Charles Sanders Peirce was a philosopher, but it is not easy toclassify him in philosophy because of the breadth of his work. (Pleaserefer to the table of contents of the entryCharles Sanders Peirce.) Logic was one of the main topics on which Peirce wrote. If we focuson logic, however, it becomes apparent that both Peirce’sconcept of logic and his work on logic were much broader than hispredecessors’, his contemporaries’, and ours. First,Peirce located logic in his large architectonic framework ofphilosophy, which is why some strongly believe that Peirce’slogic cannot be properly understood without understanding hispragmatism and his semiotics, to mention but two of his othercontributions. Even within the traditional boundaries of logic, Peircemade too many contributions to outline in a single article.

Acknowledging the nature of this next-to-impossible task, we singleout the common theme of Peirce’s various contributions to modernlogic—to extend logic, as characterized by the three differentdimensions:

1. the scope of formalism (from monadic to relations),
2. the kinds of systems (from symbolic to diagrammatic systems), and
3. semantic values (from bivalence to three values).

The main goal of this entry is not only to present Peirce’saccomplishments in each of these three extensions, but also to explorethe relations, if any, among these novel developments. The threesections of the entry will be devoted, respectively, to each of thesethree ways how the horizon of deductive logic is expanded byPeirce.

Peirce’s journey on formal deductive logic started with Booleancalculus and De Morgan’s logic of relatives. Boolean algebracreated a path to generalize Aristotelian syllogism and DeMorgan’s ambition to formalize relations opened a new territoryto conquer. However, Peirce’s predicate logic is neither amechanical expansion from the existing logics nor a simple combinationof these two. A leap made by Peirce from his contemporary logic isqualitatively substantial enough to call Peirce a founder of moderndeductive logic, as the entry explains. The first section exploresPeirce’s development of predicate logic presented in his severalwell-known papers, by locating the root of Peirce’s introductionof quantifiers and bound variables. While formal details and notationsof Peirce’s first-order logic can be overwhelming, one shouldnot lose the sight of the bigger picture by paying attention to themain motivation behind Peirce’s enterprise for a new logic.Conquering new territory—relations—with new formalnotation, Peirce’s adventure was launched into anotherdimension—a new mode of representation, that is, diagrammaticrepresentation. This is the topic of the second section. While twosystems of Peirce’s Existential Graphs (“EG”henceforth) are presented, the following perspective is at thebackground: Peirce’s EG were not invented just as randomalternatives, logically equivalent to his own predicate logicalnotation, but a reflection of Peirce’s new approach to logic andformalization. As Peirce’s tireless attempts for predicate logicbrought to us more powerful formal notation, Peirce’s search forbetter representation of relational states of affairs was pursuedbeyond his own symbolic system. Spatial, as opposed to linear,notation is still not familiar to some of us, and the second sectionintroduces the basic notational aspects of Peirce’s EG, anddiscusses the fundamental differences between EG and symbolic systems.The third section about three-valued logic examines another newenterprise of Peirce, not in syntactic notation, but in semanticvalues. Peirce scholars propose various motivations behindPeirce’s three value semantics and those different views will bebriefly discussed.

While the first contribution, that is, an extension from monadic topredicate logic, has positioned Peirce as a founder of modern logicalong with Frege, it took much longer for Peirce’s otherachievements to receive proper attention from logicians orphilosophers. The entry aims to draw a road map for Peirce’sjourney in deductive logic so that one may realize his accomplishmentsare very much connected with each other. More specifically,Peirce’s achievements in deductive logic were accumulative.After being able to formalize polyadic relations with new symbolicnotation, Peirce devised a totally new form ofrepresentation—diagrammatic systems. What we can formalize isextendedand how we can formalize what we can formalize isextended. And Peirce ventured into what our formalization representsand suggested a more fine-grained or a bigger territory of semanticvalues than binary True or False values.

• 1. From Monadic to Polyadic Logic
  • 1.1 Relations and quantification formalized
  • 1.2 Boolean tradition—algebraic and model-theoretic
• 2. From Symbolic to Iconic Representation
  • 2.1 Pragmatic maxim applied to the logic of relations
  • 2.2 Alpha system
    • Syntax
      • Vocabulary
      • Well-formed diagrams
    • Endoporeutic Reading Algorithm
    • Multiple Readings Algorithm
      • Example
    • Semantics
    • Reformulated Transformation Rules
  • 2.3 Beta system
• 3. From Bivalent to Triadic Logic
  • 3.1 Truth table of a three-valued system
  • 3.2 Why the third value?
• Bibliography
  • A. Primary Sources: Works by C. S. Peirce cited in this entry
  • B. Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. From Monadic to Polyadic Logic

Peirce and Frege, independently of each other, took us from thetraditional Aristotelian logic to modern logic—a large leap.Nobody could deny the power of formalization which has led earlytwentieth century mathematicians to surprising achievements and results.^[1] What is the essence of the leap made by Peirce and Frege? Is it justa matter of introducing new formal notations, i.e., quantifiers andvariables, so that we may easily formalize our reasoning? If so,modern logic would be just dressing up Aristotelian logic withquantifiers/variables. This would equate one of Peirce’s maincontributions in logic to the increase in formal vocabulary.

The enormous impact of the adoption of quantifiers/bound variables onthe world of logic and mathematics cannot be denied. However, thatshould not overshadow Peirce’s insight behind the new extendedformalism. This section will explore how Peirce’s convictionabout the novelty of the logic of relations led him to theintroduction of quantifiers/variables. Hence, quantification theory,according to Peirce, is not a matter of a linear extension of formalvocabulary, but an expansion into territory that is qualitativelydifferent from what Aristotelian logic covers. At the same time, weshould not forget that Peirce extended the territory of logic in thespirit of Boole’s algebra of logic.

In “An Improvement in Boole’s Calculus of Logic”(1867) Peirce hints at the need for an improvement of Boolean logic,not in the context of predicate logic, but in its inability to expressexistential statements in the context of term logic. His 1870 paper“Description of a Notation for the Logic of Relatives, Resultingfrom an Amplification of the Conceptions of Boole’sCalculus” (DNLR) reveals his ambition to marry Boole’salgebraic notation with De Morgan’s effort at relationalrepresentation. Many agree that this paper introduces essentialvocabulary of first-order predicate logic for the first time inhistory. Subsequently, in “On the Algebra of Logic” (1880)Peirce investigates two kinds of operations overrelations—relative sum and relative product—and “TheLogic of Relatives” (known as “Note B”) published inhis edited bookStudies in Logic by Members of the Johns HopkinsUniversity (1883) shows a major progress in quantification,influenced by the work of O. H. Mitchell (who was his student).Finally, Peirce’s 1885 paper “On the Algebra of Logic: AContribution to the Philosophy of Notations” has been consideredto be the place where Peirce fully presented his quantificationtheory.

Starting with DNLR, the first subsection examines Pierce’ssubsequent steps until he presented the final form of his first-orderlogic in his 1885 paper “On the Algebra of Logic: a Contributionto the Philosophy of Notations”. (For a number of manuscriptswritten between these two papers, refer to Beatty 1969; Dipert 2004:297–299; and Merrill 1978.) The second subsection locatesPeirce’s first-order predicate logic work in a largercontext.

1.1 Relations and quantification formalized

Peirce’s quantification theory is presented in a comprehensiveway together with axiomlike “icons” in his 1885 paper“On the Algebra of Logic: A contribution to the Philosophy ofNotations”. Peirce’s not-short journey to modern logicstarted with his attempt to extend the territory of formalization. Inthis, Peirce was inspired by De Morgan’s struggle for therepresentation of relations, and at the same time Peirce was empoweredwith Boolean calculus which formalizes Aristotelian term logic. Thatis, Peirce took De Morgan’s ambition as a road map for thedirection while being equipped with Boole’s method and notationfor getting there. This subsection will follow Peirce’s journeyto see how he reached the destination, by checking in at his mainstops.

The title of Peirce’s 1870 paper “Description of aNotation for the Logic of Relatives, Resulting from an Amplificationof the Conceptions of Boole’s Calculus” (DNLR) is spelledout at the beginning of the paper in the following way:

[I]t is interesting to inquire whether it [Boole’s logicalalgebra] cannot be extended over to the whole realm of formal logic,instead of being restricted to that simplest and least useful part ofthe subject, the logic of absolute terms,…The object of thispaper is to show that an affirmative answer can be given to thisquestion. (DNLR [CP 3.45])

Boolean logic needs to be “extended” if we want to coverthe entire realm of formal logic, Peirce states. What does Peirce meanby “the whole realm of formal logic”? Peirce answers:“Deductive logic can really not be understood without the studyof the logic of relatives” (1911a [CP 3.641]).^[2]

Being encouraged by Boole’s algebra of logic, but at the sametime taking his father, Professor Benjamin Peirce’s negativeview of logic seriously,^[3] Peirce explored a way to apply Boole’s method to a largerdomain of our reasoning so that relations may be formalized.

What are relations and why are they so special? Let’s comparethree sentences: “John is an American”, “John istaller than Tom”, and “John is between Tom andMary”. The first sentence has a unary predicate “is anAmerican”, the second sentence a binary predicate “istaller than”, and the third sentence a ternary predicate“is between…and…”. A unary predicate standsfor a property or quality, while a binary or ternary predicate standsfor a relation. If a first-order logical system has only unarypredicates, then we say it is monadic. Otherwise, predicate logic isassumed to have binary or other higher predicates.

When we move from monadic to polyadic logic, substantial changes takeplace. The following three changes are at the top of the list.First of all, a move from property to relation is a territoryexpansion. Noting that Aristotelian syllogisms are limited to unarypredicates, one expects polyadic logic to represent more than thereasoning involved in Aristotelian syllogism, that is, term logic.Second, monadic logic is decidable while polyadic logic isnot decidable, as Church’s theorem proved. In some sense, as theterritory is expanded, we are losing its grip.Third, achange in notation is inevitable, which necessitates modernquantification theory. How are these three important aspects arehandled by Peirce?

The realm of relations was a frontier where De Morgan did much of hiscreative and novel work on logic.^[4] However, his inquiry on the topic resides within the patterns oftraditional syllogisms.^[5] Even more importantly, De Morgan did not have enough tools toformalize this newly extended realm.^[6] Hence, not surprisingly De Morgan’s relations are ratherlimited to a certain group which fit in syllogistic reasoning. AsMerrill points out,

De Morgan develops the general logic of relations only to the pointwhere it can be used for his familiar syllogistic purposes. This meansthat he is especially interested in relations which are convertibleand/or transitive,…. (Merrill 1990: 113)

It is somewhat unclear and controversial whether Peirce’sinterest in the logic of relations started independently of DeMorgan’s work on the subject.^[7] Regardless of the origin of Peirce’s inquiry into relations,many have come to agree that it is Peirce (not De Morgan) whosuccessfully formalized the logic of relations. Merrill, a De Morganscholar, puts the matter in the following way:

The most obvious problem with this view of the proposition [DeMorgan’s way of handling relational arguments] is that it doesnot seem general enough. If we can unite two terms into a propositionby relating them, why not three or four or ten terms? DeMorgan’s concern with the relational syllogism seems to haveprecluded this generalization; but there is no reason in principle whyit could not be made. For this, though, we must wait for Frege andPeirce. (Merrill 1990: 110)

Interestingly enough, Peirce’s writings before 1870 DNLR showPeirce also attempted to solve relational arguments by traditionalsyllogistic reasoning rules,^[8] but the approach taken in DNLR is totally different—not withina syllogistic frame but by introducing Boolean algebra notation.Peirce must have realized the power of generalization that Booleannotation could provide. Boole’s algebra formalized Aristoteliancategorical syllogisms and opened a way for generalization of term logic.^[9] Peirce, who was impressed with Boole’s mathematical treatmentof Aristotelian syllogisms, not surprisingly aimed to apply thismethod to relations. In that sense, modern predicate logic started inPeirce’s 1870 pioneering work. Hence, the goal of Peirce’sproject—that is, to broaden the scope of formalization inlogic—was a main motivation for the introduction of newvocabulary for quantifiers and bound variables. If so, Peirce’sinsight early on as to the importance of reasoning involving relationsis a key element in understanding a difference between Peirce’sand Frege’s developments of first-order logic.^[10] Furthermore, the next section will show how Peirce’s obsessionwith the logic of relations led him to the invention of ExistentialGraphs.

The logic of relations formalizes a larger territory than monadiclogic, but there is a price to pay for obtaining the additionalexpressive power: While monadic logic is decidable, polyadic logic isnot. Even though we need to wait until Church’s theorem to seethe undecidability of first-order predicate logic, Peirce intuited afundamental difference between the logic of non-relatives versus thelogic of relations. Here are Peirce’s suggestive ideas over thecomparisons between monadic and relational logic:

The logic of relatives is highly multiform; it is characterized byinnumerable immediate inferences, and by various distinct conclusionsfrom the same set of premises. (1883a [CP 3.342])

And:

[T]he old syllogistic inference can be worked by machinery, butcharacteristic relative inferences cannot be performed by any meremechanical rule whatever. (1896: 330)

As Dipert correctly points out, Peirce’s remarks reveal his“understanding of the richness and difficulty whichrelations introduce into logic” (Dipert 1984a: 63).^[11]

In order to increase expressive power, Peirce left the traditionalsyllogistic pattern and brought in Boolean algebra of logic. Thefollowing comment emphasizes that Peirce’s choice of notationmarks a clear departure from De Morgan’s pursuit of relationallogic:

De Morgan’s methodology is governed by the logic of syllogismwhile Peirce’s methodology is entirely algebraic. This algebraicmodel taken over from Boole is foreign to De Morgan’s methods.This difference in methodology reflects a significant difference atthe level of definition. (Brunning 1991: 36)

And after realizing the complicated nature of the logic of relations,Peirce explored new notationbeyond Boolean calculus. Thatmove is predicted in the following passage:

The effect of these peculiarities [the non-mechanic nature of relativelogic] cannot be subjected to hard and fast rules like those of theBoolian calculus. (1883a [CP 3.342])

Here is the third aspect of a transition from monadic to polyadiclogic: The complication that relations bring in our reasoning,obviously, pushed Peirce to develop a new notational system. As therest of this subsection shows, the process made over the course of 15years—from DNLR to “On the algebra oflogic”—is rather complicated. Importantly, Peirce’sintroduction of quantifiers and bound variables could be seen as aninevitable outcome of his ambitious goal to expand the scope offormalization to cover relations, as Merrill says “Thequantification complexities of many relational statements cried outfor quantifiers” (1997: 158).

The third section of DNLR, as the title “Application of theAlgebraic Signs to Logic” says, is one of the first places whereBoolean algebraic notation and relational logic joined each other. Inthe first subsection Peirce makes it clear that the territory he aimsto cover is relational, by including polyadic predicates in thefollowing way:

(DNLR [CP 3.63–64]; the entry adopts our modern terminologyinstead of Peirce’s.)

For the rest of the third section, four kinds of algebraic signs areintroduced to be applied on these predicate letters: the inclusionsign (\(\inclusion\)), the addition sign (\(\cunion\)), themultiplication sign (juxtaposition or “,”), and theinvolution sign (exponentiation).

First, he combines the sign equality “=” and the sign\(<\) for “less than” to come up with the sign“\(\inclusion;\)” to represent inclusion:

Thus,

\[\unary{f} \inclusion \unary{m}\]

means “every Frenchman is a man”, without saying whetherthere are any other men or not. So,

\[\binary{m} \inclusion \binary{l}\]

will mean that every mother of anything is a lover of the same thing;although this interpretation in some degree anticipates a conventionto be made further on. (DNLR [CP 3.66])

Note that “\(\unary{f} \inclusion \binary{m}\)” (unlike“\(\unary{f} \inclusion \unary{m}\)”) would beungrammatical since \(\binary{m}\), being a binary predicate, cannothave an inclusion relation with a unary predicate \(\unary{f}\).

For the sign of addition, Peirce brings in Boolean sign \(+\), butwith slight variation:

The sign of addition is taken by Boole, so that

\[x + y\]

denotes everything denoted by \(x\), andbesides, everythingdenoted by \(y\).…But if there is anything which is denoted byboth the terms of the sum, the latter no longer stands for any logicalterm on account of its implying that the objects denoted by one termare to be takenbesides the objects denoted by the other. Forexample,

\[\unary{f} + \unary{u}\]

means all Frenchmen besides all violinists, and, therefore, consideredas a logical term, implies that all French violinists arebesidesthemselves. For this reason alone…I preferred to take asthe regular addition of logic a non-invertible process, such that

\[\unary{m} \cunion \unary{b}\]

stands for all men and black things, without implication that theblack things are to be taken besides the men. (DNLR [CP 3.67])

Hence, Peirce’s slightly modified addition sign, \(\cunion\) ,denotes inclusive disjunction. “\(\unary{f} \cunion\unary{u}\)” denotes all those who are either a Frenchman or aviolinist. The notation does not imply that no Frenchman is aviolinist or no violinist is a Frenchman. Even though Peirce’sexample is limited to unary predicates, we can extend the idea tobinary. Using modern notation, \[\binary{l} \cunion \binary{s} = \{\langle x, y\rangle \mid lover(x, y) \lor servant(x,y)\}.\] That is, it corresponds toa union of relations.

When the multiplication sign enters the picture, the logic ofrelations becomes powerful, and here is Peirce’s hallmark forthe interpretation of multiplication:

I shall adopt for the conception of multiplicationthe applicationof a relation, in such a way that, for example,\(\binary{l}\unary{w}\) shall denote whatever is a lover of awoman.…\(\binary{s}(\unary{m} \cunion \unary{w})\) will, then,denote whatever is a servant of anything of the class composed of menand women taken together. (DNLR [CP 3.68])

When polyadic predicates are in the picture, how to form a newrelation becomes more interesting and complicated. This is why themultiplication operation of relative product is extremely importantfor further work on relational logic. A product between two predicatesis much more interesting than an addition between two, depending onwhat kinds of predicates are involved:

1. The product between two properties is another property, being theintersection between two properties,
2. the product of a relation and a property is another new property,and
3. the product between relations produces a new relation.

Let’s try to understand Peirce’s concept of relativeproduct in terms of modern terminology:

In this modern translation, the existence of an existential quantifieris noticeable, even though Peirce himself did not mention it at all inDNLR.

Hidden quantifier implication becomes even more obvious in theoperation of involution below.

I shall take involution in such a sense that \(x^y\) will denoteeverything which is an \(x\) for every individual of \(y\). Thus\(\binary{l}^{\unary{w}}\) will be a lover of every woman. (DNLR [CP3.77])

That is, \(\binary{l}^{\unary{w}} = \{ x \mid \forall y(\textit{woman} (y) \rightarrow \textit{lover} (x,y))\}\). Here, auniversal quantifier is present!

Before we go into the details of Peirce’s quantifiers,let’s summarize algebraic signs Peirce adopted to handlepolyadic predicates:

Let’s focus on hidden but assumed presence of quantifiers in thecase of multiplication and exponentiation: \(\binary{l}\unary{w}\) isinterpreted as “a lover of some woman” and\(l^{\unary{w}}\) as “a lover of every woman”.Interestingly, in the process of introducing polyadic predicates,Peirce ends up bringing in quantifiers,some andevery. On the other hand, considering that Aristotle’ssyllogisms have two quantifiers, Peirce’s algebraic notation forquantifiers—some andevery—should notsurprise us. However, a crucial aspect of this development is thatBoole’s unsatisfactory representation of existentialpropositions (as opposed to universal propositions) pushed Peirce andhis student O. H. Mitchell to go beyond Boole’s logic.^[13] The way Peirce interprets a relativeproduct—“\(\binary{l}\unary{w}\)” meaning“lover of some woman”—allows an existentialquantifier to be expressed implicitly in terms of multiplication.

Let us see several different ways Peirce pursued to representexistential statements in a more explicit way. In the above for“\(\binary{l}\unary{w}\)”, the existential quantifier iscarried out in the way relation \(\binary{l}\) is applied to a unarypredicate “\(\unary{w}\)”, but not explicitly. There areseveral different explicit ways Peirce represents existentialstatements. One method: borrowing exponentiation (in which a binarypredicate is applied to a unary predicate, e.g.,\(\binary{l}^{\unary{w}}\)), Peirce expresses an existential statementas a contradiction of a universal statement:

Particular [existential] propositions are expressed by theconsideration that they are contradictory of universal propositions.Thus, as \(\unary{h}\bcomma(1-\unary{b})=0\) means every horse isblack, so \(0^{\unary{h}, (1-\unary{b})} = 0\) means that some horseis not black; and as \(\unary{h}\bcomma \unary{b}= 0\) means that nohorse is black, so \(0^{\unary{h}, \unary{b}} = 0\) means that somehorse is black. (DNLR [CP 3.141])

The number 1 represents the universe class and 0 the null class.However, Peirce’s notation of exponentiation whose base is 0 hassome slightly different nuance. Let’s recall Peirce’sexponentiation operation:

Then,

[Note: The base 0 denotes a relation—the null relation, whilethe exponent 0 a class—the null class.]

There is no \(y\) such that \(\textit{null-class}(y)\), since notingcould be in thenull-class. Hence, vacuously, every object inthe domain gets in \(0^{0}\). That is, the class of things which hasno relation to the null class is the universe class which isrepresented by 1. Hence, \(0^{0} = 1\).

Since nothing can bear the null-relation to every member of anynon-null class, \(0^{\unary{m}} = 0\). Hence, we get the followingexponentiation notation:

Using this result, let’s unpack Peirce’s abovequotation:

1. 1 represents the universe class and 0 the null class.(Boolean symbols)
2. “\(\unary{h}\bcomma(1-\unary{b})\)” denotesthe class of non-black horses. (Multiplication operation forintersection)
3. “\(\unary{h}\bcomma(1-\unary{b})=0\)” saysthat there is nothing that is non-black horse. I.e., every horse isblack. (by1 and2)
4. “\(\unary{h}\bcomma(1-\unary{b}) \not = 0\)”means it is not the case that every horse is black. I.e., some horseis not black. (by3)
5. Since \(\unary{h}\bcomma (1-\unary{b}) \not = 0,0^{\unary{h},(1-\unary{b})} = 0\) (by(*) above)

Similarly, “\(\unary{h}\bcomma\unary{b}=0\)” means nohorse is black. Hence, “\(\unary{h}\bcomma \unary{b} \not =0\)” says some horse is black. Therefore,“\(0^{\unary{h}\bcomma\unary{b}} = 0\)” (being theexponent being not-zero) means that some horse is black.

The method presented in DNLR [CP 3.141] is interesting for several reasons.First of all, Peirce maintains Boole’s theme that allpropositions are represented as equations. Another is that Peirceutilizes the contradictory relation between universal and existentialpropositions. Even more interestingly and importantly, Peirce bringsin his exponentiation notation, exponentiation between relation toproperty (that is, between binary to unary predicates), to expressexistential statements.

After the rather complicated presentation of existential statements,focusing on the exponent part, Peirce suggests another, much simpler,way to express existential statements by using the inequalitysign:

Particular [Existential] propositions may also be expressed by meansof the signs of inequality. Thus, some animals are horses, may bewritten \(\unary{a}\bcomma\unary{h} > 0\). (DNLR [CP 3.143])

Another method Peirce adopts is to utilize the sign \(\inclusion\) forinclusion and the sign \(\bar{ \ \ }\) for complement. That is,

However, these explicit ways to handle existential statements arelimited to unary predicates or Aristotelian syllogisms, and cannot gobeyond them. Instead, we would like to examine Peirce’smultiplication and exponential expressions between relations andproperties more carefully. Let’s recall \(\binary{l}\unary{w}\)means “a lover of some woman” while\(\binary{l}^{\unary{w}}\) “a lover of every woman”. HerePeirce’s proposal for individual terms gets in the picture sothat existential and universal quantifiers may become more explicitlyrepresented. Peirce suggests individuals be denoted by capitals (DNLR[CP 3.96]). For example, for unary predicate \(\unary{w}\), if\(\unary{w} > 0\), then \(\unary{w} = \unary{W}' \cunion\unary{W}'' \cunion \unary{W}''' \cunion \cdots\),where each of \(\unary{W}'\), \(\unary{W}''\),\(\unary{W}'''\),…denotes an individual woman.Hence,

At this point, Peirce makes a connection (i) between existentialstatements and the sign \(\Sigma\) (as logical addition) and (ii)between universal statements and \(\Pi\) (as logical multiplication),and the following passage is a precursor for the notations whichappear in subsequent papers:

where \(\Pi'\) and \(\Sigma'\) signify that the addition andthe multiplication with commas are to be used. From this it followsthat

We have now come quite close to a modern notation of quantifiers.Dipert emphasizes the significance of Peirce’s notations foundin DNLR:

C. S. Peirce was the first person in the history of logic to usequantifier-like variable binding operators (briefly in 1870, W2, 392f,predating Frege’sBegriffsschrift (1879)). (Dipert2004: 290)

Ten years later, in a more comprehensive and more survey-style paper,“On the algebra of logic” (1880), we find Peirce’sideas of DNLR emerge more tightly and more systematically. Below wewill now summarize the developments of quantification presented in“Brief description of the algebra of relatives” (1882a),“The logic of relatives” (1883a), and “On thealgebra of logic: A contribution to the philosophy of notations”(1885a plus 1885b).

First, he modifies the previous idea of representing a property interms of individual terms and extends it to relations. In DNLR (1870),“\(\unary{w}\)”, which stands for the property“being a woman”, is expressed as

(where \(\unary{W}'\),… denotes each individual woman).However, in 1882a, Peirce expresses a unary predicate usingcoefficients: For each unary predicate \(x\) and for each object \(a\)in the domain, Peirce defines the coefficient \((x)_a\) in the following way.

Continuing the example for the unary predicate \(\unary{w},\) supposethe domain has objects \(\unary{A},\) \(\unary{B},\) \(\unary{C},\)….^[14] Some object is a woman and some is not. A coefficient\((\unary{w})_a\) is defined as follows:

Then,

A unary predicate is successfully represented as a sum of individuals.Moving on to a binary predicate, a relation is modeled by a pair of objects:^[15]

A dual relative term [binary predicate], such as “lover”,“benefactor”, “servant”, is a common namesignifying a pair of objects. (1883a [CP 3.328])

And he expresses a pair of objects as “\(A:B\)”, where\(A\) and \(B\) are individual objects. Let \(\binary{l}\) stand for“a lover”. Peirce defines a coefficient for each orderedpair of objects in the following way:

Then,

Peirce’s generalization for a unary predicate \(\unary{x}\) anda binary predicate \(\binary{l}\) goes like this (1883a [CP3.329]):

This is what Peirce had in mind when he wrote “Every term[predicate] may be conceived as a limitless logical sum ofindividuals” (1880 [CP 3.217]). We take only objects whosecoefficients are 1. Suppose \(A\), \(B\), and \(D\) are women.Applying Peirce’s sign “+” as inclusive-or,\(\unary{w} = \{ A, B, D\}\). Suppose \(A\) is a lover of \(C\), and\(B\) is a lover of \(D\). Then,\(l = \{( A: C),(B: D)\}\).^[16]

The next task is how to utilize this tool to express an existentialproposition. If there is at least one woman, say \(K\), in the domain,there is at least one coefficient \(\unary{w}_k\) such that it is 1.Hence, the sum of coefficients of the individuals in the domain isgreater than 0. That is, \(\Sigma_{i} w_i > 0\). If nobody is awoman, we will get \(\Sigma_{i} w_i = 0\). In the case of theuniversal proposition that everybody is a woman, the product ofcoefficients is 0 as long as one coefficient is 0, that is, there isone person who is not a woman. That is, \(\Pi_{i} w_i = 0\). Wheneverybody is a woman, \(\Pi _{i} w_i = 1\).

While the Boolean approach for quantifiers is limited to term logic,this way of handling quantifiers is so general that we can apply itdirectly to relations as well, as the following passage states:

Any proposition whatever is equivalent to saying that some complexusof aggregates [sums] and products of such numerical coefficients isgreater than zero. Thus,

\[\Sigma_{i}\Sigma_{j} \binary{l}_{ij} > 0\]

means that something is a lover of something; and

\[\Pi_{i}\Sigma_{j} \binary{l}_{ij} > 0.\]

means that everything is a lover of something. (1883a [CP 3.351])

And Peirce proposes to drop the \(“>0”\) part:

We shall, however, naturally omit, in writing the inequality, the\(“>0”\) which terminates them all; and the above twopropositions will appear as

\(\Sigma_{i}\Sigma_{j} \binary{l}_{ij}\) and \(\Pi_{i}\Sigma_{j}\binary{l}_{ij}.\qquad\) (1883a [CP 3.351])

Getting to Peirce’s 1883 paper, we witness two significant stepstaken: One is the use of index-subscripts in a crucial way, and theother is the alternation of \(\Sigma\) and \(\Pi\). After dropping“\(>0\)” Peirce draws our attention to the role ofsubscripts:

The following are other examples:

\[\Pi_i \Sigma_j (l)_{ij} (b)_{ij}\]

means that everything is at once a lover and a benefactor ofsomething.

\[\Pi_i \Sigma_j (l)_{ij} (b)_{ji}\]

means that everything is a lover of a benefactor of itself. (1883a [CP 3.352])

The order of index-subscripts is crucial to make a distinction betweenthese two propositions. On the other hand, a difference between“Everybody loves some woman” and “There is a womaneverybody loves” relies on the order of \(\Pi\) and\(\Sigma\):

Finally in “On the Algebra of Logic: A contribution to thephilosophy of notation” (1885a) all of thesedevelopments—(i) \(\Sigma\) (sum) forsome and \(\Pi\)(product) forall, (ii) utilizing coefficients and theirsubscripts to drop denotation of individuals (e.g., from \((l)_{ij}(A_{i}: A_{j})\) to \((l)_{ij}\)), (iii) mixing \(\Sigma\) and\(\Pi\), and (iv) omitting the “\(>0\)” part—havebecome official:

In general, according to 1885a [CP 3.393]:

\(\Sigma_i x_i\) means that \(x\) is true of some one of theindividuals noted by \(i\) or

\[\Sigma_{i}x_i = x_i + x_j + x_k + \textit{etc.}\]

In the same way, \(\Pi_i x_i\) means that \(x\) is true of all theseindividuals, or

\[\Pi_{i}x_i = x_i x_j x_k, \textit{etc.}\]

And if \(x\) is a simple relation [binary predicate],

• \(\Pi_i\Pi_j x_{i,j}\) means that every \(i\) is in this relationto every \(j\),
• \(\Pi_j\Sigma_i x_{i,j}\) that every \(j\) some \(i\) or other isin this relation,
• \(\Sigma_i\Sigma_j x_{i,j}\) that some \(i\) is in this relationto some \(j\).

For example, according to 1885a [CP 3.394]:

Let \(l_{ij}\) mean that \(i\) is a lover of \(j\) and \(b_{ij}\) that\(i\) is a benefactor of \(j\).

Let \(g_i\) mean that \(i\) is a griffin, and \(c_i\) that \(i\) is achimera.

Then \(\Pi_i\Sigma_j (l)_{ij }(b)_{ij}\) means that everything is atonce a lover and a benefactor of something; and

• [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land\textit{Benefactor} (x, y)]\), meaning that everybody is a lover and abenefactor of someone.]

And \(\Pi_i\Sigma_j (l)_{ij}(b)_{ji}\) means that everything is alover of a benefactor of itself.

• [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land\textit{Benefactor} (y, x)]\), meaning that everybody is a lover of abenefactor of himself/herself.]

And we find ourselves arriving at the land of modern logic. At thesame time, we realize that the key concepts and vocabulary offirst-order logic were already formed in his previous work discussedabove. We also note that the goal outlined in the first section of the1885 paper is more or less the same as the proposal made in his 1870paper: “The first is the extension of the power of logicalalgebra over the whole of its proper realm” (1885a [CP 3.364]).Also, he almost reiterates the limit of the project which he“regrets” in 1870: “I shall not be able to perfectthe algebra sufficiently to give facile methods of reaching logicalconclusions” (1885a [CP 3.364]). That is, we should not expect afull-blown deductive system in this paper, but “I can only givea method by which any legitimate conclusion may be reached and anyfallacious one avoided” (1885a [CP 3.364]). He carries out hispromise in section 3 of the paper titled as “§3.First-intentional logic of relatives”,^[17] by suggesting a list of methods of transformation. He did not mean toclaim this list is exhaustive, but “the one which seems to methe most useful tool on the whole” (1885a [CP 3.396]). Thefollowing are some of the rules involving quantifiers:^[18]

In spite of the six-year interval between Frege’sBegriffsschrift (1879) and Peirce’s quantificationtheory in 1885, credit has been given to both logicians. We call themboth the founders of modern logic, since Peirce was not aware ofFrege’s work on the topic. Also, it should be noted that Fregepresented a logical system equipped with axioms and rules, which wasnot pursued in Peirce’s work.

1.2 Boolean tradition—algebraic and model-theoretic

Boole’s aspiration to capture logic in terms of an algebraicsystem has inspired many mathematicians and logicians who areinterested in connecting the two disciplines, logic and mathematics.As seen in the previous section, Peirce is clearly one of them. Hepushed the Boolean idea of an algebraic system further in two ways:One is to improve Boole’s representation of particularproposition (i.e., “SomeA isB”) so thattraditional Aristotelian syllogisms may fit in an algebraic system.The other is to represent not only qualities but also relations sothat a new algebraic system may reach beyond traditional syllogisms.In that process, new notations for quantifiers and variables wereinvented.

Peirce’s two-decade work made a major contribution to thealgebra of logic tradition in two important ways. First,Peirce’s introduction of quantifiers and variables itself is asignificant advance in formal logic, close to the predicate logic weknow. Second, subsequent momentous work on mathematical logic wasbuilt on the new notation and extended logic by Peirce and hisstudents, O. H. Mitchell and C. Ladd (later Ladd-Franklin). A decadeafter Peirce’s “On the Algebra of Logic”, (1885)Ernst Schröder published three volumes of mathematical logicVorlesungen über die Algebra der Logik (1890–1905).^[19] His work was carried out squarely in the Boolean algebraic tradition,and two important aspects of the book reflect Peirce’sinfluence: He adopted Peirce’s notation (over Frege’s),and the third volume is devoted to the logic of relations.Goldfarb’s insightful paper expresses these two aspects ofPeirce’s influence in the following way:

Building on earlier work of Peirce, in the third volume of hisLectures on the algebra of logic [1895] Schröderdevelops the calculus of relatives (that is, relations). Quantifiersare defined as certain possibly infinite sums and products, overindividuals or other relations. (Goldfarb 1979: 354)

As seen in the previous subsection, including relations (not justqualities) in algebraic expressions and representing the universalquantifier as products (i.e., \(\Pi\)) and the existential quantifieras sums (i.e., \(\Sigma\)) were the main output of Peirce’s twodecade tireless work. Considering Schröder’s book was themost popular logic textbook for mathematical logic students duringthat era, we can easily say Peirce’s legacy has lived on.Peckhaus, working on a delicate relation between Frege’s andSchröder’s quantification theory, locates whereSchröder’s modern quantification theory originates from:

This [Schröder’sVorlesungen über die Algebra derLogik was the result of his learning modern quantification fromFrege’sBegriffsschrift] is a simple and plausibleanswer, but it is false. Schröder never claimed any priority forhis quantification theory, but he did not take it from Frege.Schröder himself gives the credit for his use of \(\Sigma\) and\(\Pi\) to Charles S. Peirce and Peirce’s student Oscar HowardMitchell (Schröder 1891, 120–121). (Peckhaus 2004: 12)

Afterwards well-known mathematicians and logicians, Löwenheim,Skolem and Zermelo, all used Peirce-Schröder notation. Peano wasalso very much familiar with Peirce-Schröder algebraic logic.Putnam includes Whitehead in this tradition as well:

This [Whitehead’sUniversal Algebra] is a work squarelyin the tradition to which Boole, Schröder, and Peirce belonged,the tradition that treated general algebra and logic as virtually onesubject. (Putnam 1982: 298)

Interestingly enough, Putnam points out that this portion ofWhitehead’s work was prior to his collaboration with Russell,and during this early period Whitehead’s work, especially onquantifiers, mentions Peirce and his students, but not Frege. Clearlywe needed to wait until Russell drew our attention to Frege, but“it was Peirce who seems to have been known to the entire worldlogical community” (Putnam 1982: 297). Putnam’s label ofthe Peirce group as “effective” discoverers of thequantifier and Frege as a discoverer could be a resolution to theFrege-first versus Peirce-first debate.

While many have focused on the development of quantifiers, it is quitenoteworthy that Tarski drew our attention to the importance of thealgebra of relations in his 1941 paper “On the Calculus of Relations”.^[20] Just as Peirce did in his DNLR (1870) paper, Tarski acknowledges DeMorgan’s contribution: It is De Morgan who first realized thenecessity of representing relations as well as qualities and struggledover the limits of traditional logic. And Tarski gives full credit toPeirce in terms of solid advance over the calculus of relations.

The title of creator of the theory of relations was reserved for C. S.Peirce. In several papers published between 1870 and 1882a, heintroduced and made precise all the fundamental concepts of the theoryof relations and formulated and established its fundamentallaws.…In particular, his investigations made it clear that alarge part of the theory of relations can be presented as a calculuswhich is formally much like the calculus of classes developed by G.Boole and W. S. Jevons, but which greatly exceeds it in richness ofexpression and is therefore incomparably more interesting from thedeductive point of view. (Tarski 1941: 73)

This passage not only situates Peirce’s logical achievement inthe context of the algebra of logic tradition but also characterizesPeirce’s work as an extension of Boole’s and Jevons’monadic logic. (For more details about Peirce’s position inBoole’s tradition, see the entrythe algebra of logic tradition.)

Some Peirce scholars have also claimed that Peirce’s inventionof quantifiers is a product of Peirce’s own philosophy of logic,which is different from Frege’s (Brady 1997; Burch 1997; Iliff1997; Merrill 1997). Hintikka’s proposal (1997) to explain themain difference between Frege’s and Peirce’s contributionsto modern logic is quite intriguing. Tracing back to Frege’s owndistinction betweencalculus ratiocinator versusa linguacharacterica, van Heijenoort adds a new dimension to these twoopposing views of logic beyond what Frege alluded to (van Heijenoort1967: footnote 1, p. 329).^[21] While Frege emphasized a difference between propositional andquantification logic, van Heijenoort located a difference in what istaken as a totality. Boole’s tradition does not make anyontological commitment about a totality, but it “can be changedat will” (1967: 325). On the other hand, Frege’s languageis aboutthe universe. Borrowing van Heijenoort’sdistinction between Boole’s logic as a calculus andFrege’s universality of logic, Hintikka locates Peirce inBoole’s camp, calling it the model-theoretic tradition. UnlikeFrege’s view of the universe, the model-theoretic traditionallows us to reinterpret a language and thus assign differentuniverses to quantifiers. According to Hintikka, Peirce’sdevelopment of modal logic is a good piece of evidence to show howfruitful Peirce’s way of understanding quantifiers could be(Hintikka 1997). In the next section where Peirce’s graphicalsystems are introduced, we will revisit this issue.

2. From Symbolic to Iconic Representation

So far, we have argued that Peirce’s insight on relations pushedhim to extend the territory of logicfrom monadic,non-relational, propositional logicto polyadic, relational,quantification logic. This is the beginning of modern logic as we knowit. In this section, taking up a different angle of Peirce’sadventure—to extend forms of representationfromsymbolic systemsto diagrammatic systems, we present a storywhere his two different kinds of extension—one fromnon-relations to relations and the other from symbolic todiagrammatic—are connected with each other.

Peirce presented propositional logic, quantification logic, and modallogic in a graphical way, and invented three systems of ExistentialGraphs (EG)—Alpha, Beta, and Gamma, respectively. In spite ofPeirce’s own evaluation of Existential Graphs as “my chefd’oeuvre”, EG had to wait to be understood for a halfcentury until two philosophers—Don Roberts and JayZeman—produced their impressive work. In the 1980s, EG wasreceiving attention from new disciplines—computer science andartificial intelligence—thanks to John Sowa’s novelapplication of EG to knowledge representation inConceptualStructure (1984). More recently, toward the end of the twentiethcentury interdisciplinary research on multi-modal reasoning has drawnour attention to non-symbolic systems (see, e.g., Barwise &Allwein [eds] 1996 and Barwise & Etchemendy 1991) and EG, notsurprisingly, occupied the top of their list. In that context, Shin(2002) focused on differences between symbolic versus diagrammaticsystems and suggested a new way of understanding the EG system, thoughthis was criticized in Pietarinen 2006.

While Peirce mainly presented linear expressions in his officialwritings from 1870 to 1885,^[22] the notation adopted in Frege’s 1879Begriffsschriftis more iconic; it is at least not as linear as Peirce’s in theabove period. However, it is Peirce, not Frege, who invented afull-blown non-symbolic system for first-order logic—ExistentialGraphs. It is Peirce’s EG, not his linear first-order notation,which is presented as a deductive system with inference rules. As theEG system has been investigated more rigorously, philosophicalquestions involving Peirce’s invention of the system have beenraised as well. The discovery of EG’s power and novelty hasnaturally led us to other parts of Peirce’s philosophy. Why andhow did the invention of EG come about? What does EG reveal aboutPeirce’s view of logic and representation?

Many have pointed to Peirce’s theory of signs, which classifiessigns as being of three kinds—symbols, indices, andicons—as the foremost theoretical background for Peirce’s EG.^[23] For example, as will be shown below, ovals and lines, along withletters, are the basic vocabulary of Peirce’s EG. It is naturalto connect Peirce’s interest in icons with his invention ofgraphical systems, and the connection is real (Shin 2002:22–35). However, to pinpoint the features of icons and theiconic nature of Peirce’s graphical systems requires much morework than our intuition provides. Moreover, there is a big gap betweenPeirce’s discussions of icons^[24] and his invention of full-blown graphical systems; something else hasto be brought into the picture to explain how Peirce got from hisinitial ideas about icons all the way to his EG.

In a slightly different and bigger picture, van Heijenoort’sdistinction between Boole’scalculus ratiocinatorversus Frege’slingua characteristica could be relatedto the topic. Agreeing with both Hintikka’s and Goldfarb’sevaluation that Peirce belongs to Boole’s tradition, Shin findsa connection between the model-theoretic view of logic (where Booleand Peirce are placed) and EG’s birth (see Shin 2002:14–16 and Pietarinen 2006). However, Peirce’s awareness ofthe re-interpretation of language is necessary, but not sufficient,for his pursuit of a different form of representation. While theacknowledgment of the possibility of different models of a givensystem was presupposed by Peirce’s project for various kinds ofsystems, not every Boolean has presented multiple systems. Boolehimself

was quite conscious of the idea ofdisinterpretaion, of theidea of using a mathematical system as an algorithm, transforming thesigns purely mechanically without any reliance on meanings. (Putnam1982: 294)

On the other hand, Burris and Legris’s entry shows us howBoole’s algebra of logic tradition has led us to the developmentof model theory (see the entry onthe algebra of logic tradition).

2.1 Pragmatic maxim applied to the logic of relations

Without challenging these existing explanations involvingPeirce’s EG, in this entry we would like to bring in oneoverlooked but crucial aspect of Peirce’s journey to EG so thatour story may fill in part of the puzzle of Peirce’s overallphilosophy. Peirce’s mission for a new logic started with how torepresent relations, which led him to invent quantifiers and boundvariables, as we discussed in the previous section. The samecommitment, that is, to represent relations in a logical system, weclaim, was a main motivation behind Peirce’s search for a newkind of sign systems—iconic representation of relations.Peirce’s work on Euler/Venn diagrams provides us with anotherpiece of evidence to support our claim that the main motivation behindEG was to represent relations. While improving Venn systems, Peircerealizes that the following defect cannot be eliminated:

[T]he system [Venn’s] affords no means of exhibiting reasoning,the gist of which is of a relational or abstractional kind. It doesnot extend to the logic of relatives. (Peirce 1911b [CP 4.356])

Again, we do not think this isthe crucial ingredient for thecreation of EG, but one key element which works nicely together withhis theory of signs and his model-theoretic view of logic.

Peirce’s graphical representation first appears in his 1897paper “The Logic of Relatives”. After his own new linearnotation came out in 1885 as seen above, why did Peirce revisit thelogic of relations? The first paragraph of the paper provides a directanswer:

I desire to convey some idea of what the new logic is, how two“algebras”, that is, systems of diagrammaticalrepresentation by means of letters and other characters, more or lessanalogous to those of the algebra of arithmetic, have been inventedfor the study of the logic of relatives, and…. (1897a [CP3.456])

Two things should be noted. One is that diagrammatic systems are alsocalled “algebra” by Peirce. That is, according to Peirce,algebra is not limited to symbolic systems. The other is that Peircemakes it clear that two different forms of algebra carry outthe new logic, not new logics.

In thinking about the scope of the logic of relations, the questionarises: Why did Pierce feel the need for another form ofrepresentation different from the 1885 notation? “I must clearlyshow what a relation is” (1897a [CP 3.456]). The clearunderstanding of “relations”, Peirce believes, is a guidefor his excursion into different forms of logical systems. Here wewould like to draw reader’s attention to Peirce’swell-known paper “How To Make Our Ideas Clear ” (1878),where three sections are devoted to the three grades of meaning (seethe entry onPeirce’s theory of signs).

The first grade of understanding the word “relation” comesfrom our ordinary experience, and the second grade is to have a moreabstract and general definition-like understanding. According toPeirce, that is not enough to achieve a full understanding of the word“relation”. Finally, Peirce’s hallmark of thepragmatic maxim leads us to the third grade of clarity:

It appears, then, the rule for attaining the third grade of clearnessof apprehension is as follows: Consider what effects, which mightconceivably have practical bearings, we conceive the object of ourconception to have. Then the whole of our conception of those effectsis the whole of our conception of the object. (1878 [CP 5.402])

In order to understand what a relation is, we need to know whatfollows from it. Then, the question is how we know what itsconsequences are. Here is one answer given by Peirce in the 1897apaper, as far as the term “relation” goes:

The third grade of clearness consists in such a representation of theidea that fruitful reasoning can be made to turn upon it, and that itcan be applied to the resolution of difficult practical problems.(1897a [CP 3.457])

Therefore, how a relation is represented is crucial in figuring outwhat follows from a relational state of affairs. Betterrepresentations will yield more “fruitful reasoning” andhence, will be more helpful for solving practical problems. It isobvious that in the paper Peirce intends to search for moredesirable representations. Importantly, in section 4 when thethird grade of clearness of the meaning “relation” isdiscussed, diagrammatic representation of relations makes its firstappearance.

Influenced by A. B. Kempe’s graphic representation,^[25] Peirce finds an analogy between relations and chemical compounds:

A chemical atom is quite like a relative in having a definite numberof loose ends or “unsaturated bonds”, corresponding to theblanks of the relative. (1897a [CP 3.469])

A chemical molecule consists of chemical atoms, and how atoms areconnected with one another is based on the number of loose ends ofeach atom. For example, chemical atom H has one loose end and chemicalatom O has two. So, the following combination is possible, and it is arepresentation of the water molecule, H₂O:

An analogy to the logic of relations runs like this: A sentenceconsists of names (proper names or indices) and predicates, and eachpredicate has a fixed arity. For example, the predicate“love” needs two names and “give” three.Hence, the following diagrammatic representation is grammatical and itis a representation of the proposition “John lovesMary”.

Peirce created a novel and productive analogy in representationbetween chemistry and the logic of relation by adopting the doctrineof valency as the key element for the analogy, as shown in the abovetwo diagrams. Believing that this graphic style of representationwould help us conceive the consequences or effects of a given relationin a more efficient way,^[26] Peirce presents Entitative Graphs, which is a predecessor of EG.^[27]

EG keeps the representation of a relation developed here, and remainsas Peirce’s final and the most cherished notation for the logicof relations (1903a). EG consists of three parts, Alpha, Beta, andGamma, which correspond to propositional, first-order, and modallogic, respectively. After presenting the Alpha system in a formalway, we discuss the Beta system of EG focusing on Peirce’s novelideas in expanding a propositional graphic system to aquantificational graphic system. For more details, we recommend workson EG by Roberts, Zeman, Sowa, and Shin.

2.2 Alpha system

Peirce’s Alpha graphs may be drawn on a blackboard, on awhiteboard, or on a sheet of paper. The basic unit is a simplesentence without any sentential connectives, that is, negation,conjunction, disjunction or conditional, etc. The following is anexample of a basic Alpha graph, asserting that it is sunny.

When we would like to assert that it is sunny and windy, we juxtaposetwo basic Alpha graphs in the following way:

In order to make the Alpha Graph Boolean-functionally complete, all weneed is to represent negation. The following Alpha graph says that itis not the case it is sunny, by enclosing the above graph with acut:

When we have negation and conjunction, it is important to keep theorder in the right way. “It is not sunny and it is notwindy” is different from “It is not the case that it issunny and windy”. Hence, the sentence “It is not the casethat it is sunny and it is windy” is ambiguous, depending on thescope of “it is not the case”. In the case of sententiallogic, parentheses get in to prevent this ambiguity: \(\neg (S \landW)\) versus \(\neg S \land W\). Peirce’s warning follows:

The interpretation of existential graphs isendoporeutic,that is proceeds inwardly; so that a nest sucks the meaning fromwithout inwards unto its centre, as a sponge absorbs water. (Peirce1910a: 18, Ms 650)

Hence, the following Alpha graph should be readnot as“\(\neg P \land \neg Q\)”,but as “\(\neg(P\land \neg Q)\)”:

This way of understanding Alpha Graphs is not incorrect, but has givena wrong impression that the Alpha system is equivalent to a sententialsystem with two connective symbols, negation and conjunction. We allprefer having more connectives than these two, especially when we usethe language. The section explores an alternative reading of Alphadiagrams, beyond negation and conjunction only, without introducingany new syntactic device.

Below we introduce Alpha Graphs as a formal system equipped with itssyntax and semantics. These tools not being available to Peirce, thepresentation aims to show Peirce’s EG is not intrinsicallydifferent from other formal systems. At the same time, in order toplace Peirce’s graphical systems in the traditionalwell-developed discourse of logic, there will be an intermediatestage, that is, to read off Peirce’s graphs into symboliclanguage. This will make Peirce’s graphs more accessible, and atthe same time support our claim that Peirce extended forms ofrepresentations with the same scope of logic as symbolicrepresentation.

Syntax

Vocabulary

1. Sentence symbols: \(A_{1},\) \(A_{2},\)…

2. Cut

   

Well-formed diagrams

1. An empty space is a well-formed diagram.
2. A sentence symbol is a well-formed diagram.
3. If \(D\) is a well-formed diagram, then so is a single cut of\(D\) (we write“\([D]\)]”).
4. If \(D_{1}\) and \(D_{2}\) are well-formed diagrams, then so isthe juxtaposition of \(D_{1}\) and \(D_{2}\) (write“\(D_{1}\D_{2}\)”).
5. Nothing else is a well-formed diagram.

Here we present two equivalent reading methods for the system. TheEndoporeutic reading algorithm, formalized based on Peirce’s ownsuggestion (as quoted above), is a traditional way to understand EG.An alternative reading method, the Multiple reading algorithm, wasmore recently presented to approach EG in a more efficient way.^[28]

Endoporeutic Reading Algorithm

1. If \(D\) is an empty space, then it is translated into\(\top\).
2. If \(D\) is a sentence letter, say \(A_{i}\), then it istranslated into \(A_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then, \([D]\) istranslated into \((\neg \alpha)\).

4. Suppose the translation of \(D_{1}\) is \(\alpha_{1}\) and thetranslation of \(D_{2}\) is \(\alpha_{2}\).

   Then, the translation of \(D_{1}\ D_{2}\) is \((\alpha_{1} \land\alpha_{2})\).

Multiple Readings Algorithm

1. If \(D\) is an empty space, then it is translated into\(\top\).
2. If \(D\) is a sentence letter, say \(A_{i}\), then it istranslated into \(A_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then,\([D]\) is translated into \((\neg \alpha)\).
4. Suppose the translation of \(D_{1}\) is \(\alpha_{1}\)and the translation of \(D_{2}\) is \(\alpha_{2}\).
   1. the translation of \(D_{1} D_{2}\) is \((\alpha_{1}\land \alpha_{2})\),
   2. the translation of \([D_{1} D_{2}]\) is\((\neg\alpha_{1} \lor \neg \alpha_{2})\),
   3. the translation of \([D_{1}\ [D_{2}]]\) is \((\alpha_{1}\rightarrow \alpha_{2})\), and
   4. the translation of \([[D_{1}]\ [D_{2}]]\) is\((\alpha_{1} \lor \alpha_{2})\).

Each of these two readings has its own strength.^[29] The Endopreuticreading assures us that the Alpha system is truth-functionallycomplete, since it has power to express conjunction and negation.However, this traditional method has been partly responsible for thefollowing two incorrect judgments about Alpha graphs:

1. There is not much difference between the Alpha system and apropositional language with only two connectives, \(\land\) and\(\neg\), except that Alpha graphs have cuts instead of symbolicconnectives.
2. When it comes down to practical use, just as we do not want touse only two connectives in a language, we have no reason to adopt theAlpha system over propositional languages with more connectives.

Challenging these misconceptions, the Multiple readings algorithmshows that Alpha diagrams do not have to be read off as a sentencewith “\(\land\)” and “\(\neg\)” only, but canbe directly read off in terms of other connectives as well. Twoquestions may be raised:

1. Is there a redundancy in the Multiple readings method? Forexample, isclause 4(b) above dispensable in terms ofclause 3 andclause 4(a)?
2. Does this new reading show that the Alpha system is just like apropositional language with various connectives?

Let us answer these questions through the following example.

Example

The following graph is translated into the following fourformulas:

1.	\(\neg(\neg R \land \neg S)\)	Endoporeutic Reading
2.	\(R \lor S\)	4(d) of Multiple Readings
3.	\(\neg R \rightarrow S\)	3 and4(c) of Multiple Readings
4.	\(\neg\neg R\lor \neg\neg S\)	3 and4(b) of Multiple Readings

The Endoporeutic reading allows us to get the first reading only, butwe may obtain different sentences by the Multiple Readings. Of course,all of these sentences are logically equivalent. Here is aninteresting point: In the case of symbolic systems, we need to provethe equivalence among the above sentences by using inference rules.But, derivation processes are dispensable in the case of the Alphasystem when the Multiple readings are adopted.^[30] Hence, having theclause 4(b) above in addition toclause 3 andclause 4(a) is not redundant, but instead highlights a fundamental differencebetween the Alpha system and a symbolic language with variousconnectives (see Shin 2002: §§4.3.2, 4.4.4, and 4.5.3).

Since we have the semantics for propositional logic and our readingmethods translate Alpha diagrams into a propositional language, we canlive without the direct semantics. However, if one insists on thedirect semantics:

Semantics

Let \(v\) be a truth function such that it assigns t or f to eachsentence letter and t to an empty space. Now, we extend this functionto \(\overline{v}\) as follows:

1. \(\overline{v}(D) = v(D)\) if \(D\) is a sentence symbol or an emptyspace.

2. \(\overline{v} ( {\bf [}D{\bf ]})\) = t iff \(\overline{v}(D)\) =f.

3. \(\overline{v}(D_{1} D_{2})\) = t iff \(\overline{v}(D_{1}) =\)t and\(\overline{v}(D_{2})\) = t.

We also would like to emphasize that this is not the only way toapproach Peirce’s EG. For example, some claim thatgame-theoretic semantics were foreshadowed by Peirce, and thus arguefor a more dynamic understanding of EG from the game-theoretic pointof view (Burch 1994; Hilpinen 1982; Hintikka 1997; Pietarinen2006).

Peirce makes it clear his EG is a deductive system equipped withinference rules:

The System of Existential Graphs is a certain class ofdiagrams upon which it is permitted to operate certaintransformations. (1903a [CP 4.414])

The inference rules for the Alpha system are presented as follows:(1903a [CP 4.415])^[31]

Emphasizing the symmetry both in erasure versus insertion and in evenversus odd number of cuts, Shin rewrote the rules (Shin 2002:84–85):

Reformulated Transformation Rules

1. RR1: In an E-area,^[32] say, area \(a\),

   1. we mayerase any graph, and

   2. we maydraw graph \(X\), if there is a token of \(X\)

      1. in the same area, i.e., area \(a\), or

      2. in the next-outer area from area \(a\).

2. RR2: In an O-area,^[33] say, area \(a\),

   1. we mayerase graph \(X\), if there is another token of\(X\)

      1. in the same area, i.e., area \(a\), or

      2. in the next-outer area from area \(a\), and

   2. we maydraw any graph.

3. RR3: A double cut may be erased or drawn around any part of agraph.

For examples of deduction sequences, refer to Roberts (1973:45–46) and Shin (2002: 91).

2.3 Beta system

In§1.1, we showed that formalizing relations was a key motivation behindPeirce’s new logic—first-order logic. In§2.1, we established a connection between Peirce’s own pragmaticmaxim and his graphic representation of relations. Peirce did not aimto present a new logic by inventing a graphic system, but rather topresent another new notation for the logic carried out by quantifiersand bound variables. He almost took it for granted that a graphicrepresentation of relations helps us observe their consequences in amore efficient way. Hence, the Beta system may be considered to be thefinal stop of Peirce’s long journey to search for betternotation for the logic of relations, which started in 1870 at the latest.^[34]

We will not go into the formal details of the Beta system in thisentry but will instead refer to Chapter 5 of Shin, where threeslightly different approaches to Beta graphs—Zeman’s,Roberts’, and Shin’s—are discussed at a full length.While Zeman’s reading is comprehensive and formal,Roberts’ method seems to appeal to a more intuitiveunderstanding of the system. Taking advantage of the merits of thesetwo existing works, Shin developed a new reading method of Beta graphsand reformulated the transformation rules of the system.^[35] Her approach focuses on visual features of Beta graphs and highlightsfundamental differences between symbolic versus diagrammatic systems.In the remaining part of the entry, we would like to examine how theessence of the logic of relations is graphically represented in theBeta system so that the reader may place EG in the larger context ofPeirce’s enterprise.

The introduction of quantifiers and bound variables is believed to beone of the key steps of first-order logic in symbolic systems. This iswhy some logicians take Peirce’s 1885 paper “On theAlgebra of Logic: A contribution to the Philosophy of Notations”to be the birthplace of modern logic. If this is the case, then howdoes Peirce represent quantifiers and bound variables in Betagraphs?

Interestingly enough, when Peirce considered a graphic system hisfirst concern was representation of relations, not representation ofquantifiers. As we said in §3.1, Peirce presented diagrammaticrepresentation based on an analogy to chemical molecules for a fullunderstanding of relations. Hence, the arity of a predicate isrepresented by the number of lines radiating from the predicate term.Next, Peirce extends the use of a line to connect predicates:

In many reasonings it becomes necessary to write a copulativeproposition in which two members relate to the same individual so asto distinguish these members.… [I]t is necessary that the signsof them should be connected in fact. No way of doing this can be moreperfectly iconic than that exemplified in [the following graph]:

(1903b [CP 4.442])

The line connecting two predicates, representingone and thesame object, is called aline of identity by Peirce.That is, the sameness is represented visually in Beta diagrams.^[36] In the case of a symbolic language, we may adopt one and the samequantified variable-type to represent the identity. For example, theabove diagram says \(\exists x(x <A \ \land \ B < x)\), andhence, the variable-type \(x\) (roughly) corresponds to the identityline. However, the same variable-type is not sufficient for expressingthe sameness in other cases, e.g., \(\exists x(x <A \ \land \ B< x) \rightarrow \exists x (x < C)\).

The way universal and existential statements are represented in theBeta system highlights a difference between graphic and symbolicsystems. Rather than adopting one more syntactic device forquantification, Peirce relies on the following visual features:

[A]ny line of identity whose outermost part is evenly enclosed referstosomething, and any one whose outermost part isoddly enclosed refers toanything there may be.(1903b [CP 4.458]^[37])

Let us borrow the two following graphs from Roberts (1973: 51):^[38]

The first graph (where the outermost part of the line is evenly, zero,enclosed) says that something good is ugly, and the second graph(where the outermost part is enclosed once) says that everything goodis ugly.^[39]

How about the scope problem which arises when multiple quantifiers areused? In the case of a symbolic system, the linear order takes care ofthe problem. Peirce’s solution for EG is to read off anotherkind of visuality: The less enclosed the outermost part of a line is,the larger the scope that the line gets.

Roberts’ following example illustrates the scope matter nicely(1973: 52):

The first graph says

and the second

In the first graph, the line whose outermost part is oddly enclosed isless enclosed than the line whose outermost part is evenly enclosed.Therefore, the universal quantifier has larger scope than theexistential quantifier. In the second graph, it is the other wayaround.

Let us summarize three interesting features of the Beta system:

1. Relations are represented graphically, not symbolically, in the Betasystem, in terms of a line. We argued that ultimately Peirce’spragmatic maxim was behind this alternative way ofrepresentation.

2. A distinction between universal versus existential statements isrepresented by the visual fact about whether the outermost part of aline lies in an area enclosed either by an odd number or by evennumber of cuts.

3. The order of quantification is represented by the following visuality:The less enclosed a line is, the more extensive scope it has.

3. From Bivalent to Triadic Logic

Fisch and Turquette (1966) discovered three crucial pages out ofPeirce’sLogic Notebook (1865–1909, Ms 339).^[40] This shows that Peirce’s invention of three-valued sententiallogic predates by at least a decade Jan Lukasiewicz’s and EmilPost’s achievements on the same topic. The three pages containthe essential elements of triadic logic and an intriguing passageabout Peirce’s motivation behind triadic logic. If we putPeirce’s development of triadic logic in contemporary terms,Peirce seemed to be branching out to non-standard logic. If so, thisadventure would be qualitatively different from the other two we havejust discussed in the previous sections.

When Peirce developed relational logic, the territory of formalizationwas vastly expanded. New vocabulary, hence, new syntactic rules andsemantic rules, were added. Naturally we welcome the territory offormalization and, hence, theoretical justification is not needed.From sentential to relational logic—this is an extension in aliteral sense: We do not discard the previous results—hence theyare preserved—but all we do is to expand them.

On the other hand, in the case of extension to non-symbolic languages,the logic itself stays the same, without addition or subtraction, buta new form of representation is introduced. That is, what to representis not extended, but how to represent is. Some might not see the needfor various forms of representation, and might not be convinced of thenecessity of graphical systems. Nonetheless, at a theoretical level,Peirce’s EG does not demand lengthy theoretical justification.In some sense, the proof is in the pudding: Can this new graphicalsystem carry out the same task as existing symbolic systems do? If so,which system is easier to use? Which system is more efficient? Wemight not arrive at a clean consensus, but the discussions are more orless predictable.

However, when one more semantic value is added toT (true) andF (false), logic is not preserved any more. When semantics isextended or changed, the new logic is neither a monotonic expansion ofthe territory of logic nor an alternative syntactic form ofrepresentation to existing symbolic systems. Triadic logic, byintroducing one more semantic value, departs from the standard logicwhich is based on bivalence. Here, the status of the principle ofexcluded middle (“Q or not-Q”) is shaken. Sois the law of contradiction. It is the burden of any non-standardlogic to justify its being non-standard: Why the third value? What isthe third value? Unknown? If so, is it an epistemological issue?Indeterminate? If so, does this require a metaphysicalexplanation?

The first subsection summarizes Peirce’s calculus of triadiclogic and the second briefly discusses Peirce’s own motive fortriadic logic.

3.1 Truth table of a three-valued system

Three values,V,L, andF, are introduced, whereV is true,L indeterminate, andF false. Thetraditional semantic domain for sentential logic, true and false, isextended to include “indeterminate”. Based on thisextended semantic territory, Peirce presents the semantics of severalsentential operators, one unary and the others binary. ModifyingPeirce’s presentation slightly to make it more similar to ourconventional truth table style without changing content, we presentthe three operators’ truth tables.

The semantics of a unary operator, which corresponds to negation:

The semantics for six binary connectives is presented:

Why six? What is the rationale behind the semantics of theseconnectives? One way to understand them is to figure out a dominancehierarchy among the three values.

In the case of \(\Phi\),

1. if at least one isV, then \(\Phi(x, y)\) isV,
2. else if at least one isF, then \(\Phi(x, y)\) isF,and
3. else \(\Phi(x, y)\) isL.

That is,V is the most dominant,F next, andL isthe least.

Six patterns of hierarchy emerges:

Peirce’s \(\Theta\) is our familiar disjunction andPeirce’sZ conjunction.

3.2 Why the third value?

In Peirce’s own words:

Triadic Logic is that logic, which though not rejecting entirely thePrinciple of Excluded Middle, nevertheless recognizes that everyproposition,S isP, is either true or false, or elsehas a lower mode of being such that it can neither be determinatelyP, nor determinately not-P, but is at the limit betweenP and notP. (Ms 339, copied from Fisch & Turquette1966: 75)

When do we have valueL (indeterminate) for the proposition“S isP”? SometimesS, Peirce says,has a lower mode of beingP and is at the limit betweenP and notP. The crux of the matter is how to interpretthe two phrases—“lower mode of beingP” and“being at the limit betweenP and notP”.Existing literature offers two different explanations—modalityversus continuity.

Fisch and Turquette, upon the discovery of Peirce’s notes ontriadic logic, locate the root of indeterminacy in potentiality. Thatis, indeterminacy is the semantic value assigned to an unrealizedsituation; hence, we can say neither “S isP” nor “S is notP” at thispoint. Potentiality, according to this view, cannot be captured bydyadic logic. If so, Peirce’s triadic logic is directly relatedto modality talk, which Fisch and Turquette conclude:

Essentially, Peirce seems to be saying that triadic logic may beinterpreted as a modal logic which is designed to deal with theindeterminacies resulting from that mode of being which Peirce hascalled “Potentiality” and “Real Possibility”.Under such an interpretation, dyadic logic becomes a limiting case oftriadic modal logic resulting from removing indeterminacy and beingdetermined entirely by “Actuality”. (Fisch & Turquette1966: 79)

According to the modality interpretation, Peirce’s “lowermode of beingP” meansP not being actual, andPeirce’s third valueL, being potential, is “at thelimit betweenP (i.e.,T) and notP (i.e.,F)”. Later in the paper, suggesting a possible relationbetween Peirce’s triadic logic and MacColl’s implication(as opposed to material implication), the authors make an interestingremark:

Considering MacColl’s rejection of Russell’s materialimplication, it is interesting to notice also that MacColl’s“Def. 13” gives what is now called “C. I.Lewis’s strict implication”. (Fisch & Turquette 1966:83)

Even though the connection was not pursued further in their paper, onecannot help realizing that their modality interpretation is boosted bya relation to MacColl’s implication since C. I. Lewis’strict implication is a beginning of modal logic. However, equatingPeirce’s triadic logic with modal logic, the modality view needsto explain the relation between Peirce’s Gamma graph lecture in1903c (which is aboutmodality) and the triadic logic notes written in 1909 ([Ms 339] 340v, 341v, 344r). Modal logic explored in Gammagraphs is the extension of classical logic, which required newvocabulary, e.g., broken cuts and tincture. Modal logic does not haveto be non-standard. On the other hand, triadic logic does not add anyvocabulary, but brings in different interpretations, and becomesnon-standard logic. On a slightly different note, Fisch andTurquette’s suggests Peirce’s tychism (the view thatindeterminacy is part of the reality) as a motivation forPeirce’s invention of triadic logic. If so, Peirce’striadic logic is a reflection of his own metaphysics.

Challenging the modality view, Robert Lane proposes the continuityinterpretation for Peirce’s triadic logic. According to Lane,Peirce’s indeterminate valueL has nothing to do withmodality, hence, Peirce’s development of triadic logic is notanother mechanism for modal logic, but with Peirce’ssynechism—the doctrine “that all that exists iscontinuous” (c. 1897b [CP 1.172])! How does Peirce’sphilosophy of continuity justify the third value?

First, Lane makes a distinction between the principle of excludedmiddle (PEM, henceforth)being false with regard to aproposition and PEMnot being applied to a proposition. IfPEM is true or false, it means the principle is applied to it. And,Lane claims that PEM is applied to only non-general and non-modalpropositions, citing the following passages from Peirce:

anything isgeneral in so far as the principle of excludedmiddle does not apply to it and isvague in so far as theprinciple of contradiction does not apply to it. (1905: 488 [CP5.448])

an assertion is said to be made in “the mode of necessity”if, and only if, the affirmation and the denial that [sic] which is soasserted could conceivably be both alikefalse. Thus if aperson says “It willcertainly rain tomorrow”, itmay be alike false that it is certain to rain and that it is certainnot to rain. (1910b: 26–28, Ms 678)

If a proposition is either general or expresses necessity, PEM is notfalse, but is not applied. Hence, focusing on individual and non-modalpropositions, Lane draws our attention to a special nature of apredicate inL-propositions. Lane calls the kind of theproperty which results inL-propositions a“boundary-property”. Here is Peirce’s own example ofa boundary-property:

Thus, a blot is made on the sheet. Then every point of the sheet isunblackened or blackened. But there are points on the boundary line,and those points are insusceptible of being unblackened or of beingblackened, since these predicates refer to the area about S and a linehas no area about any point of it. (Ms 339: 344r, quoted in Lane 1999:294)

Those points on the boundary line are neither black nor non-black.Consider the propositions “PointO is black” and“PointO is not black” (where pointO is onthe boundary line of a black blot). Neither of them is true, butfalse, either. These are prime examples of Peirce’sL-proposition. Using Peirce’s own truth-tables in theprevious subsection, let’s compute the truth value of“PointO is black or pointO is notblack”.

Let \(\alpha\) be the value of “PointO is black”,which isL.

Note that PEM is applied, but not true, period.

Lane’s following conclusion would be welcomed by many Peircescholars:

[B]oundary-propositions were important to Peirce because continuitywas important to him;…this [the thought that an actualbreach of continuity possesses neither of the properties that are theboundary properties relative to that breach] led him to think thatboundary-propositions are neither true nor false. To accommodate suchpropositions, and thus the phenomenon of continuity, within the boundsof formal reasoning, was, I contend, the motivation behindPeirce’s experiments in triadic logic. (Lane 1999: 304)

Regardless of endorsement of the continuity talk, some might notwelcome metaphysics getting into logic. Moreover, if Peirce’ssynechism is not embraced, Peirce’s triadic logic, which Laneargues is an attempt to formalize the continuity phenomenon, mightlose its force.

The previous two sections showed that Peirce’s relational logicand graphical systems push us further both in what we do with logicand how we do logic so that we may formalize more and we may formalizein more diverse ways. Triadic logic, as explained at the beginning ofthis section, is neither just a monotonic extension to get further noran alternative to get to the same place. By extending semanticentities, we have a different logic, for example, PEM being not true.That is why we call triadic logic a non-standard logic. However,Peirce’s way of introducing the third value gives us some pause.First of all, unlike with contemporary triadic logic, Peirce does notdiscard PEM completely:

Triadic Logic…not rejecting entirely the Principle of ExcludedMiddle,… (Ms 339: 344r, copied from Fisch & Turquette 1966:75)

I do not say that the Principle of Excluded Middle is downrightfalse; (1909: 21–22 [NEM 3/2: 851], quoted in Fisch& Turquette 1966: 81)

For certain (not all) properties, because of the way things are, wefind ourselves caught at limits between clearlyP and clearlynon-P. If we want to formalize those cases as well, the thirdvalue,L, is needed to express the indeterminacy of boundarycases. Hence, Peirce himself does not think triadic logic is a newlogic, but an addition or extension of the existing dyadic logic:

The recognition [that there is an intermediate ground betweenpositive assertion andpositive negation which isjust as Real as they are] does not involve any denial of existinglogic, but it involves a great addition to it. (1909: 21–22 [NEM3/2: 851], quoted in Fisch & Turquette 1966: 81)

If we accept Peirce’s suggestion literally, his triadic logic isnot a typical form of non-standard logic but Peirce’s anotherway to extend the territory of logic, along with his relationallogic.

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• Hammer, Eric, “Peirce’s Logic”,StanfordEncyclopedia of Philosophy (Fall 2010 Edition), Edward N. Zalta(ed.), URL = <https://plato.stanford.edu/archives/fall2010/entries/peirce-logic/>. [This was the previous entry on Peirce’s logic in theStanfordEncyclopedia of Philosophy—see theversion history.]

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