)).
```
This formula says that for any x, it is false that x thunders and not lightens. By Peirce’s definition of the implication operator, ~(p ∧ ~q) is equivalent to pq; therefore, the part of the formula following the quantifier can be rewritten as an implication:
```(∀x)(thunder(x) ⊃ lightening(x)).
```
This formula says that for every x, if x thunders, then x lightens. Peirce, however, preferred to treat a nest of an unshaded oval inside a shaded oval as a single implication, which he read "If it thunders, it lightens."

[p.12] In order to make the lines of identity in their connexion with shading and its absence perfectly perspicuous, I must provide you with a bit or two of nomenclature. By an "area", then, I mean the whole of any continuous part of the surface on which graphs are scribed that is alike in all part of it either shaded or unshaded. By a "graph" I mean the way in which a given assertion is scribed. It is the general kind not a single instance. For example there is in English but one single "word" that serves as definite article. It is the word "the". It will occur some twenty or more time on an average page; and when an editor asks for an article of so many thousand of "words" he means to count each of those instances as a distinct word. He speaks loosely of instances of words as words, which they are not.

In this paragraph, Peirce draws a distinction between a graph as a general type and particular occurrences or instances of a graph. This distinction is important when Peirce talks about the rules of inference, which he called permissions because they state the conditions in which it is permissible to insert or erase an instance of a graph in any particular area.
Now in like manner a graph is one thing, and a "graph instance" is another thing. Any expression of an assertion in this particular diagrammatic syntax is an Existential Graph, of which I use the single word "graph" as a common abbreviation as long as I have nothing to do with another kind of graph. A graph then may be complex or indivisible. Thus

is a graph instance composed of instances of three indivisible graphs which assert "there is a male" "there is something human" and "there is an African". The syntactic junction or point of teridentity asserts the identity of something denoted by all three.

In modern terminology, Peirce’s indivisible graphs are called atoms. In predicate calculus, each atom consists of a single predicate with its associated arguments, which Peirce called logical subjects. In EGs, each predicate is represented by a character string such as "male", "human", or "African", and each argument or subject is represented by a line called a peg. By itself, a peg represents an existentially quantified variable, and a line of identity that connects two or more pegs corresponds to an equal sign "=" between the corresponding variables. Therefore, the above graph may be translated to the following formula:
```(∃x)(∃y)(∃z)
(male(x) ∧ human(y) ∧ African(z) ∧ x=y ∧ y=z)
```
By Peirce’s rules of 1885, this formula may be simplified by substituting one variable name for another when they are connected by an equal sign:
```(∃x)(male(x) ∧ human(x) ∧ African(x))
```
With the graph notation, the lines of identity have no names. Therefore, the rules of inference are much simpler because there is no need for equal signs between variables or rules for substituting one variable name for another.
Indivisible graphs usually carry "pegs" which are places on their periphery appropriated to denote, each of them, one of the subjects of the graph. A graph like "thunders" is called a "medad" as having no peg (though one might have made it mean "some time it thunders" when it would require a peg.

A graph or graph instance having 0 peg is medad.

A graph or graph instance having 1 peg is monad.

A graph or graph instance having 2 pegs is dyad.

A graph or graph instance having 3 pegs is triad.

A medad, which has no arguments, corresponds to a proposition in propositional logic (the name comes from the Greek me for not). A monad corresponds to a monadic predicate, which is sometimes called a property; a dyad corresponds to a dyadic predicate or a binary relation; and a triad corresponds to a triadic predicate or a ternary relation. In general, the n pegs of an n-adic predicate or n-ary relation are ordered from left to right according to their point of attachment to the character string that names the predicate or relation.

Note that by treating a medad as a special case of a predicate or relation, Peirce avoids the need to distinguish propositional logic from predicate logic. In earlier writings, he presented existential graphs in three parts: Alpha is the theory of propositional logic, which avoids any use of lines of identity; Beta is the theory of relations, which introduces lines of identity; and Gamma is the theory of modal and higher-order logic, which uses colors and specially marked boundaries to indicate which oval enclosures represent modalities other than negation. In MS 514, Peirce combines Alpha and Beta in a unified presentation of first-order logic with equality.

Every indivisible graph instance must [p.14] be wholly contained in a single area. The line of identity can be regarded as a graph composed of any number of dyads "—is—" or as a single dyad. But it must be wholly in one area. Yet it may abut upon another line of identity in another area.

The option of treating a line of identity as a concatenation of dyads of the form —is— can often simplify the translation of an EG into English or the algebraic notation for logic. The next two graphs, for example, are exactly equivalent:

In the graph on the left, the shaded area negates the connection between the lines of identity on either side. To emphasize what is being negated, the graph on the right replaces the line in the middle with the dyad —is—. Therefore, the graph may be read "Something is not something," which corresponds to the following formula:

```(∃x)(∃y)x≠y.
```
This formula and the corresponding EG imply that there exist at least two things. In the following example, the graph on the left says there exist at least three things, the one in the middle says there exist at most three things, and the one on the right says there exist exactly three things:

The formula for the graph on the left would say that there exist an x, a y, and a z, each of which is distinct from the other two:

```(∃x)(∃y)(∃z)
(x≠y ∧ y≠z ∧ z≠x).
```
The formula for the graph in the middle would say that there exist an x, a y, and a z, which may or may not be distinct, and it is false that there exists a w, which is distinct from x, distinct from y, and distinct from z:
```(∃x)(∃y)(∃z)~(∃w)
(w≠y ∧ w≠y ∧ w≠z).
```
The graph on the right, which is derived by overlaying copies of the other two, says that there exist at least three things and at most three things. The corresponding formula can be derived from the preceding two formulas with some additional editing:
```(∃x)(∃y)(∃z)
(x≠y ∧ y≠z ∧ z≠x
∧ ~(∃w)(w≠y ∧ w≠y ∧ w≠z)).
```
This example illustrates some properties of existential graphs that are not true of the algebraic notation: an EG concerning a symmetric subject can be drawn in a way that preserves the symmetry (although it might have to be drawn in more than two dimensions in order to avoid crossing lines); and conjunctions of EGs can be asserted by copying and overlaying the original EGs without adding, deleting, or rearranging any other symbols.

Thus fig. 5 denies that there is a man that will not die, that is, it asserts that every man (if there be such an animal) will die. It contains two lines of identity [the part in the shaded area and the part in the unshaded area].

It denies which fig. 6 asserts, "there is a man that is something that is something that is not anything that is anything unless it be something that will not die". I state the meaning in this way, to show how the identity is continuous regardless of shading; and this is necessarily the case. In the nature of identity that is its entire meaning. For the shading denies the whole of what is in its area but not each part except disjunctively.

The preferred reading of Figure 6, "There is a man who will not die", corresponds to the following formula:
```(∃x)(man(x) ∧ ~willDie(x))
```
Peirce used the more convoluted reading of Figure 6 to illustrate his point that a line of identity may be considered as a concatenation of arbitrarily many dyads of the form —is—. The following graph uses four instances of "is" to break a line of identity into five segments:

man—is—is—is—is—will die

Each of the five segments of the line of identity corresponds to an existentially quantified variable, and each instance of the dyad —is— corresponds to an equal sign between two variables:

```(∃x)(∃y)(∃z)(∃u)(∃v)
(man(x) ∧ x=y ∧ y=z ∧ z=u ∧ u=v ∧ willDie(v))
```
Peirce’s method of reading such graphs would produce the sentence "There is a man that is something that is something that is something that is something that will die." To form the equivalent of Figure 6, suppose that the second instance of "is" and everything following it occurs inside the shaded area. The resulting graph would correspond to the following formula:
```(∃x)(∃y)(man(x) ∧ x=y ∧
~(∃z)(∃u)(∃v)
(y=z ∧ z=u ∧ u=v ∧ willDie(v)))
```
The second and third lines of this formula expresses Peirce’s phrase, "not anything that is anything unless it be something that will not die". The first two instances of "anything" correspond to the negated existential variables z and u. Since the word "unless" contains an implicit negation, he read the last variable v as "something". To cancel the effect of the implicit negation in "unless", he inserted another negation in the phrase "will not die". In the following reading, Peirce illustrates the point that two or more subgraphs in a negative (shaded) area may be read as a disjunction of negations.
Fig. 6 may be read "there is a man that is identical with something that is not identical with anything or only with something that is not identical with anything unless it will not die".
This reading can be derived by converting the previous formula according to Peirce’s 1885 rules of inference. First, move the negation in the second line of the formula inward across the three existential quantifiers. That move has the effect of converting the existential quantifiers to universals:
```(∃x)(∃y)(man(x) ∧ x=y ∧
(∀z)(∀u)(∀v)
~(y=z ∧ z=u ∧ u=v ∧ willDie(v)))
```
Then apply DeMorgan’s rule that a negation of a conjunction is equivalent to a disjunction of negations:
```(∃x)(∃y)(man(x) ∧ x=y ∧
(∀z)(∀u)(∀v)
(~y=z ∨ ~z=u ∨ ~u=v ∨ ~willDie(v)))
```
The variables in this formula can be appended as annotations to the corresponding nouns and pronouns in Peirce’s sentence: "there is a man x that is identical with something y that is not identical with anything z or only with something u that is not identical with anything v unless it v will not die". As Peirce admits, English syntax is not regular enough to express such combinations without sounding "idiotic". Existential graphs, which are more "diagrammatic", support all these combinations in a systematic way. Peirce’s claim that the parts of an EG "are really related to one another in forms of relation analogous to those of the assertions they represent" can be formalized by a mapping or graph homomorphism from an EG to the structure of the subject it represents. Such mappings are the basis for a version of model theory, called endoporeutic, which Peirce discusses later.

[p.15] I dwell on these details which from our ordinary point of view appear unspeakably trifling, — not to say idiotic, — because they go to show that this syntax is truly diagrammatic, that is to say that its parts are really related to one another in forms of relation analogous to those of the assertions they represent, and that consequently in studying this syntax we may be assured that we are studying the real relations of the parts of the assertions and reasonings; which is by no means the case with the syntax of speech.

This discussion shows why reading Peirce’s manuscripts can be both frustrating and rewarding. He began to write this text as an elementary introduction, but he repeatedly digresses into abstruse details. Those details often contain brilliant insights, but for a novice, they can be more confusing than enlightening. One of the novices was John Dewey, who took the first semester of Peirce’s course on logic at Johns Hopkins University, but dropped the second semester with the complaint that it was "too mathematical".

A line which is composed of two or more lines of identity abutting on one another is called a "ligature". Of course it is not a graph, of itself. Or it may be regarded as a graph meaning either "nothing is anything that is anything that is" (in case the shaded end is exterior to the unshaded end) or "something is identical with something that is not identical with anything but what" (in case the shaded end lies in an area enclosed by the unshaded area where the other end is).

In the next paragraph, Peirce mentions endoporeutic, which is his "outside-in" method of determining the truth value of an existential graph. Endoporeutic is logically equivalent to model theory, which Alfred Tarski (1935, 1936)

In modern terminology, endoporeutic can be defined as a two-person zero-sum perfect-information game, of the same genre as board games like chess, checkers, and tic-tac-toe. Unlike those games, which frequently end in a draw, every finite EG determines a game that must end in a win for one of the two players in a finite number of moves. For a typical formula in logic, the complexity of the game is closer to tic-tac-toe than to chess, but in principle the game can be played with an EG of any size. In fact, the first logician to develop the technique of game-theoretical semantics was Leon Henkin (1959), who showed that it could sometimes be used to evaluate the denotation of infinitely long formulas in a finite number of steps.

In the game of endoporeutic, one player, called the proposer, tries to show that a given EG is true, and another player, called the skeptic, tries to show that it is false. The game begins with an EG whose truth value is to be determined and a Tarski-style model M=(D,R), in which D is a set of individuals and R is a collection of relations defined over D. For convenience, the model M can also be represented as an EG in which there are no negations, each individual in D is represented by a line of identity, and each tuple in R is represented by a copy of the character string that names the corresponding relation (see the method of representing relations by graphs). If the EG to be evaluated contains no negations, it would be true if the proposer could find any mapping (graph homomorphism) of the lines of identity and relations of the EG to the lines of identity and relations of the model M. Such a mapping, which embodies Peirce’s notion of "diagrammatic," formalizes the correspondence theory of truth. If no such mapping exists, the EG would be false.

The correspondence theory of truth runs into difficulties when the proposition to be evaluated contains negations or any operator that is defined in terms of negation — that includes disjunction, implication, and the universal quantifier. With endoporeutic, Peirce invented a method of peeling off negations one at a time until there are none left. If an EG to be evaluated contains negations, the two players in the game of endoporeutic must remove them according to the following rules:

• If the current EG is empty (a blank sheet of assertion), the current proposer wins, and the current skeptic loses.

• If the current EG consists of just a single negation (shaded oval), the negation is removed by unshading the outermost oval and reversing the shading of any nested areas. Then the two players switch sides: the current proposer becomes the new skeptic, and the current skeptic becomes the new proposer.

• If the current EG consists of two or more negations (shaded ovals with no graph components outside an oval), the current skeptic chooses any one of the ovals as the current EG, and all the other ovals are erased.

• If the current EG contains a subgraph g outside of any shaded area, the current proposer tries to find a mapping (graph homomorphism) of g into the model graph M. There are two possibilities:
1. If no such mapping can be found, the current proposer loses, and the current skeptic wins.
2. If one or more such mappings are possible, the proposer has the option of selecting any one. The inverse of that mapping determines an assignment of some individual in D to each line of identity in g. After the proposer has made the selection, any lines of identity that extend from g into any ovals that occur in the current EG can never be reassigned to any other individuals in D. Then the graph g is erased, leaving either a blank sheet of assertion or a sheet containing one or more shaded ovals.
Since each of these rules reduces the size of the EG, any game that starts with a finite EG must end in a finite number of moves. Since none of the stopping conditions results in a draw, either the original proposer or the original skeptic must have a winning strategy for any given EG and model M. If the proposer has a winning strategy, the EG is true in terms of M; otherwise, it is false in terms of M.

This definition of endoporeutic is based on Peirce’s writings, but the wording of the statement takes advantage of many concepts that were developed in mathematics and computer science during the twentieth century: recursive definitions, graph homomorphisms, game theory, and the game-playing algorithms of artificial intelligence. Peirce had written many pages about endoporeutic, but no one clearly deciphered them until Hilpinen noticed the similarity to the game-theoretical approach. Only after mathematicians had caught up to Peirce did they have the concepts and vocabulary for interpreting what he was trying to say.

There are three simple rules for modifying premises when they have once been scribed in order to get any sound necessary conclusion from them. Of course I do not count among these rules two recommendations which are nevertheless of the highest importance. One is to be sure to scribe every premise that is really pertinent to the conclusion one aims at. The other is to scribe them with sufficient analysis of their meaning, and not by any means to neglect abstractions which modern philosophers think most foolishly are of little or no importance or are even unreal because they are of the nature of signs. They tell us that it is we who create the law of nature! That is Real which is true just the same whether you or any collection of persons opine or otherwise think it true or not. The planets were always accelerated toward the sun for millions of years before any finite mind was in being to have any opinion on the subject; therefore the law of gravitation is a Reality.

In the following digression, Peirce expresses doubts about the absolute truth of Newtonian mechanics. All such doubts were summarily dismissed by Simon Newcomb, who had been Peirce’s superior at the U.S. Coast and Geodetic Survey, but they were vindicated by Einstein’s theory of relativity. The discrepancy in the orbit of Mercury, in fact, was part of the evidence used to confirm the general theory of relativity (Einstein 1916, 1921). As early as 1891, Peirce had suspected that the geometry of the universe was non-Euclidean, and he requested funding (which Newcomb denied) for astronomical observations to test whether the sum of the angles of a triangle is exactly 180°. That suspicion was finally confirmed during an eclipse of the sun in 1919, when a measurement of the deflection of a light ray was consistent with Einstein’s predicted value for the curvature of space caused by the mass of the sun.

[p.17]I do not say that Newton’s formulation of the law of gravitation is quite right, because when Newcomb was at work on the inferior planets, Mercury and Venus, I wrote to him and called his attention to the fact that certain motions of Mercury go to show that the attraction is not precisely inversely as the 2nd power of the distance but is rather proportional to the 2.01 power or thereabouts; I see that in his tables not only of Mercury but also of Venus he has introduced such a correction. He says he introduces it to make his tables accord with observation. He does not say that the cause of the discrepancy of observation with Newton’s law. But that is the way I can see [to think of or] account for it. I had not supposed it will be prescriptible in so circular an orbit as that of Venus. No doubt all our new formulations of laws are merely approximate; but the laws, as they really are, are Real.

The distinction between nominalism and realism is a theme that Peirce discussed reapeatedly:
Anybody may happen to opine that "the" is a real English word; but that will not constitute him a realist. But if he thinks that, whether the word "hard" itself be real or not, the property, character, the predicate, hardness is not invented by men, as the word is, but is really and truly in the hard things and is one in them all, as a description of habit, disposition, or behaviour, then he is a realist. (CP 1.27n1)
In logic, Peirce combined Boole’s symbolic approach with the work of the medieval logicians, of which Ockham was one of the greatest. In metaphysics, however, Peirce disagreed with Ockham’s nominalism and preferred the realism of Ockham’s predecessor, John Duns Scotus. But Peirce also added some important qualifications:
In calling himself a Scotist, the writer does not mean that he is going back to the general views of 600 years back; he merely means that the point of metaphysics upon which Scotus chiefly insisted and which has passed out of mind, is a very important point, inseparably bound up with the most important point to be insisted upon today. (CP 4.50)
In terms of Peirce’s categories, the nominalists were masters of Firstness and Secondness, and the important point they ignored was Thirdness, which is central to the reality of physical laws. Yet Peirce always emphasized the equal status of all three categories. In distinguishing his triads from Hegel’s, Peirce rejected the idea that "Firstness and Secondness must somehow be aufgehoben" (CP 5.91).

I will now state what modifications are permissible in any graph we may have scribed.

Peirce’s three permissions correspond to what Ockham called regulae consequentiarum or rules of inference. In another passage (CP 4.423), Peirce called them "formal "rules"... by which one graph may be transformed into another without danger of passing from truth to falsity and without referring to any interpretation of the graphs." Each permission may be viewed as a pair of rules, one of which states conditions for inserting a graph, and the other states conditions for erasing a graph. In this commentary, the insertion rules are numbered 1i, 2i, 3i; the erasure rules are 1e, 2e, 3e.

In various discussions, Peirce showed that his rules are sound. His proof by the method of endoporeutic is equivalent to Tarski’s proof of soundness by model theory. As Peirce said, there is no "danger of passing from truth to falsity". It is, however, possible to pass from falsity to truth. Rule 1e, for example, allows any graph, true or false, to be erased. Erasing a true graph cannot make a true statement false, but erasing a false graph may make a false statement true. For more detailed discussions of Peirce’s rules and proofs of their soundness and completeness, see Roberts (1973) or Sowa (1984).

Peirce’s rules, in fact, are a generalization and simplification of the rules for natural deduction, which Gerhard Gentzen (1935) independently discovered many years later. Like Peirce’s rules, Gentzen’s rules also come in pairs, one of which inserts an operator, which the other removes. Unlike Peirce’s rules, Gentzen’s rules are more numerous and more complex because they were designed for the more complex algebraic notation (which Peirce discarded in favor of EGs precisely because the graphs have a simpler structure). For both Peirce and Gentzen, the only axiom is a blank sheet of paper: anything that can be proved without any prior assumptions is a theorem.

1st Permission. Any graph-instance on an unshaded area may be erased; and on a shaded area that already exists, any graph-instance may be inserted. This includes the right to cut any line of identity on an unshaded area, and to prolong one or join two on a shaded area. (The shading itself must not be erased of course, because it is not a graph-instance.)

The proof of soundness depends on the fact that erasing graphs by Rule 1e reduces the number of conditions that might be false, and inserting graphs by Rule 1i increases the number of conditions that might be false. Rule 1e, which permits erasures in an unshaded (positive) area, cannot make a true statement false; therefore, that area must be at least as true as it was before. Conversely, Rule 1i, which permits insertions in a shaded (negative), area cannot make a false statement true; therefore, the negation of that false area must be at least as true as it was before. For a more formal proof, see Sowa (1984).

These rules apply equally well to propositional logic and predicate logic. Since EGs have no variables, the rules for dealing with variables in the algebraic notation are replaced by simpler rules for cutting or connecting lines of identity (which corresponds to erasing or inserting instances of the graph —is—). In terms of Peirce’s endoporeutic, cutting a line allows either end to be assigned independently to different individuals in a model. Therefore, cutting a line has the effect of existential generalization because it allows the two ends to be assigned to different existentially quantified variables. The option of connecting two lines in a shaded area has the effect of universal instantiation, because it allows a universally quantified variable to be replaced by an arbitrary term.

[p.18] 2nd Permission. Any graph-instance may be iterated (i.e. duplicated) in the same area or in any area enclosed within that, provided the new lines of identity so introduced have identically the same connexions they had before the iteration. And if any graph-instance is already duplicated in the same area or in two areas one of which is included (whether immediately or not) within the other, their connexions being identical, then the inner of the instances (or either of them if they are in the same area) may be erased. This is called the Rule of Iteration and Deiteration.

Peirce showed that the rules of iteration (2i) and deiteration (2e) can never change the truth value of a graph. By endoporeutic, the truth value of each graph or subgraph is determined at the point when the outside-in evaluation reaches it. If a subgraph g has the value true at that point, no copies of g can affect the truth value of the current area or any enclosed area. If g has the value false, the current area must already be false, and no copies of g in the current area or any enclosed area can make the current area true.

In other writings, Peirce gave more detail about how these rules may be applied to lines of identity as well as to whole graphs. By iteration (2i), any line of identity may be extended in the same area or into any enclosed area. By deiteration (2e), any end of a line of identity that is not attached to another line or to some relation name may be erased, starting from the innermost area in which it occurs. Iteration extends a line from the outside inward, and deiteration retracts a line from the inside outward.

3rd Permission. Any ring-shaped area which is entirely vacant may be suppressed by extending the areas within and without it so that they form one. And a vacant ring shaped area may be created in any area by shading or by obliterating shading so as to separate two parts of any area by the new ring shaped area.

A vacant ring-shaped area corresponds to a double negation; i.e., two negation signs ~~ with nothing between them. The third permission says that a double negation may be drawn around (3i) or erased around (3e) any graph on any area, shaded or unshaded. In the game of endoporeutic, a double negation causes the two players to switch sides twice; therefore, its presence or absence can have no effect on the final result. An important qualification, which Peirce discusses elsewhere, is that such a ring is considered vacant even if it contains lines of identity, provided that the lines begin outside the ring and continue to the area enclosed by the ring without having any connections to one another or to anything else in the area of the ring.

The opening phrase of the next sentence, "It is evident that", is an exaggeration, since Peirce elsewhere used many pages to prove that so-called evident conclusion.

It is evident that neither of these [p.19] three principles will ever permit one to assert more than he has already asserted. I will give examples the consideration of which will suffice to convince you of this.

Fig. 7 asserts that some boy is industrious. By the 1st permission it can be changed to fig. 8, which asserts that there is a boy and that there is an industrious person. This was asserted as fig. 7, together with the identity of some case.

Figure 7 may be translated to the following formula:
```(∃x)(boy(x) ∧ industrious(x))
```
This formula is equivalent to the following, which results from treating the line of identity as if it contained the dyad —is—, which corresponds to the equality x=y.
```(∃x)(∃y)(boy(x) ∧ industrious(y) ∧ x=y)
```
Rule 1e, which allows any graph to be erased in a positive area, has the effect of erasing a dyad —is— to transform Figure 7 into Figure 8. In the algebraic notation, that rule has the effect of erasing the equality to produce the formula that corresponds to Figure 8:
```(∃x)(∃y)(boy(x) ∧ industrious(y))
```
This formula and the graph in Figure 8 assert that there exists a boy x and something industrious y, and it leaves open the question of whether the boy and the industrious thing are the same or different.

Fig. 9 asserts either there is nothing known for certain or else there is no communication with anybody. By the same permission this can be changed to fig. 10 which asserts that no communication with anybody deceased is known for certain. But this is fully included in the state of things asserted in fig. 9.

Figure 9 corresponds to the following formula:
```~(∃x)(∃y)(knownForCertain(x) ∧ communicationWith(x,y))
```
This formula may be read "It is false that some x is known for certain that is a communication with some y." The disjunctive reading corresponds to the following formula:
```(∀x)(∀y)(~knownForCertain(x) ∨ ~communicationWith(x,y))
```
Literally, this formula may be read "For every x and y, either x is not known for certain, or there is no communication x with y." By Rule 1i, the graph —deceased may be inserted in the shaded area. By a second application of Rule 1i, the unattached end of that graph may be connected to the unattached end of the graph —communicationWith— to derive Figure 10, which corresonds to the following formula:
```~(∃x)(∃y)(knownForCertain(x) ∧ communicationWith(x,y) ∧ deceased(y)).
```
This formula may be read "It is false that some x is known for certain that is a communication with some y who is deceased."

In illustrating the applications of the Second Permission, I am obliged to notice one of the faults of the system of logic which has been taught to every generation of young men for some sixty odd generations. One of the syllogisms that they have all been taught as a sound apodictic argument called Darapti (and whose validity nobody has questioned) furnishes a fair sample of the quality of intellect of the Doctors and Regents of the most famous and proudest Universities. Here is a sample of it:

Any Phoenix would be a bird

Any Phoenix rises from her own ashes

∴ Some bird rises from its own ashes

In this digression, Peirce comments on a questionable feature of Aristotle’s system of syllogisms: the implicit assumption that every category has at least one member. With that assumption, the syllogism Darapti would be sound. Without that assumption, Peirce’s counterexample shows that Darapti is unsound because a category, such as Phoenix, which has no members would lead from true premises to a false conclusion. Peirce used this example to show the need for analyzing rules of inference in painstaking detail and for justifying each of them by a method such as his endoporeutic or Tarski’s equivalent system of model theory.

[p.20]They might try to crawl out of this absurdity by saying that they do not state the premises as

Any Phoenix there may be is a bird

Any Phoenix there may be rises from its ashes

but

Every Phoenix there is is a bird, etc.

But the reply to that (passing over the fact that Sir Wm Hamilton, lauded as the highest of authorities, insists that Any and not Every is the right word) is that by "Contradictories" they mean two propositions which, by their very meaning, can neither both be true nor both be false, and they all agree that every simple proposition has a simple contradictory, and that the contradictory of "Some S is not P" is "Any, all, or every (Greek pantos) S is P". Now if this latter implied the existence of some S, Every S is P and Some S is not P could both be false by there not existing any S. That would be a much graver fault with their logic than that which I charge against it. For I only charge that two "moods" or species of syllogism are false. (i.e. not nessesary, as they profess to be.) And curiously Aristotle never mentions these with examples as he does in all other cases; but merely says — But this letter will be long enough without discussing Aristotle and his commentary, a subject on which I should own [?] time you, interesting as it is to me.

In the following paragraph, Peirce states his main argument for existential graphs: they allow the reasoning steps to be dissected "into the greatest number of distinct steps", every one of which is justified by a proof of soundness in terms of endoporeutic. With his rules of inference and their proof of soundness, Peirce attained a level of formality and rigor that surpassed anything achieved by Frege or Russell. Although Frege has been praised for the meticulous formalization of his proof procedures, he merely assumed his axioms without proving their truth, and he stated his rules of inference without proving their soundness. As an example, Frege’s first axiom, written in Peirce-Peano notation, is
```a ⊃ (b ⊃ a).
```
To justify it, Frege (1879) gives an informal argument: This axiom "says ’The case in which a is denied, b is affirmed, and a is affirmed is excluded.’ This is evident, since a cannot at the same time be denied and affirmed." Yet many steps are required to transform that axiom into the natural language sentence that is supposed to be "evidently" equivalent. In terms of Peirce’s rules, Frege’s axiom can be proved in five steps starting with a blank sheet of paper:

Rule 3i inserts a double negation around the empty graph; another application of Rule 3i inserts another double negation around the previous one; Rule 1i inserts a into the shaded area; Rule 2i iterates a into the innermost area; finally, Rule 1i inserts b into the other shaded area. The theorem to be proved contains five symbols, and each step of the proof inserts one symbol into its proper place in the final result. Frege had eight other axioms, each of which can be proved by Peirce’s rules in a similarly short proof.

Frege also had two rules of inference, each of which can be proved as a derived rule of inference from Peirce’s rules. Following is the proof of modus ponens, which starts with an arbitrary statement p and an implication of the form pq:

Rule 2e deiterates the copy of p in the shaded area; Rule 1e erases the original copy of p, which is no longer needed; finally, Rule 3e erases the double negation to leave the conclusion q by itself. Frege’s other rule of inference is universal generalization, which allows any term t to be substituted for a universally quantified variable in a statement of the form (∀x)P(x). In EGs, the term t would be represented by a graph of the form —t, which states that something satisfying the condition t exists, and the universal quantifier corresponds to a line of identity whose outermost part occurs in a shaded area:

In the first step, rule 2i iterates (extends) the line of identity attached to t into the shaded area; Rule 1i inserts a connection between the two lines in the shaded ara; finally, Rule 3e erases the double negation. Not only does Peirce "dissect" the reasoning process into the simplest possible steps, he uses the method of endoporeutic to ensure that each step is sound.

So I will break off that and just give an illustration or two of how this Syntax of Existential Graphs works. But before doing that I wish to draw your attention, in the most emphatic way possible, to the purpose this Syntax is intended to subserve: since anybody who did not pay attention to that statement would be all but sure, not merely to mis[p.21]take the intention of this syntax, but to think that intention as contrary to what is as well he could. Namely he would suppose the object was to reach the conclusion from given premises with the utmost facility and speed, while the real purpose is to dissect the reasoning into the greatest possible number of distinct steps and so to force attention to every requisite of the reasoning. The supposed purpose would be of little consequence, and it is the fussiness of the mathematicians to furnish inventions to attain it; but the real purpose is to supply a real and crying need, although logicians are so stupid as not to recognize it and to put obstacles in the way of meeting it.

Although Peirce does not name the "stupid logicians" he is criticizing, one of them is probably Bertrand Russell, whose Principles of Mathematics he reviewed in 1903:
This is not the place to speak of Mr. Russell’s book, which can hardly be called literature. That he should continue these most severe and scholastic labors for so long, bespeaks a grit and industry, as well as a high intelligence, for which more than one of his ancestors have been famed. Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years, and that have thrown a new light both upon mathematics and upon logic, will do well to take up this book. But he will not find it easy reading. Indeed, the matter of the second volume will probably consist, at least nine-tenths of it, of rows of symbols.
In this lukewarm review, the best that Peirce could say is that the book is "a convenient introduction." Since Russell admitted in the preface that he had learned symbolic logic from Peano at a conference in 1900, he had little to say in 1903 that could impress the man who had invented most of it. As Putnam (1982) pointed out, the major developments in logic from 1890 to 1910 were written in Peirce-Schröder notation, including Zermelo’s axioms for set theory, the Löwenheim-Skolem theorem, and Hilbert’s work on the foundations of mathematics. In a letter to Lady Welby in 1904, Peirce expressed his opinion more candidly: "Russell’s book is superficial to the point of nauseating me." The later book, which Peirce correctly predicted would "probably consist, at least nine-tenths of it, of rows of symbols" was the Principia Mathematica by Whitehead and Russell. Although it was an excellent compendium, the German and Polish logicians found little in it that was new. Tarski reportedly said that it was "a step backwards."

In the following example, Peirce shows that the syllogism named Barbara can be proved as a derived rule of inference from his rules for EGs. In an early article "On the grounds of the validity of the laws of logic," Peirce (1869) gave the following argument for it:

Now, what the formal logician means by an expression of the form, "Every M is P," is that anything of which M is predicable is P; thus, if S is M, that S is P. The premise that "Every M is P" may, therefore, be denied; but to admit it, unambiguously, in the sense intended, is to admit that the inference is good that S is P if S is M.
There are two points to note about this example: First, Peirce in 1869 was already using model-theoretic arguments in terms of the truth of "anything of which M is predicable," while Frege’s arguments were nothing but paraphrases in natural language. Second, Peirce’s use of EGs for deriving the rules of inference of other systems of logic was more formal in 1909 than any argument that Whitehead and Russell gave in 1910 or even 1925.

I will now, by way of an example of the way of working with this syntax, show how by successive steps of inference to pass from the premises of a simple syllogism to its conclusion.

[p.22] The first step consists in passing to fig. 12 by the 2nd Permission

The second step is simply to erase "Any M is P" by the 1st Permission. The third step is to join the two ligatures by the 1st Permission as shown in fig. 13:

It will be observed that in iterating the major premise, I had a right to put the new graph instance at any part of the area into which I put it; and I took care to have the ligature of the minor premise touch the shaded area of iterated graph instance. Now by the 1st Permission I have a right to insert what I please into a shaded area, and without making the new line of junction leave the shaded area, I make it touch the unshaded line of identity of the major premise.

By making the iterated graph instance touch the ligature from S to M, Peirce managed to shorten the proof by one step. Pedagogically, however, his proof would have been clearer if he had not made the iterated copy touch the ligature. Then the connection of the two lines of identity would have taken two steps: an extension of the ligature into the shaded area by Rule 2i, and a connection of the two lines by Rule 1i.

Before the two lines of identity were connected, the inner copy of M could not be erased by deiteration because the two copies of M were attached to different lines of identity. But after the two lines were connected, both copies of M were attached to the same line of identity, and the inner copy of M could be erased by Rule 2e.

This gives me a right in the fourth step to deiterate M so as to give fig. 14 by the second permission.

The fifth step is to delete the M on an unshaded field giving fig. 15

while the Sixth step authorized by permission the third consists in getting rid of the empty ring shaped shaded area round the P, giving fig. 16.

[p.23] By the space that I have occupied in explaining this syntax you will surely think it is my chief work. On the contrary, it is one of the smallest, but it is the only one of which I could put you into a position to gain some understanding without writing a book about it.

This last sentence is significant: Peirce admits that the invention of the EG syntax is one of his smallest contributions, but it is the only one that can be explained in less than a book. In less than fourteen hand-written pages, he has presented as much or more than can be found in most introductory textbooks on logic. Unlike most textboks, he has not bothered to split the subject into separate sections on propositional and predicate logic. Instead, he presented both at the same time, since the same rules of inference apply to both of them when they are expressed in existential graphs. To illustrate the use of Peirce’s rules to prove a more complex theorem, consider Leibniz’s Praeclarum Theorema (splendid theorem):

((pr) ∧ (qs)) ⊃ ((pq) ⊃ (rs)).

This formula, which has four implications, may be read "If p implies r and q implies s, then p and q implies r and s." Each of the four implications maps to a pair of a shaded and an unshaded area in the corresponding EG:

The following diagram shows that a proof of Leibniz’s theorem requires seven steps, starting with an empty sheet of assertion, which is Peirce’s only axiom:

The first step in this proof is an application of Rule 3i to insert a double negation around the empty graph — i.e., creating a vacant ring-shaped area on a blank sheet of assertion. That operation must be the first step in the proof of any theorem, since no other rule can be applied to an emtpy sheet. The second step in proving a theorem would insert some hypothesis into the shaded area by Rule 1i. In this case, an instance of the graph corresponding to (pr)∧(qs) is inserted. Most proofs from an empty sheet would begin with an application of those two rules. The remainder of the proof would iterate subgraphs from the shaded area into the inner unshaded area by Rule 2i and apply more rules to transform the result into the desired conclusion.

In this case, the third step of the proof is an application of Rule 2i to iterate a copy of the graph for pr. (An exactly equivalent proof would be obtained by iterating a copy of the graph for qs.) The fourth step applies Rule 1i to insert a copy of q into the shaded area nested three levels deep.

After only four steps, the graph looks almost like the desired conclusion, except for a missing copy of s inside the innermost area. Since that area is unshaded, it is not permissible to insert s directly by Rule 1i. Instead, Rule 2i is used to iterate a copy of the graph for qs. The next step applies Rule 2e to deiterate the unwanted copy of q in the shaded area. Finally, Rule 3e erases the vacant shaded area to derive the conclusion, QED.

In the Principia Mathematica, which Whitehead and Russell (1910) published 13 years after Peirce discovered his rules, the proof of the praeclarum theorema required a total of 43 steps, starting from five nonobvious axioms. One of those axioms was redundant, but the proof of its redundancy was not discovered by the authors or by any of their readers for another 16 years. All that work could have been saved if Whitehead and Russell had read Peirce’s writings on existential graphs. Unfortunately, the British journal Mind is one of several that had rejected Peirce’s article of 1897. That article, which was finally published in 1906 and which appears in Peirce’s Collected Papers, was ignored by nearly all 20th-century logicians.

You will ask perhaps "If all one has to do is to avail oneself of those 3 permissions, how is it that mathematics (which is nothing but deductive reasoning) is so difficult and demands high genius? There are several circumstances which go to clearing up this. The first of these is that the mathematician is not supplied in advance with a definite list of premises; nor is he asked whether or not a definite conclusion can or cannot be drawn. His usual first approach to a problem is something like the following entirely fanciful situation which serves to illustrate what one of his difficulties is like. An astronomer comes to a mathematician and says, "I want to consult you about something." — But hold! I can perfectly well substitute a historical case about which I am fully informed.

Commentary to be continued.

Toward the end of October 1604 the astronomer Tycho Brahe died and left a mass of observations including continued measurements of the apparent places of the planet Mars extending over 15 years. Kepler who was a remarkable mathematician and who had had the advantage of training in observations under Tycho, had possession [p.24] of the MSS and had continued the observations of Mars some years longer so as to make the series extend over 20 years; and it devolved upon him to take this measurements of the Latitude and Longitude of Mars (remarkably fine observations considering they were made with the naked eye) and by means of them to construct tables by which the Lat. and Long. of Mars could be calculated for any future time. Of course it was assumed that Mars would continue to move as it has been moving and therefore one could calculate just what its Lat. and Long. were at any instant during these 20 years, except at those times of each year when it had been too close to the sun to be observed.

Commentary to be continued.

I note that John Stuart Mill in his Logic says that Kepler only had to make a general description of facts known in the shape [?] of observation. But Mill was a constant writer of reviews who had at the same time almost the responsibility of governing India on his shoulders & it would have been beyond human powers for a man every three months to turn out an article of high literary excellence in the Westminster Review, and conduct the business of India House and add to that any profound study of logic. He had evidently no conception of what Kepler [p.25] had to do.

He had before him the latitudes and longitudes. But since Tycho was a single observer at a first station he could make no observation of parallax that is the third coordinate of Mars’s position, its distance from the observer. For observations of its position when on work on [?] the horizon. Anyway the smallest angle visible, — the minimum visibile is about one minute of arc and the greatest parallax of Mars is about 1/9 of that. Kepler it is true found an ingenious method of measuring the distance of Mars or any planet from the earth and from the sun. But it requires the theory of the motion of the planet to be complete, or nearly so, first.

In short, Kepler’s reasoning was not, and could not have been, purely mathematical. It was, on the contrary the greatest piece of Inductive reasoning ever yet conducted. Had the parallaxes, or distances of Mars from the earth been known to Kepler with the requisite degree of accuracy, it must have saved Kepler that marvellous piece of reasoning by which he ran down the truth like an indefatigable detective, with hardly [?] a wasted day. It would have been a great loss to students of reasoning. The Lunia Lectùrina were not understood in 1610 as they are today, for Pascal was only [p.26] born in 1623, or thereabout; and his theorem, that if set points are taken on a conic section and straight lines are drawn through the 1st and the 2nd, the 2nd and 3rd, and so round to a line through the 6th and 2nd, then the intersection of the first and fourth of these lines, that of the 2nd and fifth, and that of the 3rd and 6th will all lie on one straight line, no matter how the original six points are chosen (whether they are consecutively passed in going round the curve or not). This proposition, which is the foundation, one may say, of the modern theory of Conics, would have been unknown to him. But Kepler was [aware of the] "foci" of such curves, and he would undoubtedly have made great advances in the subject, had he been in search of the law of the motion of Mars with knowledge which rendered it a purely "deductive" problem that is a matter of necessary reasoning.

Commentary to be continued.

However I have after all been rather side tracked by choosing this example; for I wished to show that mathematical problems of a new kind are generally first presented in a form which is simply a bewildering mass of facts into which the mathematician has to dive and fetch up first the question to be asked and then with great subtlety pick out the appropriate premises. Then the second difficulty is that mathematical problems are apt to be so fearfully complicated (for example there are over 80 equations in the [p.27] moon’s longitude) that it requisites a most capacitive brain to embrace the question.

Finally there comes a difficulty in many problems which has had one to divide mathematical reasoning into the corollarial deductions and the theorematic deductions. The terms "corollary" and "theorem" have no definite meanings and never have. The original "theorems" of geometry were those propositions that Euclid proved, while the corollaries were simple deductions from the theorems inserted by Euclid’s commentators and editors. They are said to have been marked the figure of a little garland (or corolla), in the origin. But I use the adjectives which I form from the words "theorem" and "corollary" with exact meanings.

In other discussions, Peirce gave the following elaborations:
My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the corollarial and the theorematic, because the corollaries affixed to the propositions of Euclid are usually arguments of one kind, while the more important theorems are of the other. The peculiarity of theorematic reasoning is that it considers something not implied at all in the conceptions so far gained, which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. Euclid, for example, will add lines to his diagram which are not at all required or suggested by any previous proposition, and which the conclusion that he reaches by this means says nothing about. I show that no considerable advance can be made in thought of any kind without theorematic reasoning. When we come to consider the heuretic part of mathematical procedure, the question how such suggestions are obtained will be the central point of the discussion. (MS L75, pp. 95-96)

Any Corollary (as I shall use the term) would be a proposition deduced directly from propositions already established without the use of any other construction than one necessarily suggested in apprehending the annunciation of the proposition. (NEM 4:288)

Any Theorem (as I shall use the term) would be a proposition pronouncing, in effect, that were a general condition which it describes fulfilled, a certain result which it describes in a general way... will be impossible, this proposition being capable of demonstration from propositions previously established, but not without imagining something more than what the condition supposes to exist. (NEM 4:289)

Peirce considered this distinction "a matter of extreme importance for the theory of cognition" (NEM 4:46). In modern computer science, this distinction has proved to be the most serious obstacle to the development of true "artificial intelligence." Automated theorem provers today are far superior to any human being in corollarial reasoning, but the major challenge is theorematic reasoning, in which they cannot compete with a good high school student. A century ago, Peirce had a deeper insight into the kinds of reasoning that would be difficult for AI than many people who are today working in the field.

The ultimate premises of geometry are called by present day geometers "hypotheses", because the mathematicians, as such, do not accept any responsability for their truth. They are of three kinds, definitions, axioms, and postulates. The axioms are, in my opinion, all false, if one insists on their rigid accuracy, in all cases. The "postulates" were originally understood to be premises expressing that certain lines could be drawn, though everybody knew that they could not, exactly. But in my opinion it is far better to consider them as statements that space contains certain kinds of places. For instance, the two old postulates that a straight line can be drawn [p.28] from any point to any other ant that a straight line can be "produced" (that is lengthened) at either end; I would superside by the one postulate that, considering a line as a place, or "locus", as mathematicians have universally considered it since Descartes, "an unlimited straight line is through every pair of points, or places without parts". (Euclid’s definition of a straight line is that it is a line that lies "evenly" between its extremities; by which I suppose he means, perhaps a little vaguely, that there are points from which such a line would appear as a point, or from the modern standpoint, it is a line whose shadow, if the source of light were a point on the line, would be a moins. It is a question whether this is the better definition (as I decidedly think) or whether we ought to say that a straight line might be the path of a particle, not acted on, during the motion, by any force. The definition that a straight line is the shortest distance between two points ought I think to be regarded as an axiom presumably only approximately true.)

Now one of the great difficulties of geometry is that no propositions of the kind I should call a "theorem" can be proved without introducing subsidiary lines or surfaces, that are not mentioned either in the propositions to be proved nor in the previously proved propositions. The right to assume these subsidiary loci is derived from the postulates.

I pass over crowds of points deeply interesting to anybody who cares to explore [p.29] these fields, and come to another division of deductive reasonings, — that into what I call necessary deductions and probable deductions. All deductions are necessary reasonings in the sense that the conclusion must be true so long as the premises are so. But I use the expression "probable deduction" as a convenient abbreviation of "deduction of a probability". Probable deductions include all the logically sound parts of the doctrine of chances, otherwise called, the calculus of probabilities. This includes so much of that doctrine as could safely be made the basis of the business of insurance.

In this paragraph, Peirce emphasizes a point that he made in one way or another throughout his career: the three fundamental kinds of reasoning are deduction, induction, and abduction. That distinction applies to every kind of reasoning, formal or informal, by highly trained scientists or by the proverbial "man in the street," and even by computers, dogs, or bees. Probable deduction is not a kind of reasoning that differs in any way from logical deduction; it is merely logical deduction applied to probabilities. Fuzzy reasoning does not differ in any way from logical reasoning; when done correctly, it is merely logical reasoning about continuity, which Peirce considered one of most important subjects for logical, philosophical, and scientific study.

There is a lot more in the books, — particularly in Laplace’s book, which is the base of all ninteenth century works on the subject — Laplace being the idol of the French mathematicians, — there is I say a lot of it that is utter rot. He says a probability expresses in part knowledge and in part ignorance. This statement is a fair specimen of the loose thoughts of the book. Laplace’s mathematics is sometimes clumsy, but [p.30] it is correct as long as his premises mean anything. But when he attempts to define anything at all difficult he writes utter nonsense. In the sense in which he means it, that which express ignorance is utterly worthless & is no part of true science. If two possibilities, he says, are "également possibles" their probabilities are equal, and if two events that are mutually exclusive have equal probabilities, the probability is double that of either, that one or other will occur. "What is the probability that the inhabitants of Saturn have red hair?" asked Mill in the first edition of his logic that it is red, that it is not red are "également possibles" since we are absolutely ignorant about it, is true. For possibility that a thing may be admits of no more or less. If it is possible, that is, if we do not know that it is not so, which is certainly the case if we are utterly ignorant, then the two are "equally possible" in the only sense this phrase can have, that we don’t know anything aginst the truth of either. But how would an insurance company fare who should try to do business on such a basis? A basis for business has got to be knowledge and not ignorance?

As a specimen of Laplace’s results I would mention something he deduces from his principle of the "également possibles" and which is copied into all the books of the [p.31] subject, — all the usual ones, — to this day. Namely, Laplace says that if a man on occasions entirely new to him sees a phenomenon equally new on every one of those occasions up to N occasions (N being any whole number) then the probability is N+1/N+2 that the same phenomenon will occur on the next such occasion. I say this is nonsense. That is trying to conclude by mathematical reasoning that which requires a radically different kind of reasoning. And what proves that it is nonsense is that if N=0, the probability is 1/2. That is to say that on a wholly new occasion it would be a reasonable thing to make an even bet that an unheard of event would take place. That is the nonsense that results from trying to reason mathematically on matters of fact on the basis of pure ignorance. Laplace was renouned for lack of sound good sense, and his doctrine about these inverse probabilities, if it is correct, is a basis for business. But there can be no such basis except experience: the idea of deducing any matter of fact from anything but knowledge is absurd.

Commentary to be continued.

[p.32] Now you will ask me "How do you define probability?" I will define it in a concrete example. Suppose I say "I have a die and owing to its being somewhat ill made, instead of the probability of its turning up six at any one throw being 1/6, or 0.16 2/3, as it should be, the probability of that event is only 0.16".

In "A Theory of Probable Inference," Peirce wrote "A probability is a fraction where the numerator is the frequency of a specific kind of event, while its denominator is the frequency of the genus embracing that event." (CP 2.474, 1883)

In the definition of probability that he contributed to the Century Dictionary (1890), Peirce wrote "The ratio of the number of favorable cases to the whole number of equally possible cases, or the ratio of the number of occurrences of the event to the total number of occasions in the course of experience. This number is called the probability or chance of the event." (4:4741 col. 1)

## References

All bibliographical references have been moved to the combined bibliography for this web site.