On the Algebra of Logic

Comments on Peirce's Paper of 1885

by John F. Sowa

For more background on the history of logic and Peirce's contributions, see the articles by Putnam (1982), Quine (1995), Dipert (1995), and Hintikka (1997). Click on any of the dates highlighted in blue to see the full reference.

1. Historical Background

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 of 1870, Codd's relational algebra and the SQL language leave the existential quantifier implicit and require a double negation to express universal quantification.

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 logic, which he called the Begriffsschrift (concept writing), but no one else ever used it.

The modern notation for predicate calculus is based on Peirce's notation, which in turn was an extension of Boolean algebra. Like Boole, Peirce adopted mathematical symbols to represent logical operators. 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:

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.

Quantifiers

Model Theory

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) developed many years later. As Peirce said, logic is "the formal science of the conditions of the truth of representations" (CP 2.229); model theory and endoporeutic are two logically equivalent methods for defining those conditions. Before Peirce, the most detailed development of the model-theoretic approach was by William of Ockham. To see the similarity, compare the following quotations from Ockham, Peirce, and Tarski:

Tarski's advance over Ockham consists in defining the truth of a formula in terms of his formal notion of model. But Peirce made an equivalent advance by defining the truth of an EG by his formal method of endoporeutic.

The rule of interpretation which necessarily follows from the diagrammatization is that the interpretation is "endoporeutic" (or proceeds inwardly) that is to say a ligature denotes "something" or "anything not" according as its outermost part lies on an unshaded or a shaded area respectively.

Although Peirce's endoporeutic is logically equivalent to Tarski's model theory, the proof of equivalence was not discovered until Hintikka (1973, 1985) developed game-theoretical semantics as a simpler, more elegant method than Tarski's. Then Hilpinen (1982) showed that Peirce's endoporeutic is equivalent to the techniques of game-theoretical semantics. Sowa (1984) adopted game-theoretical semantics to define the model theory for conceptual graphs, which are based on Peirce's EGs. For their introductory textbook, Barwise and Etchemendy (1993) adopted game-theoretical semantics as a simplified method for teaching model theory.

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:

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.

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.

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 ⊃ (ba).
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.

References

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